The Ultimate SAT Math Strategies Guide Created by Sherman Snyder Fox Chapel Tutoring Pittsburgh, PA 412-352-6596 Go to Success Model.

The Ultimate SAT Math Strategies Guide

Created by Sherman SnyderFox Chapel Tutoring

Pittsburgh, PA412-352-6596

Go to Success Model

Math SAT Success Model

Math Definitions & Concepts

Test Taking Tips

Math Strategies

Student Success

Return to Introduction

SAT Test Taking Tips

Two Rules

Back to Success Model

Math Definitions and Concepts The Top 25


Math StrategiesMath Topics


Absolute ValueBack to Top 25

Definition: How far a number is from zero. An alternative definition is the numeric value of a quantity without regards to its sign. The absolute value of a number is always positive or zero. The symbol “|….|” is used to denote absolute value of a quantity.

Applications: • Values: |6.5| = 6.5; |- 3.2| = 3.2; |0| = 0• Solving equations: |x - 5| = 3• Solving inequalities: See math strategy• Graphs of functions: See math strategy

Arc

Definition: An unbroken part of the circumference of a circle. An arc can be measured by its length or by its central angle. When measured by its central angle, the arc has the same degree measure as the central angle.

arccentral angle

Applications: • Finding the length of an arc• Finding area of a sector• Finding internal angles of an isosceles triangles with one vertex

at the central angle

isosceles triangle

Back to Top 25

Average (arithmetic mean)

Applications: • Usually involves values expressed in terms of variables, not

numerical values. See math strategy• Note: You will never be asked to calculate the mean of a list of

numbers. Such questions always ask for the median, not the mean of the list.

Definition: The most commonly used type of average on the SAT

sum of values number of values

average (arithmetic mean) =

Back to Top 25

Average Speed

Applications: • Word problems that involve the motion of an object• Caution: If a question involves the motion of an object at two

different rates and asks for the overall average speed of the object, the correct answer will be the average of the two given rates if and only if each segment of motion occurs over the same time period. If the motion of each segment occurs over the same distance, the above definition of average speed must be applied.

Definition: The total distance traveled by an object divided by the total time traveled

total distance traveledtotal time

Average speed =

Back to Top 25

BisectorBack to Math

Definitions

Definition: A line segment, line, or plane that divides a geometric figure into two congruent halves.

Applications: • Most common application involves angle bisectors.

angle bisector

Central Angle

Definition: An angle whose vertex is at the center of a circle. The measure of a central angle is also the measure of the arc that the angle encloses.

Applications:• See

Applications: • Finding the length of an arc• Finding area of a sector• Finding internal angles of an isosceles triangles with one vertex

at the central angle

• Note: You will never be asked questions about inscribed angles

isosceles triangle

central angle

inscribed angle

Back to Top 25

Diagonal

Definition: A line segment joining two non-consecutive vertices of a polygon. In the figure, the three dashed lines are diagonals

Applications: • Finding the number of diagonals in a polygon of “n” sides

(see example)• Finding the number of possible triangles formed by all

diagonals from one vertex of the polygon

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Digit

Definition: The set of integers from “0” to “9” in the decimal system that are used to form numbers.

The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Note: The number zero is contained in the set of digits

Applications: • Formation of integers

Back to Top 25

Directly Proportional

Definition: A relationship between two variables in which the ratio of the value of the dependent variable to the value of the independent variable is a constant. If y is proportional to x, then y/x is a constant. This can be written in equation form as y =kx where k is a proportionality constant.

Applications: • See math strategy• Proportions, ratios, and probability are closely related in many

applications. See math strategy

Back to Top 25

Distance Between Points

Applications: • Any question that contains the words distance, points, and

number line requires the application of the above definition.

Definition: The distance between two points on a number line is the absolute value of the difference between the two points. The order of subtraction does not affect the result.

3-4 0

Distance = |3 - (-4)| = |7| = 7or

Distance = |-4 - 3| = |-7| = 7

Distance = 7

Back to Top 25

Divisor

Definition: • A number or quantity to be divided into another number or

quantity (the dividend)• A number that is a factor of another number

Applications: • Questions involving long division and remainders.

See math strategy• For some questions the word “divisor” can be replaced with the

word “factor”.

Back to Top 25

Factor

Applications: • See math strategy

Definition: A factor of a number or expression, N, is a number or expression that can be multiplied by another number or expression to get N. When a number or expression is written as a product of its factors, it is said to be in factored form.

Example: (2)(4)(15) = 120 Example: (x + 1)(x + 2) = x2 + 3x +2

Factors Factors

Back to Top 25

Function

Definition: A special relationship between two quantities in which one quantity, the argument of the function, also known as the input, is associated with a unique value of the other quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The notation f(x) is said “F of X”. An example of a function is f(x) = 2x, a function which associates with every number twice as large.


Back to Top 25

Inversely Proportional

Applications: • Questions that begin with the words “If “y” is inversely

proportional to “x” and…”• Questions that contain a table of “x” and “y” values that have a

constant product.

Definition: The product of the value of the independent variable and the value of the dependent variable is constant. Can be written as k = xy, or y = k/x. The relationship between “x” and “y” can be expressed graphically as

Back to Top 25

Median


Definition: The middle number in a sorted list of numbers. Half the numbers are less and half the numbers are greater. If the sorted list contains an even number of values, the median is the average of the two numbers in the middle of the list.

Example: 2, 3, 3, 6, 8, 9, 9

Example: 2, 3, 3, 3, 5, 6, 7, 9

Median = 4

Back to Top 25

Multiple

Definition: The product of an integer by an integer. For example, the multiples of 4 are 4, 8, 12, 16, 20, …..For any positive integer there are an infinite number of multiples.

Applications: • Finding the value of a term in a repeating sequence.• Variety of questions that require understanding of the multiple

definition

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Percent

Definition: A ratio that compares a number to 100. Percent means “out of one hundred”. For example:

10% means 10/100, 750% means 750/100, “k%” means k/100


Back to Top 25

Percent Change


Definition: The amount of change in a quantity divided by the original amount of the quantity times 100%.

% change = amount of changeoriginal amount

x 100%

Back to Top 25

Probability

Definition: The likelihood of the occurrence of an event. The probability of an event is a number between 0 and 1, inclusive. If an event is certain, it has a probability of 1. If an event is impossible, it has a probability of 0.

Applications: • Elementary probability• Probability of independent/dependent events• Geometric probability

Back to Top 25

Proportional

Applications: • Proportions, ratios, and probability are closely related in many

applications. See math strategy

Definition: An equation showing that two ratios are equal • Two variables are proportional if their ratio is constant.

If a is proportional to b, then a/b is a constant.Can be written in equation form as a = kb where k is a proportionality constant.

Back to Top 25

Rate

Definition: A rate is a ratio that compares two quantities measured with different units. For example, the speed of a car is a rate that compares distance and time.

Applications: • When the word “rate” is contained in a question, create a ratio

of the two given quantities identified in the question. Such questions usually vary the value of one of the given quantities and ask for the value of the second quantity that will maintain the given rate. To solve efficiently, create a proportion of the two ratios and solve for the unknown quantity. See math strategy

Back to Top 25

Sector

Applications: • Area of sector • Length of arc AB

Definition: A sector of a circle is the portion of a circle bounded by two radii and their intersected arc.

Sector

Back to Top 25

Sector



Sector

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Sector



Sector

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Sequences

Definition: A sequence is an ordered set of numbers. Four types of sequences on the SAT.• Arithmetic sequence: A sequence of numbers that has a

common difference between each number. 3,7,11,15,19,23• Geometric sequence: A sequence of numbers that has a

common ratio between each number. 3, 6,12, 24, 48, 96• Repeating sequence: A sequence of numbers that form a

repeating pattern. See math strategy• “Other” sequence: A sequence that does not fit any of the

above three categories. A formula is usually provided that can be used to determine each value of the sequence.

Applications: • Any or all of the above types of sequences will be found on

every SAT. However, the sequence names used above will never be found in any SAT questions. Instead, a description of the sequence is used. Bottom line….know the sequence definitions.

Back to Top 25

Similar Triangles

Definition: Two triangles are similar if and only if all pairs of corresponding angles are congruent and all pairs of corresponding sides are proportional.

76

4

3.5 3

2

Applications: (See figure at right) • When a smaller triangle is completely

inside a larger triangle such that corresponding angles are congruent or one pair of corresponding sides are parallel, the two triangles are similar. Congruent

angles

Back to Top 25

Slope of a Line

Applications: • Slope of a line when two points are known• Identification of the “x” or “y” value of a point when the

coordinates of a second point are known and the slope of the line is given.

• Slope of a line parallel or perpendicular to another line• Linear relationships or functions that ask for the change in the

value of a quantity as the independent variable is changed.

Slope = ∆y∆x

= = y2 - y1

x2 - x1 riserun

Definition: Slope is a measure of the tilt or steepness of a line. Slope is calculated as the vertical distance divided by the horizontal distance between two points.

Slope is also a measure of the amount that the dependent variable (often “y”) changes as the independent variable (often “x”) changes by one unit.

y

x

Back to Top 25

Venn Diagram

Definition: A diagram (usually made of circles) that shows all possible relations between sets.

Applications: • Venn diagrams (2 sets): See math strategy• Venn diagram (3 sets): See math strategy

Back to Top 25

Zero

Definition: Zero is an even integer (thus it is divisible by 2) that is neither positive nor negative. As a result, zero is the smallest non-negative number. Zero is also the smallest of 10 digits. Zero is a whole number, a rational number, and a real number. Division by zero results in an undefined value.

Applications: • Questions that ask for the number of integers, the number of

even integers, or the number of positive integers that are contained in a solution set.

• Questions that ask for a specific value of “x” for which a function is not defined.

Back to Top 25

Questions You Can Count On

• A figure that is rotated, flipped, reflected, taken apart, unfolded is usually either question 3, 4, or 5 in the 20 multiple choice section of math.

• Parallel lines cut by one or more transversals: See strategy

• Tangent lines to a circle: See strategy

• The “If…..then what is the value?” question: See strategy

• Equation of a line or slope of a line perpendicular to another line: See strategy

• Formation of even/odd numbers: See strategy

Learn More

Back to Tips


• Average (arithmetic mean) questions: See strategies

• Sequence questions

• Two types of definition questions• Substitution into an expression: See strategy• Words in quotations: See strategy

• Rate and or ratio questions: See strategy

• Rules of exponents: See strategy

• System of equations: See strategy

• Questions that contain an inequality: See strategy

Previous Learn More

Back to Tips


• Area of irregular shapes and area of sectors

• Counting problems including the number of ways to pair objects: See strategy

• Geometric probability: See strategy

• Probability of events occurring: See strategy

• Use of function notation and function translations or reflections: See strategy

• Percentage questions: See strategies

• Long division and remainder questions: See strategies

Previous Learn More

Back to Tips


• Overlap of data sets (Venn diagram applications): See strategy

• Patterns of number or shapes/objects: See strategy

• Similar shapes (usually triangles): See strategy

• Directly or indirectly proportion questions

• Absolute value equation or inequality: See strategy

• Median of a list of numbers: See strategy

• Creation of a cost equation for the purchase of an item or service

Previous

Back to Tips

The Two Test Taking Rules

• Keep it simple. View each question through the lens of simplicity, not the lens of complexity. The math portion of the SAT is not a two headed monster. With good reasoning skills and an understanding of basic math definitions and content, every question can be solved with little difficulty. Having this mindset will often lead to increased confidence.

• Answer the question. Make sure you answer the question being asked, not the question being assumed. Before choosing an answer, read the last half of the last sentence. If the questions asks for the cost of three pounds of bananas, do not choose the per pound cost. If a question asks for the value of the “y” variable, do not choose the value of the “x” variable. If the question asks for the value of the largest of three consecutive integers, do not choose the smallest integer. If the questions asks for the value of “4x”, do not choose the value of “x”. Answer the question being asked!

Back to Tips

The Three Questions

• What piece of information do I need? This is a crucial question to ask. SAT questions are asked in ways that are more abstract than a typical math question. The answer to this question will ensure you are heading down the correct path toward the answer.

• What do I do with the information? This is the math step that usually requires using a formula.

• What is the strategy for finding this information? This is where most students have difficulty. A good strategy is usually needed at this point. If none can be identified, students will go to Plan B (substitution of answers, elimination and guess), or skip the question.

Back to Tips

Test Day Tips

• Replace calculator batteries. Replace the batteries in your calculator (usually four AAA batteries) with fresh, out of the package batteries. Do not replace with the batteries that are rolling around in your desk drawer…..the ones that should have been tossed out the last time you replaced batteries.

• Take a watch to the testing center. You do not have control over the amount of time for each test section. However, with a watch, you are in a position to control the use of your time. If the testing room has a clock on the wall, your watch may not be needed.

• Have your admission ticket and photo ID. This a common sense issue.

• Prepare a survival kit. In a lunch bag, pack bottled water and many snacks. Include one chocolate bar to be consumed between sections seven and eight of the ten section test. Fatigue will be high at this point during the test. Eat the chocolate bar for a burst of energy and tough it out until the end.

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Test Day Tips

• Take plenty of No. 2 wood pencils. Mechanical pencils are not permitted apparently due to cheating issues.

• Proctors are not your friend. The test proctor is there to make a few bucks on a Saturday morning. They are not there to help you in anyway. They are prone to making mistakes with the timing of sections, have been observed talking on the phone causing noise issues, and often have a nasty disposition. They are not your friend!

• Four math sections…do not panic. The SAT is comprised of ten sections: three writing, three reading, three math, and one “experimental section”. The experimental section will be an additional writing, reading, or math section that will not be part of your final score. The experimental section is not identified. Do your best on all sections!

• Bubble in the student-generated response answers: Some students forget to bubble the answers.

Back to Tips

Previous

What Study Guides Will Never Reveal

• Be prepared to reason: Math content is plentiful in study guides, however math strategies are virtually nonexistent. To be successful on the SAT, reasoning skills are as important as having basic math content knowledge and basic computational skills.

• Answer the easy questions first. All questions are equally weighted. Do not try the hard questions first. Attempt the questions in the order they are presented.

• Basic calculations should be done without a calculator: Calculators are absolutely, positively not needed for the SAT, however, you should absolutely, positively use one…..sparingly. Avoid using the calculator for basic addition and multiplication operations, especially those involving negative numbers. Student calculator input errors often lead to costly mistakes that are absolutely avoidable.

• Complex computational skills not required: The SAT is a test of quantitative reasoning skills, not computational skills. With strong reasoning ability, only basic calculations are needed to answer most questions.

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• No need to memorize formulas: There is no need to memorize formulas….they are all provided. If a formula is needed and is not contained on the list of formulas at the beginning of each math section, then the formula will be provided in the text of the question. The bottom line is this….if you believe a formula is needed to solve a specific problem and the formula is not provided, look for an alternative way (and often more efficient way) to solve the problem.

• Never enter a value for “pi” into your calculator: Entering “pi’ into your calculator will often result in a close approximation to the correct answer, not the exact answer. Solve questions in terms of “pi”, especially the student-produced response questions that require exact answers.

• Cross multiplication is your best friend: The solution to many questions is made easier by using cross multiplication. Look for opportunities to use it.

• Need to know math definitions: Definitions are not provided. You are expected to know all math definitions. Examples include slope of a line, average (arithmetic mean), percent, percent change, average speed, etc.

Back to Tips

Learn MorePrevious


• The words “arithmetic” and “geometric” sequence are not used: Students are not expected to know the definition of these sequences, as suggested by study guides. Instead of using the words “arithmetic” and “geometric” sequences, SAT questions describe the characteristics of these sequences.

• Do not need to use permutations or combinations: Although both topics are discussed in most study guides, you can always use Fundamental Counting Principles to solve counting problems.

• Inscribed shape questions: When a shape is inscribed inside a second shape, their centers always coincide. This is often useful when developing a strategy to solve this class of questions.

• Never asked to calculate the average of a list of numbers: When a list of values is provided, analysis of the median (sometimes mode) is always asked. Do not be fooled into making a lengthy calculation of the mean of a list of numbers….it is never asked for.

• Never asked to find the domain of a function: This topic is discussed in study guides, however, it is not found on the SAT reasoning test. More likely to find this topic on the SAT math subject test.

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Previous

The Usual Study Guide Tips

• Use the figure when figuring. All figures are drawn to scale unless stated otherwise. Use this to your advantage. If there is a note stating the figure is not drawn to scale, you must stick to the facts when drawing conclusions about the answer.

• Student produced response answers must be non-negative rational numbers. All non-negative integers (including zero) and all fractions are acceptable answers.

• Guess on student generated response questions. No penalty is given for missing a student produced response question. If the answer is not known, take a guess.

• To guess or not to guess. There is a ¼ point penalty for each missed multiple choice question. The conventional wisdom is to guess if one answer choice can be eliminated. My recommendation is to guess if two of five choices can be eliminated.

Back to Tips

Math StrategiesTable of Contents

Number and OperationsOrdering of Negative NumbersLinear ProportionalityVenn Diagrams (2 sets)Venn Diagrams (3 sets)Ratios and their MultiplesRatios, Proportion, ProbabilityRateCounting Problems The Pairing StrategyLong Division and RemaindersPercent ChangeDealing With PercentagesRepeating SequencesConsecutive IntegersEven/Odd Integer Creation

AlgebraUsing New Definitions: Type 1Using New Definitions: Type 2Solving Simple InequalitiesEquivalent StrategySystem of EquationsMatching GameFactoring StrategyWord problemsBasic Rules of ExponentsAdditional Rules of ExponentsAbsolute Value InequalitiesCreation of Math Statements Parabolas Single Term DenominatorsMaking Connections

Geometry and MeasurementDividing Irregular ShapesLine Segment Length in SolidsPutting Shapes Together3-4-5 Triangle30-60-90 Triangle45-45-90 TriangleDistance Between Two PointsMidpoint Determination in x-y CoordinateMidpoint Determination on Number Line Exterior Angle of a TrianglePerpendicular LinesInterval Spacing - Number LineTriangle Side LengthsSimilar Triangle PropertiesThe Slippery SlopeParallel Lines and TransversalsTangent Line to a Circle

Data Analysis, Statistics, and Probability Average (Arithmetic Mean) Median of Large ListsElementary ProbabilityProbability of Independent EventsGeometric ProbabilityThe Unit CellIt’s Absolutely Easy!

FunctionsUsing Function NotationReflections - x axisReflections - y axisReflections - Absolute ValueTranslations - Horizontal ShiftTranslations - Vertical ShiftTranslations - Vertical StretchTranslations - Vertical Shrink

The Basics All the Equations You Need!The Important Definitions You Need!



Lesson 1Algebra Strategies

Using New Definitions: Type 1 Using New Definitions: Type 2 Solving Simple Inequalities Equivalent Strategy System of Equations Matching Game Factoring Strategy Word problems Basic Rules of Exponents Additional Rules of Exponents Absolute Value Inequalities Creation of Math Statements Parabolas Single Term Denominators Making Connections

Back to Success Model Back to Math Topics


Lesson 2Geometry and Measurement

Strategies

Dividing Irregular Shapes Line Segment Length in Solids Putting Shapes Together 3-4-5 Triangle 30-60-90 Triangle 45-45-90 Triangle Distance Between Two Points Midpoint

Determination in x-y Coordinate Midpoint

Determination on Number Line Exterior Angle of a Triangle Perpendicular Lines Interval Spacing - Number Line Triangle Side Lengths Similar Triangle Properties The Slippery Slope Parallel Lines and Transversals Tangent Line to a Circle


Lesson 3Number and Operations

Strategies

Ordering of Negative Numbers Directly Proportional Venn Diagrams (2 sets) Venn Diagrams (3 sets) Ratios and their Multiples Ratios, Proportion, Probability Rate Counting Problems The Pairing Strategy Long Division and Remainders Percent Change Dealing With Percentages Repeating Sequences Consecutive Integers Even/Odd Integer Creation




Lesson 4Functions Strategy

Using Function Notation Reflections - x axis Reflections - y axis Reflections - Absolute Value Translations - Horizontal Shift Translations - Vertical Shift Translations - Vertical Stretch Translations - Vertical Shrink



Lesson 5Data Analysis, Statistics, and Probability Strategies

Average (Arithmetic Mean) Median of Large Lists Elementary Probability Probability of Independent Events Geometric Probability The Unit Cell It’s Absolutely Easy!


All The Equations You Need!

Strategy: Great news! The equations on this page are the only ones you need to be successful on the SAT.

Return to Table of Contents See example of strategy

Reasoning: If the equation is not on this page, you do not need to use it. Hooray! Examples include quadratic formula, combinations, permutations, equation of a line or circle, surface area and volume of a cone, pyramid, or sphere. If one of these equations is needed to solve a problem, it will be provided.

Application: There are plenty of questions on the SAT for which these formulas are used. To save time when taking the SAT, it is recommended that you memorize these basic formulas.

Area of rectangle = lw

Area of Circle = π r2

Circumference of Circle = 2π r

Area of triangle = ½ bh

Volume of rectangular solid = lwh

Volume of cylinder = π r2h

Pythagorean theorem c2 = a2 + b2

30 - 60 - 90 Triangle Click for details

45 - 45 - 90 Triangle Click for details

All The Equations You Need! Example 1

Question: Under construction

Return to Table of Contents Return to strategy page See another example of strategy

What essential information is needed?

What is the strategy for identifying essential information?:

Solution Steps

All The Equations You Need! Example 2


Return to Table of Contents Return to strategy page Return to previous example


What is the strategy for identifying essential information?

Solution Steps

The Important Definitions You Need!

Strategy: These definitions are extremely important for you to memorize. Unlike formulas, definitions are not provided on the SAT.


Reasoning: Students often consider these definitions to be formulas. They are not formulas! Formulas are derived in geometry using proofs.

Application: These definitions are extremely valuable resources when solving a variety of problems on the SAT. The definition of empty set, integer, positive and negative numbers, even and odd numbers, digits, and percentages are also important to know.

Average speed = total distance traveledtotal time

Average (arithmetic mean) = sum of valuesnumber of values

Click for more details

Percent change = amount of changeoriginal amount

X 100%


Slope = ∆y∆x

= = y2 - y1

x2 - x1 riserun


Distance between two points = 2 1x x

The Important Definitions Example 1





Solution Steps

The Important Definitions Example 2





Solution Steps

Ordering of Negative Numbers

Strategy: Visualize the position of a single negative value or a list of negative values as they would appear on a number line


Reasoning: As you move left on a real number line, the values get smaller. This property is especially useful when ranking negative numbers.

Application: Any question that requires you to rank the values of negative values from smallest to largest or vice-versa. Also useful when assigning values to positions on a number line.

-7 -4 -1-10

A B C D E

A

-⅜ -¼ -⅛-½

B C D E

On the number line shown below, which letter best represents the location of the value -2/5?

Click to see answer

On the number line shown below, which letter best represents the location of the value -5/2?

Click to see answer

Ordering of Negative Numbers Example 1

Question: If a < 0, which of the four numbers is the greatest?A) a + 2 B) 2a + 2C) 4a + 2 D) 8a + 2E) It cannot be determined from the

information given



Solution Steps

Return to strategy page See another example of strategyReturn to Table of Contents

Ordering of Negative Numbers Example 2





Solution Steps

Directly Proportional

Strategy: Often given values for “x1” and “y1” and asked to find value for “x2” when given “y2”. Use the following proportion:

Reasoning: The ratio of y:x is constant for any two points. Click to see properties of directly proportional

Application: Any relationship that can be expressed as ratios. In addition to points on a line, examples include amount of ingredients in recipes, number of marble colors in a container, and segment lengths of a number line.

y

x

“y” is directly proportional to “x”

(x2, y2)

Properties of a directly proportional include the following: 1) Graph of “y” versus “x” is linear and passes

through the origin. Has the form of y = kx.2) Slope of line is the ratio of y:x for any point

on the line3) Slope of line is equal to proportionality

constant “k”.

(x1, y1)constant

2

2

1

1 x

y

x

y

Back to Definition

y = kx

kxy

constantx

yk


Directly Proportional Example 1

Question: A machine can produce 80 computer hard drives in 2 hours. At this rate, how many computer hard drives can the machine produce in 6.5 hours?

Return to strategy page See another example of strategy

What essential information is needed? Rate of computer hard drives per hour.

What is the strategy for identifying essential information?: Ratio the number of computer hard drives to the number of hours required to produce them. With this ratio, create a linear proportion to answer the question.

Solution Steps

1) Create a ratio representing rate of computer hard drive production:

80 hard drives2 hours

2) Create a linear proportion to solve for number of hard drives produced in 6.5 hours:

80 hard drives2 hours

“n” hard drives6.5 hours

=

3) Solve for ‘n”: 2n = (80)(6.5) n = 260

Return to Table of Contents

Directly Proportional Example 2

Question: If y varies directly as x, and if y = 10 when x = n and y = 15 when x = n + 5, what is the value of n?

Return to previous example

What essential information is needed? A link between y and x that can be used to solve for n.

What is the strategy for identifying essential information? The ratio y/x is a constant. Create a proportion and solve for n.

Solution Steps

1) Create a linear proportion to solve for n.

10n

15n + 5

=

2) Solve for n using cross multiplication:

15n = 10(n + 5)

15n = 10n + 50 5n = 50 n = 10

Return to strategy pageReturn to Table of Contents

Venn Diagram (2 sets)

Strategy: To determine the overlap (intersection) of members in two groups (sets), use the following approach:Step 1: add the total number of

members from both groups Step 2: subtract the sum consisting of

the total number of members in one group only and both groups from the number of members in step 1

See example of strategy

Reasoning: By eliminating the overlap of members, the sum of three numbers in the Venn diagram will equal the total number of members being counted.

Application: Used when members of two or more groups (sets) have common members.

18 22 10

Total number of students = 50

Number of students that study math only:

40 – 22 = 18

Number of students that study history only:

32 – 22 = 10

Number of students that study history = 32

Number of students that study math = 40

Number of students that study math and history = 22

Step 1 40 + 32 = 72

Step 2 72 – 50 = 22

Math History

18 + 22 + 10 = 50

Back to Definition

Back to Frequent

Questions

Return to Table of Contents

Venn Diagram (2 sets) Example 1

Question: The Venn diagram to the right shows the distribution of students who play football, baseball, or both. If the ratio of the number of football players to the number of baseball players is 5:3, what is the value of n?

What essential information is needed? Connection between the number of players in each sport to “n”, the number of players that participate in both sports.

What is the strategy for identifying essential information?:Use the properties of Venn diagrams and proportions to find the value of “n”

Solution Steps

Football Baseball

28 14n

1) Create a proportion of the number of football players to baseball players

n + 28n + 14

53

=

2) Solve for “n” using cross multiplication: 5n + 70 = 3n + 84

2n = 14 n = 7


Venn Diagram (2 sets) Example 2

Question: The 350 students at a local high school take either math, music, or both. If 225 students take math and 50 take both math and music, how many students take music?

What essential information is needed? Connection between the multitude of given information and the unknown quantity.

What is the strategy for identifying essential information? Use the properties of Venn diagrams to help “visualize” the given information.

Solution Steps

Math Music

175 m50

1) Create an appropriate Venn diagram to help visualize the given information.

2) Find the value of m, the number of students that take music only

175 + 50 + m = 350 m = 125

3) Find the value of m + 50, the number of students that take music

m + 50 = 125 + 50 = 175

Return to previous exampleReturn to strategy pageReturn to Table of Contents

Venn Diagrams (3 sets)

Strategy: When analyzing the overlap of three data sets, it is important to understand the meaning of each section of the resulting Venn diagram (see example)

Reasoning: The interpretation of data in each section is determined by the rules of logic

Application: Data sets in which there is overlap of members of two or more sets. Applications include student choices of school classes or sport activities, and overlapping properties of various real numbers

3

7

4 5

6 8

9

Football

Soccer

Baseball

The number of athletes that play all three sports = 3

The number of athletes that play two sports only = 16

The number of athletes that play one sport only = 23

The number of athletes that play two sports. Example: football and baseball = 10

The number of athletes that play football only (6), baseball only (8), or soccer only ( 9)

Back to Definition

See example of strategyReturn to Table of Contents

Venn Diagrams (3 sets) Example 1





Solution Steps

Venn Diagrams (3 sets) Example 2





Solution Steps

Ratios and Their Multiples

Strategy: When given the total number of several different objects and a ratio that describes their distribution, create an equation to find the exact number of each object. (click to see example)

Reasoning: For discrete objects like marbles, bowling balls, and people, the total number of each object in the group must be a multiple of their respective ratio value.

Application: Questions that ask for the distribution of angles in a triangle or the distribution of objects among containers.

A jar contains a total of 30 red, yellow, and blue marbles. The number of each marble color in the jar follows the ratio 3 red: 2 yellow: 1 blue. How many of each color are there in the jar.?

3x + 2x + x = 30 marbles

6x = 30 marbles

x = 5 blue marbles

2x = 10 yellow marbles

3x = 15 red marbles

Total = 30 marbles


Ratios and Their Multiples Example 1

Question: The measures of the interior angles in a triangle are in the ratio 9:4:2. What is the measure of the largest angle in the triangle?

What essential information is needed? The measure of each individual angle.

What is the strategy for identifying essential information? Create and solve an equation using the angle ratios and the fact that the sum of the interior angles is 180 degrees in a triangle.

Solution Steps

1) Create equation using ratio values

9x + 4x + 2x = 180 degrees

2) Solve equation for “x”. Multiply by nine to find measure of largest angle.

9x + 4x + 2x = 180 degrees

15x = 180 degrees

x = 12 degrees

9x = 108 degrees


Ratios and Their Multiples Example 2

Question: Cookies are distributed within four separate jars in the ratio of 7:5:3:1. The total number of cookies contained in the four jars is 48. How many cookies are contained in the jar with the greatest number of cookies?

What essential information is needed? The number of cookies in each jar.

What is the strategy for identifying essential information? Create and solve an equation using the given ratios and the fact that the total number of cookies contained in the four jars is 48.

Solution Steps

1) Create equation using ratio values

7x + 5x + 3x + x = 48 cookies

2) Solve equation for “x”. Multiply by seven to find measure of largest angle.

7x + 5x + 3x + x = 48 cookies

16x = 48 cookies

x = 3 cookies

7x = 21 cookies


Ratios, Proportions, Probability Connections

Strategy: When the whole consists of two parts and the parts are expressed as a ratio of each other, there are several connections between ratios, proportions, and probability that are useful to solve a variety of problems.

Reasoning: For the example shown to the right, three out of every four marbles in the can are blue. To maintain this ratio, the total number of marbles in the can must remain a multiple of four. As a result, the probability of selecting a blue marble is ¾.

Application: Problems involving lengths of line segments, rate/time, areas and perimeters, sizes of angles

The ratio of red to blue marbles is 1 to 3.

Connection 1: The total number of marbles in the can must be a multiple of four marbles (1 + 3 = 4).

Connection 2: The probability of randomly selecting a blue marble from the can is ¾.

Connection 3: To maintain this ratio when adding to or removing marbles from the can, a proportion should be used.

Back to Definition

Back to Frequent

Questions


Ratios, Proportions, Probability Example 1

Question: During the month of February (28 days) the city of Pittsburgh had two days on which it snowed for every five days on which it did not snow. For the month of February, the number of days on which it did not snow was how much greater than the number of days on which it snowed?

What essential information is needed? Need to determine the number of days in which it snowed and the number of days in which it did not snow.

What is the strategy for identifying essential information?: Use proportions to determine essential information.

Solution Steps

1) Set up a proportion using the following strategy: For every seven days (2 + 5 = 7) during the month of February, it snowed 2 days. Find the number of days it snowed.

27

n28

= n = 8 days of snow

2) Find the number of days in which it did not snow.

28 days - 8 days = 20 days

3) Subtract the result of Step 1 from the result of Step 2

20 days – 8 days = 12 days greater


Ratios, Proportions, Probability Example 2

Question: The ratio of almonds to cashews in a mixture is 2:3. How many pounds of almonds are there in a seven pound mixture of almonds and cashews.

What essential information is needed? The number of pounds of almonds required to maintain proper mixture ratio.

What is the strategy for identifying essential information? Use proportions to determine essential information.

Solution Steps

1) Set up ratio of almonds to mixture.

2 pounds almonds + 3 pounds cashews = 5 pounds mixture

2 pounds of almonds5 pounds of mixture

Ratio:

2) Create proportion to solve problem.

25

n pounds of almonds7 pounds of mixture

=

5n = 14Cross multiply

145

n = pounds of almonds


Rate Strategy

Strategy: For all questions that require the rate of two quantities to be held constant, create a proportion to solve for the new value of one quantity when the value of a second quantity is changed a given amount.

Reasoning: A proportion is an equation stating that two ratios are equivalent.

Application: Any question that requires the rate to be held constant. Examples of constant rate include speed of an object, rate of work, rate of flow of a liquid, rate of growth of money, etc.

Definition: A rate is a ratio that compares two quantities measured with different units. For example, the speed of a car is a rate that compares distance and time.

Note: When you read the word rate in a question, think ratio!


Back to Definition

Rate Strategy Example 1



Solution StepsQuestion: The rate of motion of a baseball is k feet per 2 seconds. In terms of k, how many seconds will it take a baseball to move k + 50 feet if the rate of motion is constant?A) B) C)

D) E)

1002

k

k

1002

100

2 k

k2

50

k2

100


Rate Strategy Example 2

Question: The rate of flow of water from a hose is 4 gallons per 20 seconds. At this rate, how many gallons of water can the hose provide in 5 minutes?



Solution Steps


Counting Problems

Strategy: Use “Fundamental Counting Principles” (FCP) and reasoning to solve many counting problems that do not involve pairing of objects. For pairing problems, see Handshake/ Pairing strategy.

Reasoning: FCP represent a broad class of counting principles that include permutations and combinations. Some counting problems will have constraints. Such problems, along with reasoning, can be solved using these principles.

Application: Any problem asking you to figure the number of ways to select or arrange members of a group. Examples include numbers, letters of the alphabet, or officers of a club.

Fundamental Counting Principles: If one event can happen in n ways, and a second, independent event can happen in m ways, the total number of ways in which two events can happen is n times m.

A restaurant uniform consists of a hat, shirt, and pants. If a worker has two hats, four shirts, and three pair of pants to choose from, how many uniforms can the worker create?

Step 1: Choice of a hat, shirt, or pants is independent of each other .

Step 2: Multiply the number of each together to find the total number of uniforms.

2 x 4 x 3 = 24 uniforms

Back to Frequent

Questions


Counting Problems Example 1

Question: Five individual pictures of the Jones family consists of the Jones parents and each of the four Jones children. The individual pictures are to be arranged vertically on a living room wall. How many arrangements of pictures can be made if the parent picture must be placed at the top of the arrangement?

What essential information is needed? The number of ways the five pictures can be arranged vertically on the wall.

What is the strategy for identifying essential information?: Use fundamental counting principles.

Solution Steps

1) Determine the number of arrangements of pictures. Take into account there is a constraint: the top picture must be the Jones parents.

2) Multiply each number together to find the total number of arrangements

Top position → 1 picture to choose

Second position → 4 pictures to choose

Third position → 3 pictures to choose

Fourth position → 2 pictures to choose

Fifth position → 1 picture to choose

1 x 4 x 3 x 2 x 1 = 24 arrangements


Counting Problems Example 2

Question: A certain restaurant offers ice cream specials that consist of two scoops of ice cream and one topping. If there are four toppings to choose from and four flavors of ice cream, how many different ice cream specials can be created if the two scoops of ice cream must be different flavors?

What essential information is needed? A special consists of two groups → the number of toppings and the number of ways to pair up four flavors of ice-cream.

What is the strategy for identifying essential information? Use fundamental counting principles to identify the number specials.

Solution Steps

1) Determine the number of ways to pair scoops of ice cream if there are four flavors to choose from.

2) Multiply the number of toppings (4) and number of pairs of flavors (6) to find the total number of ice cream specials

4 x 6 = 24 specials

Vanilla StrawberryChocolate Peach

1 2 34 56


The Pairing Strategy

Strategy: The total number of ways to pair “n” objects is equal to ½n(n -1).

Reasoning: For a total of “n” objects, each object can be paired with “n -1” other objects. However, each pair is shared by two objects. Click to see an example of the total number of handshakes exchanged by 6 people.

Application: Examples include determining the total number of games played in a sport league, or the number of ways a two scoop ice cream cone can be created from a known number of available flavors.

Alternative Solution: Total number of handshakes can be found by addition of the number of handshakes exchanged by each individual person.

5 + 4 + 3 + 2 + 1 + 0 = 15 handshakes

½n(n -1) = ½(6)(5) = 15 total handshakes shared by a group of 6 people

n = 6 people

n - 1 = 5 handshakesper person


The Pairing Strategy Example 1

Question: In a baseball league with 6 teams, each team plays exactly 4 games with each of the other 5 teams in the league. What is the total number of games played in the league?

What essential information is needed? How many games are played between the eight teams.

What is the strategy for identifying essential information?: Find the number of games played between the 6 teams using the handshake problem strategy. Multiply the result by 4 to account for the fact that each team plays exactly 4 games with each of the other 5 teams.

Solution Steps

1) Find the number of games played between the 6 teams

½(6)(5) = 15 individual games played without repeats

2) Multiply by 4 to account for the fact that each team plays exactly four games with each of the other 5 teams

Total number of games played: 15 x 4 = 60 games


The Pairing Strategy Example 2

Question: How many diagonals can be drawn inside a regular polygon with 6 congruent sides.

What essential information is needed? The total number of diagonals drawn from the 6 vertices of the polygon.

What is the strategy for identifying essential information? Use the pairing strategy with modifications. Polygons have sides that do not require lines connecting adjacent vertices. To account for this, multiply the total number of vertices “n” by “n - 3” rather than “n - 1”. Total number of diagonals is ½n(n - 3).

Solution Steps

n = 6 sides n -3 = 3

diagonals

½n(n - 3) = ½(6)(6 - 3) = 9 diagonals can be drawn in a regular polygon

with 6 sides

Back to Diagonal Definition


Long Division and Remainders

Strategy: Find a value for the unknown variable k by adding the given divisor to the given remainder. Process the value found for k as specified in the question. Divide this result by the new divisor to find the desired remainder. Click to see a review of long division.

Reasoning: Long division questions always involve analysis of the remainder, not the quotient. All long division questions provide a value for the divisor and remainder. By choosing a value of 1 for the quotient, a value for the dividend (unknown variable k) can be easily and quickly found.

Application: Any long division question that expresses the dividend as a variable rather than a numerical value.

137

1

-076

dividend divisor x remainderquotient +=

13 = 7 x 1 + 6

Back to Frequent

Questions

Example: When the positive integer k is divided by 7, the remainder is 6. What is the remainder when k + 8 is divided by 7 ?


Back to Divisor

Definition

Long Division and Remainders Example 1

Question: When d is divided by 9, the remainder is 7. What is the remainder when d + 4 is divided by 9?

What essential information is needed?Find a number for d that satisfies the requirements. Add 4 to d, divide by 9, and find the remainder.

What is the strategy for identifying essential information? Add remainder to the divisor. This will quickly provide a possible value for d.

Solution Steps

Find a possible value for d by adding the remainder to divisor:

d = 7 + 9 = 16

The new remainder is 2

Add d = 16 to 4:d + 4 = 20

Divide 20 by 9:20 / 9 = 2 with remainder 2


Long Division and Remainders Example 2

Question: When n is divided by 7, the remainder is 5. What is the remainder when 3n is divided by 7?

What essential information is needed?Find a number for n that satisfies the requirements. Multiply n by 3, divide by 7, and find new remainder.

What is the strategy for identifying essential information? Add remainder to the divisor. This will quickly provide a possible value for n.

Solution Steps

Find a possible value for n by adding the remainder to divisor:

n = 5 + 7 = 12

Multiply n = 12 by 3:3n = 36

Divide 36 by 7:36 / 7 = 5 with remainder 1

The new remainder is 1


Dealing With Percentages

Strategy: When a percentage is quoted as a number or variable, express the percentage as a ratio with the percentage in the numerator and the number 100 in the denominator.

Reasoning: Percentages are expressed as a ratio of a number over 100 in mathematics. This strategy will avoid issues related to expressing a percentage as a decimal when the given percentage is a variable rather than a numerical value.

Application: Any question that contains a percentage expressed as a variable.

10 % should be written as

k % should be written as

100

10

100

k

Note: If a question expresses percentages as a numerical value only, it is okay to use the decimal form of a percentage.

Back to Definition

Back to Frequent

Questions


Dealing With Percentages Example 1

Question: If k% of 60% of 180 is 54, what is the value of k?

What essential information is needed? A mathematical statement is needed that properly describes the given information and provides a way to solve for the value of “k”.

What is the strategy for identifying essential information?: Two strategies are required: • Creation of Mathematical Statements• Percentages Strategy

Solution Steps

1) Create a mathematical statement that properly expresses k%

2) Solve for “k” using algebra

k% should be expressed as k100

Math statement is:k

10060

100x x 180 = 54

k100

60100x x 180 = 54

Eliminate zero’s

k(6)(18) = 5400Multiply by 100

k = 50



Question: If the length of a rectangle is increased 40% and the width is decreased 40%, how does the new area compare to the original area?

What essential information is needed? Rectangle lengths and widths that meet the percent change requirements.

What is the strategy for identifying essential information? Start with convenient length and width values. Apply the required percentage changes to each value. Calculate new rectangle area and compare to original value.

Solution Steps

1) Choose convenient values for length and width

2) Apply percentage changes

• Note: A square is a rectangle. Great shape to use for area calculations

• Convenient original area is 100. Use length of 10 and width of 10

New length = 10 + 4 = 14

New width = 10 - 4 = 6

3) Calculate new area and compare

New area = (14)(6) = 84

Area is reduced by 16%



Question: What is ½ percent of 8?

What essential information is needed? Need to convert ½ percent into an appropriate form to answer question.

What is the strategy for identifying essential information? Use percentage strategy. Express percentage as a fraction over 100 rather than decimal form.

Solution Steps

1) Express percentage in proper form

2) Determine answer to question

• Recommended form is:

½100

= 1200

• Multiply recommended form by 8

1200

x 8 =

25

1 125

= .04

Return to example 1Return to strategy pageReturn to Table of Contents

Percent Change

Strategy: Percent change is defined as the amount of change in the quantity divided by the original amount of the quantity times 100%.

Reasoning: This a well known definition in mathematics. Mostly used in chemistry and physics.

Application: Can be used for any question involving percent increase or decrease.

Caution: Do not divide amount of change by the final amount

% change = amount of changeoriginal amount

x 100%

Back to Definition


Percent Change Example 1

Question: Elliot’s height at the end of third grade was 48 inches. His height at the end of sixth grade was 60 inches. What was the percent change in Elliot’s height? a) 12 b) 15 c) 20d) 25 e) 30

What essential information is needed? The change in height is essential to determining percent change.

What is the strategy for identifying essential information?: Determine the change in height from the end of third grade to the end of sixth grade using subtraction.

Solution Steps

1) Determine the change in Elliot’s height

Change in height = height at end of 6th grade - height at end of 3rd grade

Change in height = 60 inches - 48 inches

Change in height = 12 inches

2) Determine the percent change in Elliot’s heightPercent change =

12 inches48 inches

x 100%

Percent change = 25%


Percent Change Example 2

What essential information is needed? The change in projected population is essential to determining percent change.

What is the strategy for identifying essential information? Using the function equation, determine the population in 1990 and 2005. Subtract the two values to determine the change in population.

Solution Steps

Question: For the years 1990 to 2005, the function above expresses the projected population of Mathville. What is the projected percent increase in population of Mathville from 1990 to 2005?

P(t) = 500t + 25,000

1) Determine the population in 1990 and 2005 using function equation.

P(t) = 500t + 25,000

P(0) = 500(0) + 25,000 = 25,000

P(15) = 500(15) + 25,000 = 32,500

2) Determine the percent change in population from 1990 and 2005.

Percent change =

7,50025,000

x 100%

Percent change = 30%

Change in population = 7,500 people

Note: t = 0 for 1990 and t = 15 for 2005


Repeating Sequence

Strategy: For any sequence that repeats, the value of the last term before the sequence repeats will always be repeated for any multiple of its term number.

Reasoning: The letter “T” is the last letter before the sequence repeats. “T” appears as the 4th, 8th, 12th,.. 20th,….40th term value. Term numbers that are a multiple of 4 will always have the letter “T” as its value for this sequence.

Application: Used when any sequence of numbers or objects repeat. Examples include numbers or letters, days of the week, hours on the clock, remainders from long division.

A C F T A C F T A C…….

4th term 8th term

2nd term 6th term 10th term

The term number of letter “C” will always be the following:

4n + 2 where “n” is an integer value and 2 is the remainder when the term number is divided by the multiple 4

Back to Definition


Repeating Sequence Example 1

Question: If the day of the week is Friday and it is assigned the value of one, what day of the week would be assigned the value one hundred?

What essential information is needed? Identify the appropriate multiple number for the repeating sequence.

What is the strategy for identifying essential information?: Identify the day of week at end of cycle, apply the multiple of seven to this day, identify the day assigned the value of one hundred.

Solution Steps

1) Identify the day at end of cycle

2) Find the remainder when one hundred is divided by the value seven

• If Friday is day one of the cycle, Thursday is the end of the weekly cycle and is assigned the value of seven

• Apply multiple of seven to Thursday

1007

= 14 with a remainder of 2

3) Identify day assigned the value of one hundred• For remainder of two, day one hundred

is two days beyond Thursday → Saturday


Repeating Sequence Example 2

Question: A pattern consisting of three red circles, two blue circles, three yellow circles, and three green circles was painted side by side along the perimeter of a rectangular box. If the color of the last painted circle was blue, which of the following could be the total number of circles painted on the box?a) 80 b) 83 c) 86d) 89 e) 92

What essential information is needed? Multiple number for sequence and possible remainders for a blue circle

What is the strategy for identifying essential information? Use repeating sequence principles to identify essential information

Solution Steps

1) Identify multiple number for sequence

2) Identify possible remainders for blue circle

3 red + 2 blue + 3 yellow + 3 green = 11

• Add total number of circles in pattern:

• Multiple number is 11 for sequence

• Blue circles are located at positions four and five in sequence.

• Correct choice is a value that is 4 or 5 greater than a multiple of 11

• Correct choice is (11)(8) + 4 = 92

• Conclusion: Third green circle is always a multiple of 11 in sequence.


Using New DefinitionsType 1

Example: For all positive integers x, let x @ be defined to be (x+1)(x+2). What is the value of 4@ ?

Strategy: Read and apply the new definition carefully before choosing answers.

Reasoning: The new definition will typically break down to a simple application involving basic math operations.

What does x@ mean? How do I determine a value?

4@ = (4+1)(4+2)

4@ = (5)(6)

4@ = 30

Apply the definition in given form

Operation is easy to apply for any value of “x”

Final answer

Caution: Do not foil (x+1)(x+2). More efficient to apply definition in factored form.

Back to Frequent

Questions


Using New DefinitionsType 2

Example: A positive integer is said to be “bi-factorable” if it is the product of two consecutive integers. How many positive integers less than 100 are bi-factorable?

Strategy: Read and apply the new definition carefully before choosing answers. Note the defined word is in “quotations” and there is no math expression as in Type 1.

Reasoning: Requires reasoning to apply the intended meaning due to lack of a math expression as in Type 1. Type 2 “New Definition” questions are usually more difficult to solve than Type 1.

What does the definition “bi-factorable” mean? How

do I determine a value?

1 x 2 = 2

2 x 3 = 6

8 x 9 = 72

9 x 10 = 90

Smallest integer less than 100 that is “bi-factorable”

Largest integer less than 100 that is “bi-factorable”

Result: There are nine positive integers less than 100 that are “bi-factorable”

Back to Frequent

Questions


Using New Definitions Example 1

Question: Let <x> be defined as the sum of the integers from 1 to x, inclusive. What is the value of <53> - <50>?

What essential information is needed? Find the value of each quantity and perform the subtraction operation.

What is the strategy for identifying essential information?: Carefully apply the definition of <x> to each quantity. Look for opportunities to simplify the solution process through cancellation of like terms.

Solution Steps

Apply the definition to each quantity:<53> = 53+52+51+50+49+…+1

<50> = 50+49+…+1

Look for cancellation opportunities:<53> - <50> =

(53+52+51+50+49+…) – (50+49+…)<53> - <50> = 53+52+51

<53> - <50> = 156

Note: No calculator needed due to cancellation of like terms. Without cancellation strategy, problem would be consume too much time.



Question: Let ©(x) be defined as ©(x) = (10-x) for all values of x. If ©(b) = ©(2b-2) what is the value of b?

What essential information is needed? Find the value of b that satisfies the given equation.

What is the strategy for identifying essential information? Carefully apply given definition to the expressions on each side of the equation. Set both expressions equal to each other and solve for b using simple math operations.

Solution Steps

Apply definitions to each expression:©(b) = 10-b

©(2b-2) = 10-(2b-2)

Set both expressions equal to each other and solve:

10-b = 10-(2b-2) Distribute (-)10-b = 10-2b+2 Subtract 10

-b = -2b+2 Add 2bb = 2



What essential information is needed? Find the value of b that satisfies the given equation.

What is the strategy for identifying essential information? Carefully apply the given definition using the values in each answer choice.

Solution Steps

Apply definitions to each expression:

Set both expressions…

Question: For positive integers a and b, let a b be defined by a b = ba . Which of the following is equal to 243.A) 3 5 C) 9 27 E) 81 3B) 5 3 D) 3 81

Return to Table of Contents Return to strategy page Return to example 1

Solving Simple InequalitiesBack to

Frequent Questions

Example: For all values of x, what must be true about the value of “n” in the inequality k – n < k + 2?

Strategy: Always solve the inequality directly by eliminating like terms and/or factors before analyzing answer choices.

Reasoning: By eliminating like terms or factors, the inequality often simplifies to one of the answer choices. Without simplification, each answer choice typically requires time consuming analysis to determine correct choice.

Recommended Solution

Step 1: Eliminate like terms by subtraction

k – n < k + 2

Step 2: Solve for “n” n > - 2

Caution: Do not choose values for “k” and use guess and check methods. Can be time consuming.


Solving Simple Inequalities Example 1

Question: For all values of x, what is a possible value of x that satisfies the inequality x + 5 > x + 7?

What essential information is needed? All possible values of x that will make the left expression greater than the right expression.

What is the strategy for identifying essential information?: Look for like term cancellation opportunities that eliminate the need to do time consuming guess and check steps.

Solution Steps

2) Evaluate remaining terms of inequality:

5 > 7This result is impossible

The correct answer is the empty set.

1) Cancel x term from both sides of inequality:

x + 5 > x + 7

Note: Cancellation of like terms by subtraction provides a clear result to analyze.


Solving Simple Inequalities Example 2

Question: If a + b > a - b, which of the following statements must be true?a) b < a b) a < b c) a = bd) b > 0 e) a > 0

What essential information is needed? From answer choices it is clear a method is needed to condense the number of variables to one on each side of the inequality.

What is the strategy for identifying essential information?: Look for like term cancellation opportunities that reduce the number of variables and eliminate the need to do time consuming guess and check steps.

Solution Steps

1) Simplify inequality by elimination and consolidation of like terms

Correct answer choice is “d”

a + b > a - b

b > -b Add “b” to both sides

2b > 0

b > 0

Eliminate “a” from both sides

Divide “b” from both sides


Equivalent Strategy

525 xx

Example: What is equivalent to the following equation?

Strategy: When a question asks for an equivalent form of an equation or expression, review all answer choices for guidance on ways to process the given equation/expression.

Reasoning: Equations or expressions can be expressed in an infinite number of equivalent forms. The answer choices often provide valuable guidance on how to transform the equation or expression into the correct answer choice. Click to see equivalent forms

Equivalent Forms

xx 525 15

25

x

x

xxx 102 101 x

All of the above are equivalent forms of the original equation. Answer choices on the SAT will typically include one of the above equivalent forms and four incorrect choices.


Equivalent Strategy Example 1

What essential information is needed? Guidance on how the expression should be transformed into “correct” equivalent form

What is the strategy for identifying essential information?: Review answer choices for guidance on “correct” equivalent form.

Solution StepsQuestion: For x ≠ 0, which of the following is equivalent to

a) 6x b) 12x c) 24xd) 6x2 e) 12x2

?

8

14

3

x

x

1) Review answer choices for clues

Conclusion: Answer choices suggest equivalent form requires elimination of fractions in numerator and denominator

2) Eliminate fractions by multiplying numerator by reciprocal of denominator

1

8

4

3

814

3xx

x

x 2

26x



Question: If k is a positive integer, which of the following is equivalent to 2k + 2k + 2k + 2k ?

a) 24k b) 4k c) 42k

d) 2k+2 e) 2k+4

What essential information is needed? Need clues that better define equivalent form of expression.

What is the strategy for identifying essential information? Review answer choices for guidance on correct solution path.

Solution Steps


2) Simplify radical using proper rules

Conclusion: Answer choices suggest equivalent form requires simplification of radical expression

2k + 2k + 2k + 2k = 4(2k)

22(2k)

2k+2



What essential information is needed? Need clues that better define equivalent form of expression.

What is the strategy for identifying essential information? Review answer choices for guidance on correct solution path.

Solution Steps


Conclusion: Answer choices suggest equivalent form requires squaring of radical

2) Square radical using proper rules

3) Transform equation and factor

Question: For all x > -2, which of the following expressions is equivalent to ?

a) x + 2 = 10x b) x + 2 = 20x c) x + 2 = 10x2 d) x + 2 = 20x2 e) x(100x - 1) = 2

xx

25

2

xx 102 21002 xx

2100 2 xx

2)1100( xx


System of EquationsBack to

Frequent Questions

Example: What is the value of “w” in the following system of equations?

Strategy: Solve a system of equations using elimination method or by reasoning. Do not use substitution .

Reasoning: A system of three or more equations takes considerable time to solve using substitution methods. The questions are typically designed to be quickly solved by reasoning or by elimination of unwanted variables by the elimination method.

3w = x – y + 4 w = z – x – 92w = y – z + 11

w = 1

Strategy: Use elimination method. Reasoning method not practical without more information about the values of or relationships between the variables.

3w = x – y + 4 w = z – x – 92w = y – z + 11

6w = 6


System of EquationsExample 1

Question: At a used book sale, Hillary paid $5.25 for 2 paperback books and 3 hardback books, while Ally paid $6.75 for 4 paperback books and 3 hardback books. At these prices, what is the cost, in dollars, for 3 paperback books?

What essential information is needed? The unit price for a paperback book.

What is the strategy for identifying essential information?: Can use system of equations to develop two cost equations. An alternative method is to apply reasoning skills.

Solution Steps

1) Solution using reasoning skills

• The only difference between Hillary’s book order and Ally’s book order is the number of paperback books purchased. • Ally spent $1.50 more than Hillary to purchase 2 additional paperback books.

2) Find the unit cost for paperback books

Unit cost = $1.50/2 paperback books

Unit cost = $0.75

3) Find the cost for 3 paperback books

Total cost = $2.25


System of EquationsExample 2

Question: In the system of equations below, what is the value of x + y?

x + y - 4z = 400x + y + 6z = 1200

What essential information is needed? Need a value for the expression x + y or separate values of x and y.

What is the strategy for identifying essential information? Use elimination to determine value of expression x + y.

Solution Steps

1) Subtract second equation from first equation and solve for the value of z:

x + y - 4z = 400x + y + 6z = 1200

-10z = -800z = 80

2) Substitute the value of z into first equation and solve for x + y:

x + y -4(80) = 400x + y -320 = 400

x + y = 720


The Matching Game for Equalities

Example: If k is a constant and 2(kx + 4) = 6x + 8 for all values of x, what is the value of k?

Strategy: When two equivalent expressions are set equal to each other, match corresponding terms and solve for the unknown constant.

Reasoning: Terms on each side of the equal sign can be easily matched and common factors and/or terms can often be eliminated. This will allow the possibility of quickly identifying the value of the unknown constant.

2(kx + 4) = 6x + 8

2kx + 8 = 6x + 8

Equivalent expressions

Distribute

“k” is unknown constant

2kx + 8 = 6x + 8Match

corresponding terms

Set equal and solve for “k”

2kx = 6x 2kx2x

6x2x

=

k = 3


The Matching Game Example 1

Question: If xy2 + 5 = xy + 5, which of the following values of y are solutions to the equation?I -1 II) 0 III) 1 a) I only b) II only c) III onlyd) II and III only e) I, II, and III

What essential information is needed? All possible values of “y” that make the left side of equation equal to the right side.

What is the strategy for identifying essential information? Look for like term and common factor cancellation opportunities that eliminate the need to do time consuming guess and check steps.

Solution Steps

1) Cancel like terms from both sides of equation.

xy2 + 5 = xy + 5

2) Cancel common factors from both sides of equation.

xy2 = xy

3) Evaluate y2 = y for possible solutions

Solutions are 0 and 1.

Subtract 5

Divide out “x”

Correct answer choice is d


The Matching Game Example 2

Question: In the equation below, k and m are constants. If the equation is true for all values of x, what is the value of m?

(x + 6)(x – k) = x2 - 4x + m

What essential information is needed? Need value of “m” that will make expression on right side of equal sign equivalent to the expression on left side.

What is the strategy for identifying essential information? Match terms in expression on left side of equal sign to corresponding terms in expression on right side.

Solution Steps

1) Convert expression on left side to trinomial form by distributing:

x2 - kx + 6x - 6k = x2 - 4x + mx2 - (k – 6)x - 6k = x2 - 4x + m

2) Match like terms on each side:x2 - (k – 6)x - 6k = x2 - 4x + m

m = - 6k -(k – 6) = -4

3) To solve for “m” need value of “k”

-(k – 6) = -4 -k + 6 = -4 k = 10

m = - 6km = - 6(10)m = - 60


Factoring Strategy

Strategy: If an expression is in factored form, generally leave it that way. If an expression can be factored, it is always to your advantage to factor it.

Reasoning: Working in factored form provides opportunities to quickly reason through problems with little computation.

Example: Can be factored Strategy

Given equation is in factored form. Reason through problem in this form.

Example: In factored form

Strategy

What conditions must be true for the following expression to be odd?

I. a is oddII. b is oddIII. a + b is odd

a2 +ab

Reason through problem with the expression in factored form

a(a + b)

For the following expression, what is the largest integer value for which the expression is positive?

(4a - 2)(4 - a)

Back to Definition


Factoring StrategyExample 1

Question: If x2 – y2 = 92 and x + y = 23, what is the value of x – y?

What is the essential information needed?: Need values for x and y. Better approach is to directly find a value for the expression x – y.

What is the strategy for identifying essential information?: x + y and x – y are factors of x2 – y2 . Write x2 – y2 in factored form. Divide the value of x2 – y2 by the value of x + y.

Solution Steps

x2 – y2 = (x + y)(x – y)

1) Write in factored form

92 23 ?

2) Solve for x - y

x – y = 92 23

x - y = 4


Factoring StrategyExample 2

Question: If (x + 2)(x – 5) < 0, how many integer values of x are possible?

What is the essential information needed?: Need to identify specific integer values of x that result in a value less than zero for the left side of the inequality.

What is the strategy for identifying essential information?: It can be reasoned that the two linear binomial factors on the left side of the inequality describe a parabola. Use the properties of parabolas to determine answer.

Solution Steps

When considered a parabola, two properties are useful to answer question:1) The parabola opens upward2) The parabola has roots at x = -2 and

x = 5

There are six integer values between -2 and 5 that result in a value less than zero

-2 5{-1, 0, 1, 2, 3, 4}


Word Problems

How many days are there in h hours and m minutes?

Strategy: Use a two-step strategy to solve most word problems:1) Eliminate choices that do not properly model the situation (often obvious). 2) Eliminate choices that do not provide proper units (dimensions)for the solution.

Reasoning: By reasoning, some choices will obviously not appear to be proper solutions. Of those remaining, some will likely have wrong or inconsistent units.

144024

mh

mh

144024

mh 144024

mh

144024

mh

144024

Step 1: Both minutes and hours should be smaller than days, not greater. Likely need to divide both terms in answer by a number or variable.

No

No

No

No

Yes

Step 2: To end with units of days, divide hours by 24 hours per day. Also, divide minutes by 1440 minutes per day.

This choice properly converts hours and minutes into days.


Word Problems Example 1

Question: Water from a leaking roof is collected in a bucket. If n ounces of water are collected every m minutes, how many ounces of water are collected in z minutes?

What essential information is needed? Need to establish relationships between the given variables that provide dimensionally correct answer.

What is the strategy for identifying essential information? Determine the proper units of final expression that are consistent with question being asked. Create an expression that is consistent with the required units.

Solution Steps

1) Determine units of correct answer• Final answer represents quantity

of water collected• Units of final answer should be

ounces

2) Arrange the three variables in proper way that provides correct units

ouncesminute

minutesxUnits of minutes cancel - ounces remain

Replace units with corresponding variables

nm

(z) = nzm


Word Problems Example 2

Question: In a certain grocery store, there are b stockcases with c shelves in each stockcase. If a total of d cans is to be stored on each of the shelves, what is the number of cans per shelf?

What essential information is needed? Need to establish relationships between the given variables that provide dimensionally correct answer.

What is the strategy for identifying essential information? Determine the proper units of final expression that are consistent with question being asked. Create an expression that is consistent with the required units.

Solution Steps

1) Determine units of correct answer

• Final units should be cans per shelf

2) Divide the total number of cans (d) by the total number of shelves

b stockcases x c shelvesstockcase

= bc shelves

Number of cans per shelf = dbc


Rules of ExponentsBasic Rules

Back to Frequent

Questions

Strategy: When the bases of two powers are the same in an equation, use these three basic rules to combine the two powers into a power with a single base. The value on the right hand side of the equation should be converted into a power with the same base as the power on the left hand side of the equation.

Reasoning: The three basic rules of exponents evolve from the fundamental definition of “exponentiation” that states: xa means “x” multiplied “a” times.

Example: If x and y are positive integers and (23x )(23y) = 64, what is the value of x + y?

Product of Two Powers Rule:

Quotient of Two Powers Rule:

Power of a Power Rule:

bab

a

xx

x

baba xxx

baba xx

Caution: The product and power rules are often confused for one another.


Basic Rules of Exponents Example 1

Question: If p and n are positive integers, and 32p = 2n , what is the value of p/n?

What essential information is needed? Need to establish a relationship between expressions on left side and right side of equal sign that clarify the relationship between p and n.

What is the strategy for identifying essential information?:Use rules of exponents to covert 32 to a power with a base of 2.

Solution Steps

1) Convert 32p to a power with base 2

32p = 2n

(25)p = 2n

25p = 2n

2) Set exponents equal to each other and solve for p/n.

5p = n

pn

15

=


Basic Rules of Exponents Example 2

Question: If 28x+2 = 643 , what is the value of 4x?

What essential information is needed? Need to establish a relationship between expressions on left side and right side of equal sign that clarify the relationship between the two exponents.

What is the strategy for identifying essential information? Use rules of exponents to convert 64 to a power with base 2.

Solution Steps

1) Convert 643 to a power with base 2.

28x+2 = 643

28x+2 = (26)3

28x+2 = 218

2) Set exponents equal to each other to solve for the value of “4x”

8x + 2 = 18

8x = 16

4x = 8

Note: No need to solve for “x”. Can solve directly for the value of 4x.


Rules of ExponentsAdditional Rules

Strategy: Use these additional rules of exponents when needed. Of the four additional rules, the negative exponent and rational (fractional) exponent rules are utilized most.

Reasoning: When an equation contains a variable with a negative exponent and rational exponent, follow a two step process to isolate variable:

Application: Questions with expressions that contain negative exponents and/or rational exponents.

Negative Exponent Rule:

Zero Exponent Rule:

Power of a Product Rule:

nn

xx

1

10 x

aaa yxxy

Rational (fractional) Exponent Rule:

a

ba b xx

1) Convert the negative exponent to a positive exponent using rule2) Raise both sides of equation to the reciprocal of the rational exponent.


Additional Rules of Exponents Example 1

Question: Positive integers a, b, and c satisfy the equations a-½ = ¼ and b-¾ = ⅛. What is the value of a + b?

What essential information is needed? The values of a and b are needed.

What is the strategy for identifying essential information?: Use negative exponent rule and raise both sides of each equation to the reciprocal of the rational exponent.

Solution Steps

1) Apply negative exponent rule to each equation

2) Raise both sides of each equation to the reciprocal of the rational exponent

a-½ = ¼ 1

a½ = ¼

a½ = 4

b-¾ = ⅛1b¾

= ⅛

b¾ = 8

(a½ )2 = 42

a = 16

(b¾ )4/3 = 84/3

b = 16

a + b = 32


Additional Rules of Exponents Example 2

Question: If 4-y/2 = 16-1 , then y = ?

What essential information is needed? Need to directly solve for the value of “y”

What is the strategy for identifying essential information? Use negative exponent rule first. Solve for value of “y” by converting powers on both sides of equation to the same base.

Solution Steps

1) Apply negative exponent rule to both sides of equation

2) Convert to same powers

4-y/2 = 16-1 1

4y/2= 1

164y/2 = 16

4y/2 = 16

4y/2 = 42

y2

= 2

y = 4


Absolute Value Inequalities

Strategy: To solve absolute value inequalities quickly, use a three step approach: 1) Using given information eliminate choices

representing the wrong solution type 2) Remove absolute value, evaluate positive

solution, and eliminate choices3) With remaining choices evaluate negative

solution and choose correct answer

Reasoning: Absolute value inequalities have properties that can be used to eliminate wrong choices.

Example: A manufacturer produces picture frames between 28 and 42 inches in width. If x represents the size, in inches, of the picture frames produced by the manufacturer, which of the following represents all possible values of x ?

28 < x < 42

| x – 35 | < 7

x – 35 < 7 x < 42

x > 28

Possible Solution Types:

Possible Inequality:

Solution details for | x – 35 | < 7

28 < x < 42

+/-(x – 35) < 7

x – 35 > -7

x < 28 or x > 42

| x – 35 | > 7

Example of Solution:

x < a or x > ba < x < b

Remove absolute value:

Positive solution:

Negative solution:

Overall solution:

Back to Definition

Back to Frequent

Questions


Absolute Value Inequalities Example 1

Question: For a certain airline company, the weight of pilots must be between 140 and 200 pounds. If w pounds is the acceptable weight of a pilot for this airline company, which of the following represents all possible values of w?a) │w - 170│= 30 b) │w + 140│< 60c) │w - 170│> 30 d) │w -170│< 30e) │w - 140│< 60

What essential information is needed? The correct answer must be the solution to 140 < w < 200.

What is the strategy for identifying essential information?: Use the absolute value strategy to identify answer

Solution Steps

1 & 2) Remove absolute value sign, solve positive solution, eliminate choices that do not meet the solution w < 200.a) w = 200 Not a solution b) w < -80 Not a solution c) w > 200 Not a solution d) w < 200 Possible solution e) w < 200 Possible solution

3) Evaluate negative solution

d) w - 170 > -30 w > 140 Solution

e) w - 170 > -60 w > 110

Not a solution


Absolute Value Inequalities Example 2

Question: A certain manufacturer of pencils requires all pencils to meet a length specification between 6.9 and 7.0 inches inclusive. If x is the length of a pencil that meets the specification, which of the following represents the length of pencils that do not meet the specification?a) │x - 6.0│< 1.0 b) │x - 6.0│> .05c) │x - 6.0│> 1.0 d) │x - 6.95│> .05e) │x + 6.0│> 13.0

What essential information is needed? The correct answer will be the solution to x < 6.9 or x > 7.0

What is the strategy for identifying essential information? Use the absolute value strategy to identify answer

Solution Steps

1 & 2) Remove absolute value sign, solve positive solution, eliminate choices that do not meet the solution x > 7.0

a) x < 7.0 Not a solution b) x > 6.5 Not a solution c) x > 7.0 Possible solution d) x > 7.0 Possible solution e) x > 7.0 Possible solution

3) Evaluate negative solutionc) x - 6.0 < -1.0 x < 5.0

Not a solution

d) x - 6.95 < -.05 x < 6.9 Solution

e) x + 6.0 < -13.0 x < -7.0 Not a

solution


Creation of Math Statements from Words

Strategy: Use the information in the table to the right to translate words into mathematical expressions and equations.

Reasoning: These are common words that are utilized in questions. When properly translated, the solution to a question is usually straightforward.

Words Symbol Translation

Is, the same as, is equal to

= Equals

Sum of, more than, greater than

+ Addition

Less than, difference, fewer

- Subtraction

Of, product, times × Multiplication

For, per ÷ Division

Example: If three times a number x is twelve less than x, what is x ?

Translation: 3x = x – 12Solution: x = -6


Creation of Math Statements from Words Example 1

Question: If ¾ of 3x is 15, what is ½ of 6x?

What essential information is needed? Create a math statement that properly describes the given information.

What is the strategy for identifying essential information?: Use the table of words to convert the given information into the proper math statement. Recognize that ½ of 6x equals 3x. Solving for the value of 3x will provide correct answer to question.

Solution Steps

1) Create the proper math statement

2) Solve for the value of “3x”

¾ · 3x = 15

¾ of 3x is 15times equals

¾ · 3x = 15 multiply by 43

[¾ · 3x] = [15] 43

43

5

3x = 20 Correct answer


Creation of Math Statements from Words Example 2

Question: Which of the following expresses the number that is 15 less than the product of 4 and x + 1?a) -4x + 14b) -4x + 16c) 4x - 11d) 4x - 13e) 4x - 15

What essential information is needed? Create a math statement that properly describes the given information.

What is the strategy for identifying essential information? Use the table of words to convert the given information into the proper math statement.

Solution Steps

1) Create the proper math statement from given information

Product of 4 and x + 1

2) Simplify the math statement to match answer choices

4(x + 1)

15 less than product of 4 and x + 1

4(x + 1) - 15

4(x + 1) - 15 Distribute 4

4x + 4 - 15 Subtract 15

4x - 11 Correct answer


The Parabola

Strategy: Many questions about the parabola (sometimes called “the quadratic function”) require an understanding of the impact of constants “a”, “b”, and ‘c” on the graph of f(x).

Reasoning: 1) The coefficient or constant “a” directly influences the x2 term of the function. When f(x) = x2, the parabola opens up. When f(x) = -x2, the parabola opens in the opposite direction or down. 2) The constant “c” is the function value for f(0) = “c”. This is the definition the y-intercept. 3) The impact of “b” is more complicated and usually not important.

Example: The quadratic function f is given by f(x) = ax2 + bx + c, where “a” is a positive constant and “c” is a negative constant. Which of the figures could be the graph of f?

Standard form of a parabola f(x) = ax2 +bx + c

“a” positiveopens up

“a” negativeopens down

“c” positivepositive “y” intercept

“c” negativenegative “y” intercept

Click to show answer


The Parabola Example 1

Question: The quadratic function f is given by f(x) = ax2 + bx + c, where “a” is a positive constant and “c” is equal to zero. Which of the figures could be the graph of f?

What essential information is needed? Need to know the effects of constants “a” and “c” on the graph of a parabola.

What is the strategy for identifying essential information?: Use parabola strategy to determine effects of “a” and “c”.

Solution Steps

A

D

B C

E

“a” positive“c” positive

“a” positive“c” negative

“a” positive“c” zero

“a” negative“c” zero


What is the correct choice? (click to verify choice)


The Parabola Example 2

Question: The quadratic function f is given by f(x) = ax2 + bx + c, where the product “ac“ is a positive constant. Which of the figures could be the graph of f?

What essential information is needed?Need to know the effects of constants “a” and “c” on the graph of a parabola.

What is the strategy for identifying essential information? Use parabola strategy to determine effects of “a” and “c”.

A B C

D E


“a” positive“c” negative



“a” negative“c” negative


Solution Steps

“ac” = positive


Single Term Denominator Equations

Strategy: When an expression contains two or more variable terms in the numerator and a single variable term in the denominator, expand the expression by placing each term in the numerator over the variable in the denominator

Reasoning: The expression will often easily simplify into the form required to directly answer the question.

5xx

yx

265

=+

5xx

yx

265

= -

yx

15

=

Example: If , what is the value of ? 5x + y x

265

= yx

Alternative Solution: This problem can also be solved using cross multiplication. Although the algebra is straightforward, students often struggle to isolate the answer when a ratio is required. Try it!

5x + y x

265

=


Single Term Denominator Example 1

What essential information is needed? Need values of each variable or find way to simplify the expression using the given ratio values.

What is the strategy for identifying essential information? Use single term denominator strategy to simplify the expression without the need to identify values of each variable.

Solution StepsQuestion: What is the value of if and ?

7x + y + z yy

x = 14 z

y = 5

1) Expand the expression

7xy

yy

zy

+ +

2) Substitute given ratio information and simplify

yy

1=x y

114

= zy

5=

Note: The value of each ratio is given

7[ ]+ 1 + 5114

+ 1 + 5 12

6.5


Single Term Denominator Example 2

What essential information is needed?Need values of each variable or find way to simplify the expression using the given ratio values.

What is the strategy for identifying essential information? Use single term denominator strategy to simplify the expression without the need to identify values of each variable.

Solution StepsQuestion: If , what is the value of ?

6

76

y

yx

y

x


Dividing Irregular Shapes in Polygon Shapes

x

y

5

450

Incorrect strategy

Note: You are setting a “trap” when a shape is divided into a trapezoid.

Triangle

Trapezoid

Rectangle

Triangle

Example: Which of the following represents the area of the five-sided figure shown to the right?

Correct strategy

Strategy: Always divide irregular polygon shapes into rectangles (or squares) and right triangles. Do not divide the shape into trapezoids or parallelograms. Click to see the animation.

Reasoning: The area and perimeter of rectangles and right triangles are usually easy to determine from the given information. In particular, right triangles can be solved using Pythagorean theorem or properties of 30-60-90 and 45-45-90 triangles.


Dividing Irregular ShapesExample 1

What essential information is needed? Sides AB and BC easy to determine. Need to divide figure into shapes that will provide an efficient way to find the length of segment AC

What is the strategy for identifying essential information?: Divide the shape into a rectangle and right triangle.

Solution Steps

Question: In the figure above, what is the perimeter of triangle ABC?

A

B

C

4

3

8

6

Figure not drawn to scale 1) Divide the shape into a

rectangle and right triangle (see original figure) 4

9

4

9

A

C

2) Determine the length of each side of triangle ABC• Determine length of sides AB and BC

from properties of 3-4-5 triangleAB = 5 and BC = 10

• Determine length of side AC from Pythagorean Theorem

9749AC 22


Dividing Irregular ShapesExample 2

Question: In the rectangle above, the sum of the areas of the shaded region is 14. What is the area of the unshaded region?


What is the strategy for identifying essential information? Divide the shape into a rectangle and right triangle.

Solution Steps

xyx

xy


Line Segment or Diagonal Length in a Geometric Solid

a

b

c

Right Triangle

Line Segment

c2 = a2 + b2

Pythagorean Theorem

Example: In the figure shown to the right, the endpoints of the line segment are midpoints of two edges of a cube of volume 64cm3. What is the length of the line segment?

Strategy: To find the length of a diagonal or a line segment that connects two edges of a geometric solid, create a right triangle within the solid that uses the unknown segment as the hypotenuse. Click to see the animation.

Reasoning: By finding a right triangle within the solid, Pythagorean Theorem can be used to find the segment or diagonal length.

Helpful Hint: The diagonal of any cube is equal to the cube side length times √3 Caution: Does not apply for rectangular solids (shoe box shape)


Line Segment Length in Solid Example 1

Question: What is the volume of a cube that has a diagonal length of 4√3?

What essential information is needed? Side length of the cube is needed to find the volume.

What is the strategy for identifying essential information?: Use the properties of a cube, the diagonal length, and Pythagorean theorem to find the side length.

Solution Steps

1) Establish relationships between cube diagonal length and side length using properties of a cube

a

a√2

a

a

• Let “a” be the side length of cube

• The longer side length of right triangle found using properties of

45-45-90 triangle

4√3

2) Apply Pythagorean theorem to find side length a2 + (a√2)2 = (4√3)2

a = 4

Volume = a3 = 43 = 64


Line Segment Length in Solid Example 2

What essential information is needed? A connection between given side lengths, the center of solid, and the midpoint of AB

What is the strategy for identifying essential information? Half the length of diagonal BD is equivalent to the desired distance. Use Pythagorean theorem.

Solution Steps

Question: In the figure above, if AB = 24, BC = 12, and CD = 16, what is the distance from the center of the rectangular solid to the midpoint of AB?

A

C

B

D

E

1) Diagonal BD is the hypotenuse of right triangle BCD. Find the length of BD.

A

C

B

D

E

24

12

16

Can easily find the length of BD by recognizing that triangle BCD is a multiple of the 3-4-5 triangle. The length of BD is 20. (12-16-20)

2) Half the length of diagonal BC is 20/2= 10 (shown in white on diagram)


Putting Shapes Together

Strategy: When asked to piece together several regular shapes into one shape, sum together the areas of individual pieces. The final shape will have the same area as the sum of the individual pieces.

Reasoning: The area must be conserved provided there is no overlap when the individual pieces are combined into one shape. Click to see the animation of the correct choice.

Which of the shapes below could be made from the three individual shapes shown above?

Area = 9 Area = 10 Area = 8

Area = 2Area = 3

Area = 5

Total area of three shapes = 10

Unit Area = 1 block


Putting Shapes Together Example 1

Question: Page under construction




Solution Steps

Putting Shapes Together Example 2





Solution Steps

3-4-5 Triangle

Strategy: Recognizing the 3-4-5 right triangle in a figure can save time and reduce the possibility of error when determining side lengths of a triangle.

Reasoning: Recognizing triangles as 3-4-5 do not require calculation of the third side using Pythagorean Theorem. Triangles with common multiple lengths of a 3-4-5 are similar to the 3-4-5.

Application: Look for right triangles with side lengths that are multiples of 3-4-5. Common examples include 6-8-10, 9-12-15, 12-16-20, and 15-20-25 triangles. Use similar triangle properties to determine unknown side lengths, not Pythagorean Theorem.

3

45

6

810

9

1215


3-4-5 Triangle Example 1

What essential information is needed? Side length BC is needed to find the triangle area.

What is the strategy for identifying essential information?: Can use Pythagorean theorem, however, more efficient to use properties of 3-4-5 triangle.

Solution Steps

Question: In the figure above, what is the area of ∆ABC?

100

80A

B

C1) Use properties of 3-4-5 triangle to find length of BC

• Side CA has a length of 80. This is a multiple of four (4 x 20 = 80)

• Side AB (hypotenuse) has a length of 100. This is a multiple of five (5 x 20 = 100)

• Conclusion: Side BC is a multiple of 3 and will have a length of 60. (3 x 20 = 60)

2) Calculate the area of ∆ABC

Area = ½(base)(height) = ½(80)(60)

Area = 2400



What essential information is needed? The length of side XZ is needed to find perimeter.

What is the strategy for identifying essential information? ∆XYZ is a right triangle. Can use Pythagorean theorem, however, it is easier and more efficient to use 3-4-5 triangle relationships.

Solution Steps

Question: In the figure above, what is the perimeter of ∆XYZ?

x

y

z

55

33

1) Use properties of 3-4-5 triangle to find length of XZ

2) Calculate the perimeter of ∆XYZ

• Side YZ has a length that is a multiple of three (3 x 11 = 33)

• Side XY has a length that is a multiple of five (5 x 11 = 55)

• Conclusion: Side XZ is a multiple of four and will have a length of 44. (4 x 11 = 44)

Perimeter = XY + YZ + XZ Perimeter = 55 + 33 + 44 Perimeter = 132


30-60-90 Triangle

Reasoning: This relationship is derived by splitting an equilateral triangle into two congruent 30-60-90 triangles. The relationships between sides are derived using Pythagorean Theorem. The formula for this relationship is found on the SAT formula sheet.

Application: Consider using for any triangle that has a 300 or 600 angle. Also, use for any right triangle that has a 300 or 600 angle.

Strategy: If the leg of a right triangle is expressed in terms of , the triangle is likely a 30-60-90. The coefficient associated with the is the length of the shorter leg. The hypotenuse is twice the length of the shorter leg.

3

3

Coefficient

600

300

5√3

510

Note: The 30-60-90 triangle is not a 3-4-5 triangle



What essential information is needed? A connection between side lengths that justifies calling triangle ABC a right triangle.

What is the strategy for identifying essential information?: Use properties of 45-45-90 triangle or 30-60-90 triangle to establish connection to right triangle.

Solution Steps

Question: In triangle ABC shown above, the length of side BC is half the length of side AB. The length of side AC is 4√3. What is the length of side AB?

C A

B

1) Identify connection to right triangle

2) Use properties of 30-60-90 triangle to find length of AB

• Triangle side BC = ½ side AB

• Triangle side AC has length 4√3

Conclusion: ∆ABC is a 30-60-90 triangle

• Side BC is short leg of triangle• Side AC is long leg of triangle• Side AB is hypotenuse of

triangle3) Determine length of side AB

AC = 4√3 BC = 4 AB = 2 x 4 = 8



What essential information is needed? Need a connection between side length AB (value of 4), AD (base of ∆ABD), and BD (height of ∆ABD)

What is the strategy for identifying essential information? The altitude of an equilibrium triangle divides the triangle into two 30-60-90 triangles. Use properties of 30-60-90 triangle to make connection.

Solution Steps

Question: Equilateral triangle ABC has a side length of 4. If BD is an altitude of ∆ABC, what is the area of ∆ABD?

AD

B

C

4

1) Find the length of AD (base of ∆ABD) and length of BD (height of ∆ABD)

2) Find the area of ∆ABD

Note: ABD is a 30-60-90 triangle with angle BAD = 600 and angle ABD = 300 Conclusion: Side AD = 2; half the length of hypotenuse ABConclusion: Side BD = 2√3; √3 times the length of the short side AD

Area = ½(base)(height)

Area = ½(2)(2√3)

Area = 2√3


45-45-90 Triangle

Reasoning: This relationship is a property of the 45-45-90 triangle. It can be derived using Pythagorean Theorem. The formula for this relationship is found on the SAT formula sheet.

Application: Consider using for any triangle that has a 450 angle . Also, any right triangle that is isosceles will be a 45-45-90 triangle.

Strategy: If the hypotenuse of a right triangle is expressed in terms of √2 , the triangle is likely a 45-45-90. The coefficient associated with the √2 is the length of each triangle leg.

Coefficient

450

450

5

5

5√2



What essential information is needed? Side length of square is needed to calculate area.

What is the strategy for identifying essential information?: Most efficient strategy is to recognize that the diagonal of a square divides the square into two congruent, isoceles triangles. Each triangle is a 45-45-90.

Solution StepsQuestion: In the figure below, what is the area of the square?

101) Use properties of 45-45-90 triangle to find side length

2) Calculate area of square

(Side length ) √2 = 10

Area = (side length)2

Side length = 10√2

(10)(√2)

Area = (10)(√2)

Area = 50

= 1002



What essential information is needed? Need to make a connection between the value of DC and the value of BC.

What is the strategy for identifying essential information? The two right triangles share a common side AC. Use properties of 30-60-90 and 45-45-90 triangles to make connection.

Solution Steps

Question: In the figure above, if DC = 2√6, what is the value of BC?

B

C

A

D

450

300

1) Find the length of AC using properties of 30-60-90 triangle

2) Find the length of BC using properties of 45-45-90 triangle

Note: AC is twice the length of AD and DC is √3 times the length of AD

AD(√3) = DC = 2√6

AD = 2√6√3

= 2√2

Conclusion: AC = 2(2√2) = 4√2

Note: BC is √2 times the length of AC

BC = (4√2)(√2)

BC = 8


Distance Between Two Pointsx-y Coordinate Plane

Strategy: Draw the x-y coordinate, plot the points, and find a right triangle. Calculate the distance as shown.

Reasoning: As shown to the right, the distance formula is an outcome of applying Pythagorean Theorem in the x-y coordinate plane. The distance “formula” is not given on the SAT formula sheet.

Application: Multitude of problems involving lines and points in the x-y coordinate plane. See examples for specific applications.

212

212 yyxxd

x2 - x1

y2 - y1

(3, 3.5)

(-5, -2.5)

10100

5.25.353 22

d

d

d = 10


Distance Between Two Points Example 1

Question: If points A (6, 2), B(12, 2), and C(9, 9) are endpoints of triangle ABC, what is the perimeter of the triangle?

What essential information is needed? Need to find the length of each side of triangle ABC.

What is the strategy for identifying essential information?: A quick sketch of the triangle reveals an isosceles triangle with the non-congruent side AB parallel to the x-axis. The remaining two sides are congruent and require use of the distance formula to find side length.

Solution Steps

1) Find the length of side AB using distance formula for a number line

2) Find the length of congruent sides AC and BC using distance formula for x-y coordinate plane

d = │12 - 6│ = 6

212

212 yyxxd

582969 22 BCAC dd

3) Find the perimeter of triangle ABCPerimeter = 6 + √58 + √58

Perimeter = 6 + 2√58

A(6, 2) B(12, 2)

C(9, 9)


Distance Between Two Points Example 2





Solution Steps

Midpoint DeterminationNumber Line

Strategy: The midpoint (xm) between two endpoints on a number line is found by averaging the two endpoints.

Reasoning: The midpoint is equidistant from either endpoints. This is consistent with the properties of the average (mean) of two numbers.

Application: Number line applications that requires the determination of midpoint or endpoint values. The midpoint “formula” is not given on the SAT formula sheet.

Midpoint

5.12

74

221

xxxm

xm

5.5 5.5

- 4x1

7x2

0

xm = 1.5


Midpoint Determination Example 1

Question: If 3n and 3n+4 are end points on a number line, what is the midpoint?a) 3n+1 b) 3n+2 c) 3n+2.5 d) 3n+3 e) 41(3n)

What essential information is needed? Find the point that is located midway between the two endpoints.

What is the strategy for identifying essential information?: Use the midpoint determination strategy for finding midpoint on a number line

Solution Steps

1) Find the sum of the two endpoints

3n + 3n ·34 Factor 3n

3n (1 + 34 ) = 3n (1 + 81)

2) Divide the sum by two to find midpoint

82(3n )2

= 41(3n )

82(3n )

3n + 3n+4 Expand 3n+4



Question: If x - 2 and y are endpoints on a number line and x + 6 is the midpoint, which of the following expressions represents y? a) xb) x + 2c) x + 12d) x + 14e) x + 16

What essential information is needed? Find the endpoint that has x + 6 as the midpoint when x - 2 is the other endpoint.

What is the strategy for identifying essential information? Apply the midpoint determination strategy to find the endpoint “y”.

Solution Steps

1) Apply the midpoint strategy to set up the solution.

x + 6 =(x - 2) + y2

2) Solve for the endpoint “y”

x + 6 =(x - 2) + y2

Cross multiply

2(x + 6) = (x - 2) + y Simplify and solve for “y”

2x + 12 = x - 2 + y

x +14 = y


Midpoint Determinationx-y Coordinate Plane

Strategy: The midpoint (xm , ym ) between two endpoints on the x-y coordinate plane is found by averaging the x-coordinates and y-coordinates of the two endpoints.

Reasoning: The midpoint of each x-y coordinate point is equidistant from either endpoint. This is consistent with the properties of the average of two numbers

Application: In addition to the x-y coordinate, questions could ask for the midpoint on a number line. Some questions will give the midpoint and one end point and ask for the unknown end point. The midpoint “formula” is not given on the SAT formula sheet.

12

68

221

m

m

x

xxx

Midpoint

(xm , ym )

x1 + x2

y1 + y2

(6, 6)

(-8, -4)

12

64

221

m

m

y

yyy

(-1, 1)



Question: In the x-y coordinate plane, the points (2, 8) and (12, 2) are on line m. The point (7, y) is also on line m. What is the value of y?

What essential information is needed? A method for determining the value of “y”

What is the strategy for identifying essential information?: Can use two known points to find the equation of line m and use equation to find y. Equation of line not on SAT formula sheet. As a result, likely not the most efficient approach. As an alternative, midpoint analysis can be used.

Solution Steps

1) Midpoint analysis of “x” values

2) Find the midpoint of 2 and 8

Conclusion: The “y” value must be the midpoint of 2 and 8

72

122

mx

The “x” value of 7 is the midpoint of 2 and 12

52

82

my

“y” value = 5

Note: Same result using equation of line……less efficient method.



Question: In the x-y coordinate plane, the midpoint of AB is (2, 3). If the coordinates of point A are (-1, 1), what are the coordinates of point B?

What essential information is needed? Need to connect coordinates of endpoint to the coordinates of midpoint.

What is the strategy for identifying essential information? Use the midpoint formula to connect the coordinates of endpoints to the midpoint.

Solution Steps

1) Find the endpoint by using the midpoint formula

Coordinates of endpoint are (5, 5)

221 xx

xm

2

21 yyym

2

12 2x

2

13 2y

214 x 216 y

52 x 52 y


Exterior Angle of a Triangle

Strategy: Any exterior angle of a triangle is equal to the sum of the two remote interior angles

Reasoning: The sum of the two remote interior angles is supplementary to the third interior angle. Likewise, the exterior angle is supplementary to the third interior angle.

Application: This strategy is a useful way to save time and potential calculation errors when an exterior angle of any triangle is needed.

450

750

x0

Exterior angle

Remote interior angles


Exterior Angle of a Triangle Example 1

What essential information is needed? A strategy is needed to connect the known angle values to the unknown variables.

What is the strategy for identifying essential information?: Can easily find the value of x + y using exterior angle of triangle strategy. Can also find the value of y. From alternate interior angles, x = z.

Solution Steps

Question: In the figure above, line m is parallel to line k. What is the value of z?

x0 y0

1100

m

k

z0

1000 1) Find the value of x + y

2) Find the value of y

3) Find the value of z

1100 is an exterior angle; x and y are the remote interior anglesConclusion: x + y =1100

y is a linear pair with angle 1000

Conclusion: y = 800 and x = 300

From alternate interior angles, z = x

Conclusion: z = 300


Exterior Angle of a Triangle Example 2

What essential information is needed? Need a connection between the given angle value of 950 and the unknown angle variables.

What is the strategy for identifying essential information? The given angle of 950 is an exterior angle to both triangles.

Solution Steps

Question: In the figure above, what is the sum of a + b + c + d?

a0 b0

d0 c0

950

1) Find the value of a + b

2) Find the value of c + d

950 is an exterior angle; a and b are the remote interior anglesConclusion: a + b = 950

950 is an exterior angle; c and d are the remote interior anglesConclusion: c + d = 950

3) Find the value of a + b + c + d

a + b + c + d = 2(950 )

a + b + c + d = 1900


Perpendicular Lines

Strategy: The slopes of perpendicular lines are opposite reciprocals of each other.

Reasoning: This is a fundamental relationship developed in coordinate geometry

Application: All questions involving perpendicular lines require comparison of slopes

l

q

ql mm

1

Back to Frequent

Questions


Perpendicular Lines Example 1

Question: In the xy-plane above, the equation of line m is 4x + 3y = 12. Which of the following is an equation of a line that is perpendicular to line m?a) y = x + 3 b) y = -4x + 3 c) y = 4x - 3 d) y = ¾x + 6e) y = -¾x - 6

What essential information is needed? The slope of line m is needed to determine the slope of line perpendicular to line m

What is the strategy for identifying essential information?: Slope of line m can be determined from equation of line m or directly from figure.

Solution Steps

42

2 4

1) Slope of line m

2) Equation of line perpendicular to line m

= - 43

• Slope using figure

• Slope using equation of line m

Slope = ∆y∆x

4 - 00 - 3

=

4x + 3y = 123y = -4x + 12

y x + 12= - 43

• Correct choice is y = ¾x + 6

• Line must have slope = ¾


Perpendicular Lines Example 2

What essential information is needed? Need to identify a line perpendicular to line q and determine the slope of the new line.

What is the strategy for identifying essential information? Draw a line from origin to point of tangency. This line is a radius and is perpendicular to line q.

Solution Steps

Question: Line q is tangent to the circle at the point (4, -3). What is the slope of line q?

(4, -3)

q

1) Find slope of new line

2) Find the slope of line q

• Slope of a line that passes through origin can be determined from the ratio of y/x for any point on the line.

• Slope of new line is -¾

• Slope of line q is the opposite reciprocal of slope of new line

Slope of line q is 43


Interval Spacing

Strategy: The interval spacing on a number line is found by a two-step process:1. Determine the distance between two

known points on the number line2. Divide the distance by the number of

intervals separating the two known points (Caution: Do not divide by the number of tick marks)

Reasoning: By design, the number line has equal distance between each tick mark on the line

Application: Used to identify an unknown coordinate on number line. Also used to identify the value of specific term in an arithmetic sequence.

3 18 23

What is this value?

2.5

|18 - 3|6

= 2.5

18 + 2(2.5) = 23


Interval Spacing Example 1

Question: The value of each term of a sequence is determined by adding the same number to the term immediately preceding it. The value of the third term of a sequence is 4 and the value of the eighth term is 16.5. What is the value of the tenth term?

What essential information is needed?The common value added to each term of the sequence.

What is the strategy for identifying essential information? Use interval spacing strategy to identify the common value. Add twice this value to the eighth term to find value of tenth term.

Solution Steps

1) Find the common value.

16.5 - 45 intervals

= 12.5 5 intervals

= 2.5

2) Add twice the common value of 2.5 to the eighth term value of 16.5.

Tenth term = 16.5 + 2.5 + 2.5

Tenth term = 21.5


Interval Spacing Example 2

Question: On the number line above, what is the value of point P? a) 2n+½ b) 2n+¾ c) 3·2n

d) 3·2n+1 e) 3·2n+2

What essential information is needed? The interval spacing can be used to find the value of “P”.

What is the strategy for identifying essential information? Find the interval spacing by dividing the difference of the two endpoints by the number of intervals (six). Multiply the interval spacing by three and add to the value of the left endpoint.

Solution Steps2n+1 2n+2 P

1) Find the interval spacing

2) Find the value of “P”

2n+2 - 2n+1 Expand the powers

2n ·22 - 2n ·21 Common factor is 2n

2n (22 - 21) Simplify 22 - 21 2n (2) Divide by six intervals

2n (2)6

= 2n 3

Interval spacing

2n+1 + (3)2n 3

= 2n+1 + 2n Expand the powers and factor2n ·21 + 2n = 2n (21 + 1)

3∙ 2n Value of point “P”

3


Triangle Side Lengths

Strategy: The 3rd side of any triangle is greater than the difference and smaller than the sum of the other two sides

Reasoning: A side length of 15 would require the formation of a line, not a triangle. A side length of 3 would also require the formation of a line, not a triangle

Application: Given two sides, choose the smallest or greatest integer value of third side. Given three sides as answer choices, which will not form a triangle.

9

6

3 < x < 15

9 6

15

9

63


Triangle Side Lengths Example 1

Question: If the side lengths of a triangle are 8 and 23, what is the smallest integer length of the third side?a) 14 b) 15 c) 16d) 30 e) 31

What essential information is needed? The smallest possible length of the third side of the triangle

What is the strategy for identifying essential information?: The third side of a triangle must be greater than the difference of the given two sides of the triangle.

Solution Steps

1) Find the smallest possible length of the third side

2) Determine the smallest integer length of third side of triangle

Length of third side > 23 - 8

Length of third side > 15

Smallest integer length is 16


Triangle Side Lengths Example 2

Question: Each choice below represents three suggested side lengths for a triangle. Which of the following suggested choices will not result in a triangle?a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12)d) (5, 6, 7) e) (6, 6, 11)

What essential information is needed? The range of possible triangle side lengths for each answer choice.

What is the strategy for identifying essential information? Evaluate the first two numbers of each answer choice using triangle side length strategy. Test the third number of each answer choice by comparing to range of possibilities based on first two numbers.

Solution Steps

1) Determine range of possible side lengths using first two numbers

2)Test third number of each answer choice

a) 5 - 2 < x < 5 + 2 3 < x < 7b) 7 - 3 < x < 7 + 3

c) 8 - 3 < x < 8 + 3

d) 6 - 5 < x < 6 + 5

e) 6 - 6 < x < 6 + 6

4 < x < 10

5 < x < 11

1 < x < 110 < x < 12

a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12)

d) (5, 6, 7) e) (6, 6, 11)

Correct answer choice is “c”

yes

yes

no

yes

yes


Similar Triangle Properties

Strategy: Under construction

Reasoning:

Application:

Back to Frequent

Questions


Similar Triangle Properties Example 1

Question: In the figure to the right, what is the value of “a” ?



Solution Steps

x

x

a

4

3

8


Similar Triangle Properties Example 2



Solution Steps

Question: In the figure to the right, , , , and What is the length of ?

DEAC || 2BD 4DA 3DE

AC

ED

C

B

A


The Slippery Slope

Strategy: When given linear equations as answer choices and a question about the amount of change in the “y” variable as the “x” variable is changed a given amount, use the properties of slope to quickly select the correct choice.

Reasoning: Slope is a measure of the amount of change in the “y” value when the “x” value is changed by one unit. The constant in the equation has no impact on the amount of change in the dependent variable value.

Application: Any question

a) d = 50t - 100

e) d = -500t + 10000

b) d = 40t + 1000

c) d = 40t + 100 d) d = -50t + 1000

If d represents the distance measured in meters from a particular coffee shop and t is time measured in minutes, which of the following equations describes the greatest increase in distance from the coffee shop during the period from t = 5 minutes to t = 8 minutes?

Caution: Do not calculate distance values by direct substitution into each equation. Use properties of slope to quickly determine answer. Click for correct choice.


The Slippery Slope Example 1

Question: The table to the right gives the value in dollars of five different investments at t years after the investment was started. The value of which investment falls the greatest amount during the period t = 4 to t = 9 ?



Solution Steps

Investment Value at t Years

A -30t + 50

B -10t - 50

C -10t + 50

D 10t - 50

E 30t - 50


The Slippery Slope Example 2





Solution Steps

Using Function Notation

Strategy: Replace the variable in the function expression (right side of equal sign) with the value, letter, or expression that has replaced the variable (usually x) in the function notation (left hand side of equal sign)

Reasoning: Function notation is a road map or guide that directly connects the “x” value for a given function with one unique “y” value.

Application: Function notation can be applied in many different ways on the SAT. See examples for details. Function notation is commonly used to describe translations and reflections of functions. See Table of Contents for additional strategies that use function notation.

Function notation such as f(x), g(x), and h(x) are useful ways of representing the dependent variable “y” when working with functions. For example, the function y = 2x + 5 can be written as f(x) = 2x + 5, g(x) = 2x + 5, or h(x) = 2x + 5.

Introduction

Important Note: Function notation is not a mathematical operation. See example of commonly made mistake.

Back to Definition

Back to Frequent

Questions


Using Function Notation Example of Common Mistake

Question: At a certain factory, the cost of producing control units is given by the equation C(n) = 5n + b. If the cost of producing 20 control units is $300, what is the value of “b”?

Common mistake: Function notation should not be used as a math operation. C(n) should be replaced with 300 when n = 20. Do not multiply 300 and 20 as in a math operation.

Correct use of function notation: C(n) is replaced with 300 when n is replaced with 20 in the function equation.

Solution Steps for Commonly Made Mistake

1) Replace “C” with 300 and replace “n” with 20

C(n) = 5n + b

300(20) = 5(20) + b

6000 = 100 + b

b = 5900 (incorrect answer)

C(n) = 5n + b

Correct Solution Steps

300 = 5(20) + b 300 = 100 + b

b = 200 (correct answer)

Return to Table of Contents Return to strategy page See example of strategy

Using Function NotationExample 1

Question: If f(x) = x + 7 and 5f(a) =15, what is the value of f(-2a)?

What essential information is needed? The value of “a” is needed to determine the value of f(-2a).

What is the strategy for identifying essential information?: Use the given information and properties of function notation to identify the value of “a”. Use this value to evaluate f(-2a).

Solution Steps

1) Find the value of “a”

Given 5f(a) = 15 Divide both sides by 5

Result f(a) = 3

Given f(x) = x + 7 Evaluate f(a)

f(a) = a + 7 = 3

Result: a = -4

2) Use a = -4 to find f(-2a)

f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8)

f(8) = 8 + 7

f(-2a) = 15



Question: The graph of y = f(x) is shown to the right. If the function y = g(x) is related to f(x) by the formula g(x) = f(2x) + 2, what is the value of g(1)?

What essential information is needed? The math expression g(1) from which the value of g(1) can be determined

What is the strategy for identifying essential information? Find the expression for g(1) by substitution and the value of g(1) using the graph of y = f(x).

Solution Steps

y = f(x)2

2

-2

-2

1) Find the expression for g(1)

g(x) = f(2x) + 2

g(1) = f(2) + 2

2) Find the value of f(2) from the graph of y = f(x)

f(2) = 2

g(1) = 2 + 2 g(1) = 4



Question: Using the table to the right, if f(3) = k, what is the value of g(k)?

What essential information is needed? The value of “k” is needed to find g(k).

What is the strategy for identifying essential information? Use the table of function values to find “k”. Once known, find g(k) using the table of function values.

Solution Steps

x f(x) g(x)

1 3 8

2 4 10

3 5 8

4 6 6

5 7 4

1) Find the value of “k” using table.

f(3) = k

2) Find the value of g(5) using table.

f(3) = 5

g(5) = 4



Question: If f(x) = x + 8, for what value of x does f(4x) = 4?

What essential information is needed? Need to determine the value of “x” that satisfies f(4x) = 4.

What is the strategy for identifying essential information? Use function notation principles to determine an expression for f(4x). Set the expression equal to the value of 4.

Solution Steps

1) Determine an expression for f(4x)

2) Set the expression for f(4x) equal to 4 and solve for the value of “x”

f(x) = x + 8

f(4x) = 4x + 8

f(4x) = 4x + 8 = 44x + 8 = 44x = -4

x = -1


Function Reflectionsx - Axis

Strategy: The reflection of a function y = f(x) around the x-axis is easily performed by graphing the opposite (negative) of each y-value. Using function notation, this can be communicated as y = - f(x).

Reasoning: The reflection of a function around the x-axis can be viewed as a mirror image of the original reflection. Imagine the x-axis as a flat mirror that reflects and produces an image of the original function on the opposite side of the x-axis.

Application: x-axis reflections can be performed for any function using the strategy described above.

y = f(x)

y = - f(x)Reflection of f(x)


Function Reflections: x - Axis Example 1

Question: If point (a, b) is reflected over the x-axis, what are the coordinates of the point after the reflection?

What essential information is needed? Must determine which, if any, coordinate signs will be affected.

What is the strategy for identifying essential information?: For an x-axis reflection, use the function notation y = -f(x) as a guide.

Solution Steps

A reflection over the x-axis is described by y = -f(x). To accomplish the reflection, change the sign of the y-coordinate only.

Correct answer is (a,-b)

Note: Do not get confused by the original sign of the y-coordinate. If the original sign is “-y”, the reflected point will have the sign “+y”.


Function Reflections: x - Axis Example 2





Solution Steps

Function Reflectionsy - Axis

Strategy: The reflection of a function y = f(x) around the y-axis is easily performed by graphing the opposite (negative) of each x-value. Using function notation, this can be communicated as y = f(-x).

Reasoning: The reflection of a function around the y-axis can be viewed as a mirror image of the original reflection. Imagine the y-axis as a flat mirror that reflects and produces an image of the original function on the opposite side of the y-axis.

Application: y-axis reflections can be performed for any function using the strategy described above.

y = f(x)y = f(-x)Reflection of f(x)


Function Reflections: y - Axis Example 1

What essential information is needed? Must determine which, if any, coordinate signs will be affected.

What is the strategy for identifying essential information?: Helps to recognize that f(x) = f(-x) describes a reflection about the y - axis.

Solution Steps

1) Reflect f(x) about the y - axis ( click to show reflection)

2) Identify the point for which f(x) = f(-x)

Question: For the graph of the function f shown above, for what point does f(x) = f(-x)?

(-1, 0)

(0, 1)

(2, 2)

(5, 0)

• The only point that remains the same after reflection is the y intercept

f(0) = 1 and f(-0) = 1Correct choice is (0, 1)


Function Reflections: y - Axis Example 2





Solution Steps

Function ReflectionsAbsolute Value

Strategy: The absolute value of function y = f(x) is easily created by graphing the opposite (negative) of each y-value that is negative on the original function. Using function notation, this can be communicated as y = |f(x)|.

Reasoning: The absolute value of a function is a reflection of y = f(x) around the x-axis for those intervals of x that have negative y values.

Application: Absolute value can be created for any function using the strategy described above.

y = f(x)y = |f(x)|

Back to Definition


Function Reflections: Absolute Value Example 1

What essential information is needed? Need to determine the effect of absolute value on the graph of f(x)

What is the strategy for identifying essential information?: The absolute value strategy should be used.

Solution Steps

A B C

D E

Question: The graph of y = f(x) is shown above. Which of the choices could be the graph of y = │f(x)│?

The absolute value reflects the graph of y = f(x) about the x- axis for intervals of “x” where f(x) < 0.

Correct answer is choice C


Function Reflections: Absolute Value Example 2





Solution Steps

Function TranslationsHorizontal Shift

Strategy: A horizontal shift of a function y = f(x) is easily performed by sliding the function right or left parallel to the x-axis a specified distance. Using function notation, a shift to the right of 2 units can be communicated as y = f(x-2). A shift to the left of 4 units can be communicated as y = f(x+4)

Reasoning: A horizontal shift described by y = f(x-2) has the same y-value at x = 2 as the original function f(x) at x = 0.

Application: Horizontal shifts can be performed for any function using the strategy described above.

y = f(x)

y = f(x-2)

y = f(x+4)

2

2


Function Horizontal Shift Example 1

Question: The graph of y = f(x) is shown to the right. Which of the following could be the graph of y = -f(x+1) ? Click to see answer choices

What essential information is needed? Need to interpret the impact of -f(x+1) on the original function y = f(x).

What is the strategy for identifying essential information? Use the function notation strategy and the properties of function reflections and translations to choose the correct answer.

Solution Steps


A

Horizontal shift lefty = f(x+1)

E

Horizontal shift leftx-axis reflection

y = -f(x+1)

C

Horizontal shift righty = f(x-1)

D

Horizontal shift rightx-axis reflection

y = -f(x-1)

B

x-axis reflectiony = -f(x)

-1 2

y = f(x)


Function Horizontal Shift Example 2





Solution Steps

Function TranslationsVertical Shift

Strategy: A vertical shift of the function y = f(x) is easily performed by sliding the function up or down parallel to the y-axis a specified distance. Using function notation, a shift down of 2 units can be communicated as y = f(x)-2. A shift up of 4 units can be communicated as y = f(x)+4

Reasoning: A vertical shift described by y = f(x)-2 decreases the y-value by 2 units for each value of x on the original function y = f(x).

Application: Vertical shifts can be performed for any function using the strategy described above.

y = f(x)

y = f(x)- 2

2

y = f(x)+4


Function Vertical Shift Example 1

Question: The figure to the right shows the graph of function f(x) in the x-y coordinate plane. If the area between f(x) and x-axis is 10, what is the area between the function f(x)+2 and x-axis ?



Solution Steps

y = f(x)

5


Function Vertical Shift Example 2





Solution Steps

Function TranslationsVertical Stretch

Strategy: A vertical stretch of the function y = f(x) is easily performed by multiplying each y-value by a specified amount greater than one. Using function notation, a vertical stretch of 2 units can be communicated as y = 2f(x).

Reasoning: A vertical stretch described by y = 2f(x) multiplies each y-value by 2 units for each value of x on the original function y = f(x).

Application: Vertical stretches can be performed for any function using the strategy described above.

y = f(x)

y = 2f(x)

Multiply each y-value by 2


Function Vertical Stretch Example 1

What essential information is needed? Need to understand the impact on y = f(x) when f(x) is multiplied by 2.

What is the strategy for identifying essential information?: y = 2f(x) describes a vertical stretch. Apply the properties of a vertical stretch to y = f(x).

Solution Steps

Question: The graph of y = f(x) is shown above. Which of the choices could be y = 2f(x)?

A B C

D E

A vertical stretch multiplies each “y” value on f(x) by two. As a result, the x-intercepts remain the same on y = 2f(x).

The correct answer choice is E


Function Vertical Stretch Example 2





Solution Steps

Function TranslationsVertical Shrink

Strategy: A vertical shrink of the function y = f(x) is easily performed by multiplying each y-value by a specified amount between zero and one. Using function notation, a vertical shrink of ½ units can be communicated as y = ½f(x).

Reasoning: A vertical shrink described by y = ½f( x) multiplies each y-value by ½ units for each value of x on the original function y = f(x).

Application: Vertical shrinks can be performed for any function using the strategy described above.

y = f(x)

y = ½f(x)

Multiply each y-value by ½


Function Vertical Shrink Example 1





Solution Steps

Function Vertical Shrink Example 2





Solution Steps

Average (Arithmetic Mean) Problems

Strategy: Apply the basic definition of average (arithmetic mean) to solve this class of problems.

Reasoning: Information will typically be given for the average and the number of values. The sum of values will be always be needed to reason through question and will typically consist of an expression with unknown variable(s).

Application: 1) Problems that ask for an unknown value when given remaining values in the list and the average value of the list. 2) Problems that provide the average of a list of numbers, removes a number from the list, gives the new average, and asks for the value of the removed number.

sum of values number of values

average =

Caution: You will rarely be asked to find the average of a list of values. Instead, you will typically be asked to find the median of a list of values.

Often used form:sum of values =

(average)( number of values)

Back to Definition

Back to Frequent

Questions


Average (Arithmetic Mean) Example 1

Question: If the average of 6 and x is 12, and the average of 5 and y is 13, what is the average of x and y?

What essential information is needed? Need values of x and y to determine average value.

What is the strategy for identifying essential information?: Apply basic definition of average to find values of x and y separately.

Solution Steps

1) Determine the values of x and y:

6 + x2

= 12 5 + y2

= 13

6 + x = 24 5 + y = 26

x = 18 y = 21

2) Find average of x and y using basic definition of average:

18 + 212

Average =

Average = 19.5


Average (Arithmetic Mean) Example 2

Question: The average of five positive odd integers is 15. If n is the greatest of these integers, what is the greatest possible value of n?

What essential information is needed? The sum of the five positive odd integers is needed and a strategy to determine the greatest possible value of “n”

What is the strategy for identifying essential information? Apply the definition of average to find sum. Use reasoning skills to determine greatest possible value of “n”

Solution Steps

1) Find the sum of the five positive odd integers.

Sum of values = (15)(5) = 75

2) Determine the greatest possible value of “n” using reasoning skills

• The four smallest positive integers are 1, 1, 1, 1 with a sum of four.

• The greatest possible value of “n” is

n = 75 - 4 = 71


Median of Large Lists

Strategy: The middle value in a list of ascending or descending ordered values is the median. Large lists of values (more than 7 values) are usually structured in table form or bar chart form. Either form will not require rewriting of the order by the student.

Reasoning: Values provided in table form are similar to values provided in histogram form. In both forms it is easy to determine the cumulative total number of values starting with the lowest value.

Application: When values are organized in tables, questions will generally ask for the median directly or will give the median and ask for the value of an unknown variable.

Caution: Do not confuse median with mean. When presented a table of values or a list of values, the question typically requires determination of the median, not the mean.

Additional Helpful Hints

1) For an ordered list with an odd number of values, the median is the middle value. 2) For an ordered list with an even number of values, the median is the average of the two middle values.

Back to Definition

Back to Frequent

Questions


Median of Large ListsExample 1

Question: The scores on a recent physics test for 20 students are shown in the table to the right. What is the median score for the test?

What essential information is needed? When the test scores are ordered from largest to smallest, find the middle score for the list.

What is the strategy for identifying essential information?: With the test scores in table form, no additional ordering is needed. With 20 students, the median is the average of the scores of the 10th and 11th students.

Solution Steps

Score Number of Students

100 0

95 1

90 1

85 2

80 3

75 4

70 3

65 2

60 4

The 8th ,9th ,10th ,and 11th students each received a score of 75 on the test

Median score is 75

0 1123

Sum = 7

0 11234

Sum = 11

The 5th , 6th , and 7th students each received a score of 80 on the test


Median of Large ListsExample 2

Question: If the median of 10 consecutive odd integers is 40, what is the smallest integer among these integers?



Solution Steps


Elementary Probability

Strategy: Divide the number of values that meet the given criteria by the total number of values in the set.

Reasoning: This is the basic definition of probability. The probability of an event is a number between 0 and 1, inclusive. If an event is certain, the probability is 1. If an event is impossible, the probability is 0.

Application: Additional applications include finding the probability of choosing a particular object (marbles, cookies, coins) from a container with more than one type of object.

Given information:

{10, 12, 13, 18, 21, 23, 25, 29}

Question:What is the probability of choosing a prime number at random from the above set?

Essential information:1)The number of values meeting the question criteria is 32)The total number of values in the set is 8

Solution:

Probability = ⅜

Back to Definition


Elementary Probability Example 1

Question: A jar contains red, blue, and yellow marbles in the ratio 9:4:2. If a marble is selected at random, what is the probability of selecting a blue marble?

What essential information is needed? The ratio of number of blue marbles to the total number of marbles.

What is the strategy for identifying essential information?: Use the properties of ratios to determine the essential information. Use the ratio to determine the probability.

Solution Steps

1) Determine the ratio of blue marbles to total number of marbles

2) Determine the probability

• For every 15 total marbles in the jar (9 + 4 + 2 = 15) there are 4 blue marbles

• The probability can be found by using the ratio of blue marbles to total marbles.

Note: It is not necessary to know the exact number of each marble in the jar. Ratios are sufficient for probability.

Probability =415


Elementary Probability Example 2

Question: A certain bowling center has two sizes of bowling balls, twelve pounds and sixteen pounds. For every 3 twelve pound bowling balls there are 4 sixteen pound bowling balls. If a bowling ball is chosen at random, what is the probability that a sixteen pound bowling ball will be selected?

What essential information is needed? The ratio of the number of sixteen pound bowling balls to the total number of bowling balls.

What is the strategy for identifying essential information? Use the properties of ratios to determine the essential information. Use the ratio to determine the probability.

Solution Steps

1) Determine the ratio of blue marbles to total number of marbles

2) Determine the probability

• For every 7 bowling balls (3 + 4 = 7), there are 4 sixteen pound bowling balls

• The probability can be found by using the ratio of sixteen pound bowling balls to the total number of bowling balls Probability =4

7

Note: The strategy for this problem is identical to the previous example. The questions are slightly different, however both involve ratios


Probability of Independent Events

Strategy: Multiply the probabilities of the individual events together to find the overall probability.

Reasoning: Each individual first event must be paired with each individual second event. To account for the total number of outcomes meeting the given criteria (value in numerator) and the total number of possible outcomes (value in denominator), the individual probabilities must be multiplied together. Application: Popular applications include the probability of an outcome when a coin is flipped multiple times and the probability of passing multiple academic courses

Definition: Two events are independent if the outcome of the first event has no effect on the outcome of the second event

Example: David has a red, yellow, blue, and green hat. He also has a red and blue shirt. If an outfit consists of a hat and shirt, what is the probability that David will wear an all red outfit?

Solution: The probability of choosing a red hat is ¼ and the probability of choosing a red shirt is ½.

The overall probability is (¼)(½) = ⅛

Back to Definition

Back to Frequent

Questions


Probability of Independent Events Example 1

Question: Adam has a 90% chance of passing history and a 60% chance of passing calculus. What is the probability that Adam will pass calculus and not pass history?

What essential information is needed? Are these events independent of each other?

What is the strategy for identifying essential information?: It can be assumed that passing history is independent of passing calculus. The two events are independent and the individual probabilities can be multiplied together.

Solution Steps

1) Determine the probability that Adam will pass calculus

2) Determine the probability that Adam will not pass history

3) Determine the probability that Adam will pass calculus and not pass history

Overall probability = 110

610

x = 6100

= 350

Probability = 60100

610

=

Probability = 10100

= 110


Probability of Independent Events Example 2

Question: The three cards shown to the right were taken from a box of ten cards, each with a different integer from 0 to 9. What is the probability that the next two cards selected from the box will both have an even integer on it?



Solution Steps

1 5 7


Geometric Probability

Strategy: Divide the area of the smaller geometric shape by the area of the larger geometric shape.

Reasoning: For planar geometrical shapes, area is the proper quantity to compare when selecting a point inside the figure.

Application: Usually involve simple shapes such as circles, rectangles, and squares. In all cases there is a smaller shape inside the larger shape and the analysis requires calculation of shape area.

Definition: Geometric probabilities involve the use of two or more geometric figures.

Example: A small circle with radius 3 is completely inside a larger circle with radius 6. If a point is chosen at random from the large circle, what is the probability that the point will be in the small circle?

Essential information:1) Area of small circle is π(3)2 = 9π

2) Area of large circle is π(6)2 = 36π

Solution:

Probability = ¼

Back to Definition

Back to Frequent

Questions


Geometric Probability Example 1

Question: In the figure above, each of the small circles has a radius of 3 and the large circle has a radius of 9. If a point is chosen at random inside the larger circle, what is the probability that the point does not lie in the shaded area?

What essential information is needed? Need the area of the large circle and area of each of the smaller circles.

What is the strategy for identifying essential information?: Use the formula for area of a circle to find areas of each circle. To find probability, ratio the area of the shaded region to the area of the large circle.

Solution Steps

1) Find the area of each circle

2) Find the geometric probability

Area of each small circle = π(3)2 = 9π

Area of large circle = π(9)2 = 81π

Probability = 81π - 2(9π)81π

=63π81π

Probability = 79


Geometric Probability Example 2

What essential information is needed? Need to determine the area of triangle ABC and the area of the rectangle. The length of AB is needed to find both areas.

What is the strategy for identifying essential information? Side AB is twice the radius of circle C. Knowing AB, use Pythagorean theorem to find AC and CB.

Solution Steps

Question: The rectangle above with side length 4 contains circle C that has a radius of 1. If a point is chosen at random inside the rectangle, what is the probability that the point will lie in triangle ABC?

C

4 B

A

1) Find the area of triangle ABC

2) Find the probability

• Triangle ABC is a 45-45-90 triangle• AB is twice the radius of circle C

and has a length of 2• AC and CB are congruent and are

each equal to √2

Area = ½(√2)(√2) = 1

Probability = area of trianglearea of rectangle

= 1(2)(4)

Probability = ⅛


The Unit Cell

Strategy: Divide the given end of the metal strip into a smaller shape, called a “unit cell”, that can used to easily and quickly answer the question. Click to show the unit cell!

Reasoning: The unit cell is a repeating shape that comprises the entire object shape. Ten unit cells comprise the entire metal strip. Click to see calculation. The top horizontal section and the bottom notched section of each unit cell contributes 3 + 1 + 3 + 1 = 8 inches to the perimeter.

Application: Any question that provides, in the form of a figure, a representative section of a longer object.

One end of a 30-inch long metal strip is shown in the figure above. The lower edge was formed by removing a 1-in square from the end of each 3-inch length on one edge of the metal strip. What is the total perimeter, in inches, of the 30-inch metal strip?

The “Unit Cell”

1 in

3 in

1 in

1 in

The total perimeter is equal to:10 unit cells x 8-in/unit cell + 2 vertical sides x 3-

in Perimeter = 86 inches

Three “Unit Cells” shown

Leftover section Not a unit cell

30-in strip3-in unit cell

= 10 unit cells

Back to Frequent

Questions


The Unit Cell Example 1





Solution Steps

The Unit Cell Example 2





Solution Steps

It’s Absolutely Easy!

Strategy: Under construction


Reasoning:

Application:

It’s Absolutely Easy! Example 1





Solution Steps

It’s Absolutely Easy! Example 2





Solution Steps

Making ConnectionsThe “if…” Statement

Back to Frequent

Questions

Strategy: For questions that begin with “If…” and end with “what is the value of…”, or “which of the following must equal…”, find a straightforward connection that links the given information (usually an equation) to the desired answer (usually the value of an expression).

Reasoning: The questions are designed to be solved in a straightforward way, provided the connection between the given information and the desired answer is made. To find the connection typically requires out of the box thinking.

Example 1: If 4x2 = 18y = 36, what is the value of 2x2y?

Example 2: If 2x + 7y = y, which of the following must equal 4x + 12y ?

Example 1

4x2 = 18y = 36 2x2y?

Connection #1: Set 4x2 = 36. Solve for 2x2

Connection #2: Set 18y = 36. Solve for y

Connection?

Example 2

2x + 7y = y 4x + 12y

Connection: Subtract y from both sides of equation. Result is 2x + 6y = 0. Multiply both sides of equation by 2.

Connection?


Making Connections Example 1

Question: If x is positive and x(x-1) = 30, what is the value of x(x+1) ?

What essential information is needed? Need to find a connection between the factored form of the expression on the left side of the equal sign and the value of 30 on the right side.

What is the strategy for identifying essential information?: The factors on the left side are consecutive integers. Determine if the value 30 has factors that are consecutive positive integers. Note: Not necessary to foil the expression and solve as a quadratic equation x2 - x - 30 = 0

Solution Steps

1) Identify the factors of 30 that are consecutive integers:

6(6-1) = 6(5) = 30

x = 6

2) Find the value of x(x+1) for x = 6

6(6+1) = 6(7) = 42


Making Connections Example 2

Question: If x and y are positive numbers and , then what is the value of ?

What essential information is needed? Need to find a connection between the equation and the expression.

What is the strategy for identifying essential information? Solve directly for and substitute the result into the expression

Solution Steps

1) Solve directly for

2) Substitute result into expression

y

x 99 yx

9x

9x

9 yx

09 yx

yx 9

19

y

y

y

x


Parallel Lines and Transversals

Strategy: If uncertain of parallel line properties, use the diagram appearance to determine the relationship between pairs of angles. Note: This strategy is valid if and only if the figure is drawn to scale.

Reasoning: Any pair of angles will either be congruent (equal measure) or supplementary (sum to 180 degrees). Using the figure given in a question, it is usually obvious when angles are congruent. If they do not appear congruent, they are supplementary.

Application: Many questions contain parallel lines with two transversals (see example 2).

In the figure shown above, pairs of red or pairs of blue angles are congruent. A pair consisting of a red and blue angle are supplementary.

Parallel Lines

Back to Frequent

Questions


Parallel Lines and Transversals Example 1

Question: In the figure to the right, if m is parallel to n, what is the value of x ?

What essential information is needed? Determine the measures of the two remaining angles inside the triangle that contains angle x.

What is the strategy for identifying essential information?: Use the properties of parallel lines and transversals to determine the measures of the two angles.

Solution Steps

1) The two remaining angles inside the triangle are 50o (congruent to the 50o angle) and 65o (supplementary to the 115o angle). Click again to see animation of the angles.

2) Calculate the measure of angle x:x = 180 - (50 + 65)

x = 65o

50o 65o

115o

115o

50o

xo

n

m

q

p


Parallel Lines and Transversals Example 2

Question: In the figure to the right, if m is parallel to n, what is the value of x + y ?

What essential information is needed? Need to define the two remaining angles inside the triangle in terms of x and y.

What is the strategy for identifying essential information? Use the properties of parallel lines and transversals to define the measures of the two angles in terms of x and y.

Solution Steps

1) The two remaining angles inside the triangle are 180 - x (supplementary to angle x) and 180 - y (supplementary to angle y). Click again to see animation of the angles.

2) Calculate the measure of angle x:(180 - x) + (180 - y) + 55 = 180

x + y = 235o

yo xo

55o

180 - yo 180 - xo

180 - xo 180 - yo

m

n


Even/Odd Integers

Strategy: Use the table of properties to the right to determine if an operation between two integers will result in an even or odd integer.

Reasoning: These integer formation properties eliminate the need to use the “plug in a number” strategy that is often more time consuming than applying the integer properties.

Application: There is always at least one question that can be easily solved using these integer formation properties.

Addition or Subtraction

Multiplication

odd + odd = evenodd - odd = even

odd x odd = odd

even + even = eveneven - even = even

even x even = even

odd + even = oddodd - even = odd

odd x even = even

Back to Frequent

Questions


Even/Odd Integers Example 1

Question: If a + b is an even integer, which of the following must be even?a) 2a + b b) 2a - bc) ab d) (a + 1)(b + 1) e) a2 - b2



Solution Steps


Even/Odd Integers Example 2

Question: If 2a + b is an odd integer, which of the following must be true?I. a is oddII. b is oddIII. 2a2 - b2 is odd



Solution Steps


Consecutive Integers

Strategy: Express the sum of three consecutive integers, consecutive odd integers, or consecutive even integers as the sum of the expressions shown to the right.

Reasoning: When you count by one’s from any number in the set of integers, consecutive integers are obtained. If you count by two’s beginning with any even/odd integer, consecutive even/odd integers are obtained.

Application: Questions that ask for the smallest of three consecutive integers or consecutive odd/even integers when their sum is a specified value. Any question that begins with the phrase “Given three consecutive integers”.

Consecutive Integersn, n + 1, n + 2

Where n is any integer

Consecutive Odd Integersn, n + 2, n + 4

Where n is an odd integer

Consecutive Even Integersn, n + 2, n + 4

Where n is an even integer


Consecutive Integers Example 1

Question: The average of a set of 5 consecutive even integers is 20. What is the smallest of these 5 integers?

What essential information is needed? Find the sum of the 5 consecutive even integers. Use the sum to find the smallest integer.

What is the strategy for identifying essential information?: Use the definition of average to find the sum. Use the sum and the consecutive integer strategy to find the smallest integer.

Solution Steps

1) Find the sum of the 5 integers using the definition of average

2) Find the smallest integer using consecutive even integer strategy

205

valuesofsum

valuesofnumber

valuesofsumaverage

100 valuesofsum

)8()6()4()2( nnnnnvaluesofsum

100205 nvaluesofsum

16n


Consecutive Integers Example 2

Question: What is the median of 7 consecutive integers if their sum is 42?

What essential information is needed? The fourth value in a list of seven consecutive integers.

What is the strategy for identifying essential information? Use the consecutive integer strategy to find the smallest integer. Add three to the smallest integer to find the value of the fourth integer. This will be the median value.

Solution Steps

1) Find the smallest integer in a list of seven integers.

2) Find the median value by adding three to the smallest integer.

42)6()5()4()3()2()1( nnnnnnn

42217 n

3n

3 nvaluemedian

6 valuemedian


Tangent To A Circle

Strategy: If a line is drawn tangent to a circle, draw the radius of the circle to the point of tangency with the line.

(Click again to draw radius)

Reasoning: A tangent line and the radius always form a right angle at the point of tangency. The right angle relationship will be used in all applications involving tangent lines to circles.

Application: Find the slope of the tangent line when given the coordinates of the point of tangency with the circle and the center of the circle. Find the perimeter of a shape when a circle is inscribed inside the given shape.

Tangent line

Back to Frequent

Questions


Tangent To A Circle Example 1

Question: In the figure to the right, a circle is centered at the origin and is tangent to the line at point P. If the radius of the circle is 15, what is the slope of line?

What essential information is needed? The radius and line are perpendicular to each other. Find the radius slope and use the relationship that the slope of perpendicular lines are opposite reciprocals of each other.

What is the strategy for identifying essential information?:Use the radius length and the x-coordinate of point P to find b, the y-coordinate of point P. This is accomplished using Pythagorean Theorem.

Solution Steps

P(9, b)

1) Using Pythagorean Theorem, the y-coordinate, b, has a value of -12. The slope of the radius is:

9

1512

2) Find the slope of line using the relationship between the slopes of perpendicular lines. Slope of line is

3

4

9

12

09

012

4

3

341

P(9, -12)


Tangent To A Circle Example 2

Question: In the figure to the right, a circle is tangent to the side of equilateral triangle xyz and the radius equals 5. What is the perimeter of triangle xyz ?

What essential information is needed? The length of a side of the triangle.

What is the strategy for identifying essential information? The circle radius and the equilateral triangle side are perpendicular at the tangent point. Draw a right triangle and use the properties of the 30-60-90 triangle to find the side length.Click again to show the right triangle

Solution Steps

3) The perimeter is three times the triangle side length:

x

y

z

30

60

35

5

Radius

1) Using properties of the 30-60-90 triangle, the length of half the triangle side is . 2) The triangle side length is .

35

310352

3303103


Strategy Section Concluded

Return to Table of Contents Return to first strategy Return to Introduction

This is the end of the Strategy section. Please select one of the options at the bottom of this page







Sample Strategy

Strategy: The 3rd side of any triangle is greater than the difference and smaller than the sum of the other two sides


Reasoning: A side length of 15 would require the formation of a line, not a triangle

Application: A side length of 3 would also require the formation of a line, not a triangle

Sample Strategy

1) The total cost of 4 equally priced notebooks is $5.00. If the price is increased by $0.75, how much will 6 of these notebooks cost at the new rate?

(A) $7.50(B) $8.00(C) $10.00(D) $12.00(E) $14.00




Sample Strategy

2) If Jim traveled 20 miles in 2 hours and Sue traveled twice as far in twice the time, what was Sue’s average speed, in miles per hour?

(A) 5(B) 10(C) 20(D) 30(E) 40




Sample Strategy

3) In the figure below, if CD is a line, what is the value x ?

(A) 45(B) 60(C) 90(D) 100(E) 120




C Dx0

x0 x0x0

x0x0

y0

Note: Figure not drawn to scale.

Sample Strategy

4) For which of the following functions is f(-2) > f(2) ?

(A) 3x2

(B) 3(C) 3/x2

(D) x2 + 2(E) 3 - x3




Sample Strategy

5) The energy required to stretch a spring beyond its natural length is proportional to the square of how far the spring is being stretched. If an energy of 20 joules stretches a spring 4 centimeters beyond its natural length, what energy, in joules, is needed to stretch this spring 8 centimeters beyond its natural length?

(A) 10(B) 40(C) 80(D) 100(E) 120


What essential information is needed?What is the strategy for identifying essential information?

Sample Strategy

6) The average (arithmetic mean) of x and y is 10 and the average of x, y, and z is 12. What is the value of z ?

(A) 2(B) 4(C) 12(D) 16(E) 26




Sample Strategy

7) If Z is the midpoint of XY and M is the midpoint of XZ, what is the length of ZY if the length of MZ is 2 ?

(A) 2(B) 4(C) 6(D) 8(E) More information is needed to answer question




Sample Strategy

8) In the figure below, line L is parallel to line m. What is the value of x ?

(A) 110(B) 120(C) 130(D) 140(E) 150




x0

600

1100

M

L

Sample Strategy

9) If a and b are odd integers, which of the following must also be an odd integer?

(A) I only(B) II only(C) III only(D) I and II(E) II and III




I. (a + b)bII. (a + b) +bIII. ab +b

Sample Factoring Strategy Example 1

Question:




Solution Steps

Sample Factoring Strategy Example 2

Question:




Solution Steps

The Ultimate SAT Math Strategies Guide Created by Sherman Snyder Fox Chapel Tutoring Pittsburgh, PA 412-352-6596 Go to Success Model.

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