MAT 083 Elementary Algebra-Part I Page 1 MAT 093 Intensive … · 2017. 10. 2. · EXAMPLES: Translating Verbal Phrases into Algebraic Expressions Translate each of the following

MAT 083 – Elementary Algebra-Part I Page 1 MAT 093 – Intensive Elementary Algebra Section 2.4 and a portion of Section 1.1 (Bittinger CA7)

[email protected] kradermath.jimdo.com 09/2017

Topics to be Covered and Student Learning Outcomes (SLOs) After we cover Section 2.4 and a portion of Section 1.1, you should be able to:

1. Convert between fractions, decimals and percent notation. [SLO 3] 2. Translate questions involving percents into an appropriate algebraic equation. [SLO 8] 3. Solve application problems involving percents. [SLO 8] 4. Translate English phrases into algebraic expressions. [SLO 7] 5. Translate English sentences into algebraic equations. [SLO 7] 6. Use equations to model information presented in a table. [SLO 7]

Equations as Mathematical Models A mathematical model is a mathematical representation of a “real-world” situation. Equations are frequently used as mathematical models. In fact, many application problems (i.e., word problems) can be solved by translating the English problem statement into an algebraic equation (i.e., a mathematical statement). EXAMPLE: Resort Fee Translate the following problem statement into an algebraic equation:

“The price of a hotel room plus the $12 resort fee is $152.”

To begin our discussion of modeling, we will look at applications involving percent. Multiple Representations of Percent “Percent” and the percent symbol “%” mean “per hundred” or “hundredths.”

1% 0.01

100 100

nn n n

Percents, fractions and decimals are three different ways to describe the same thing. You can convert from any one of the three forms to any of the others.

We do not use the “%” symbol in algebraic expressions or equations, just like we do not use English words in algebraic expressions or equations (think of the “%” symbol as an abbreviation for the English word “percent.”) Numbers using the “%” symbol are converted to decimals or fractions.



EXAMPLE: US Currency

If you take $1 and divide it into 100 cents, each cent represents 1100

or 1% of a dollar.

Percents may be written as fractions or decimals.

75 cents 25 cents

Percent

of $1

Fraction

of $1 dollar

Decimal Notation

EXAMPLE: Converting Percent Notation to Fractions and Decimals

Write as a fraction and a decimal

Percent Fraction Decimal

100%

Slide 7

Different ways to describe the same thingPercents may be written as fractions or decimals

75

100

75%

0.75

3• 25

4• 25

3

4

25 1

100 4

25%

0.25

$1100%

1

1%100

MAT 083/093 (Bittinger CA6) - 2.4 GHK 09/2013



Converting Between Percents and Decimals Remember: Per cent means “per hundred” or “hundredths.”

75

75% 0.75100

EXAMPLE: Converting Between Percents and Decimals

Convert each of the following percents to a decimal:

8%

7.25%

175%

400%

0.003%

34

%

Convert each of the following decimals to a percent:

0.237

0.0525

1.45

6

0.0004

Converting percents to decimals: Move decimal point 2 places to

the left (i.e., divide by 100). Drop the % symbol.

Converting decimals to percents: Move decimal point 2 places to

the right (i.e., multiply by 100). Add the % symbol.



Converting Fractions to Percents METHOD 1:

Convert the fraction to a decimal using division: numerator denominator (use a calculator if necessary).

Convert the decimal to a percent by moving the decimal point two places to the right and adding the % symbol.

Round (don’t truncate) your answer if appropriate. METHOD 2:

If the denominator is a factor of 100 (i.e., the denominator divides evenly into 100), then build up the fraction until its denominator is 100.

The numerator is the percent. EXAMPLE: Converting Fractions to Percents

Smaller Piece Larger Piece

Represent the piece as a fraction of the pie.

Represent the piece as a decimal.

Represent the piece as a percent of the whole pie.

Slide 51

Different ways to describe the same thingPercents may be written as fractions or decimals

MAT 083/093 (Bittinger CA6) - 2.4

Source: www.cakedutchess.com

GHK 09/2013



EXAMPLE: Converting Fractions to Percents

Convert each of the following numbers involving a fraction to a percent. Use a calculator if necessary.

3

10

1

6

8

5

11

4

Applications Involving Percents

Use a variable to represent the unknown.

― “What” often indicates the unknown.

Translate the verbal statement into an equation.

― Convert percents to decimals. (The percent symbol “%” is an abbreviation for an English word and is not used in algebraic expressions or equations.)

― Look for keywords that help us translate.

“Of” translates into multiplication.

“Is,” “Was,” “Equals”, etc., translate to “=”.

― The percent always multiplies some quantity (number or variable).

Solve the equation.

Answer the question in a complete English sentence, using the proper units where applicable (i.e., dollars, meters, hours, etc.)



EXAMPLES: Applications Involving Percents

1. What is 13% of 72?

2. If the sales tax rate is 10.25%, how much sales tax is charged when you buy a $150 camera? NOTE: When your answer involves money, round – don’t truncate – to the nearest cent unless told otherwise.

3. 9 is 12% of what number?

4. Dr. Goode paid $9200 in state income tax, which is 5% of taxable income. What was Dr. Goode’s taxable income?



EXAMPLES: Applications Involving Percents (cont’d)

5. What percent of 60 is 27?

6. Marla took an algebra test and got 32 of the 40 problems correct. What percentage of the problems were correct? What percentage of the problems were wrong?

Helpful Hints:

1. Using equations to model application problems is one of the most important topics in an algebra course. If your solutions do not involve variables or equations, you are missing one of the most important learning objectives of this course and you are likely to find it very difficult to solve more advanced application problems in subsequent courses.

2. When solving an application problem, ask yourself if your answer seems reasonable. For

example, if you are asked to find the price of a new car, is $1730 a reasonable answer? If your answer does not seem reasonable, it is probably not correct.



EXAMPLES: Applications Involving Percents (cont’d)

7. Eric got a promotion which increased his pay from $24.00 to $26.50 per hour. By what percentage did his pay increase? (Round your answer to the nearest tenth of a percent.)

8. After giving Eric a raise, the company experiences financial difficulties and needs to reduce Eric’s salary by the same percentage. Will his salary go back to $24.00? Justify your answer.



Introduction to Application Problems As we already discussed, many application problems (i.e., word problems) can be solved by translating the English problem statement into an algebraic equation (i.e., a mathematical statement). Recall that equations consist of two algebraic expressions separated by an equal sign. The equation says that the two expressions are equal, e.g.:

5 2 3 2x x

In order to translate an English problem statement into the correct equation we must be able to write the correct expressions on either side of the equal sign. Recall that algebraic expressions consist of variables and/or real numbers connected by arithmetic operations and grouping symbols. We will learn how to recognize key words in the problem statement that tell us which arithmetic operations are used in the expressions on either side of the equal sign. EXAMPLE: Translating Problem Statements into Algebraic Equations Translate the following statement into an equation:

“The sum of a number and three is 15.” Translating Verbal Phrases into Algebraic Expressions: Some words are normally associated with specific arithmetic operations.

Addition Subtraction Multiplication Division Plus Added to Increased by More than Sum of

Minus Take away Subtracted from Decreased by Less than Difference of

Times Twice Multiplied by Of (when used with a fraction or percent) Product of

Divided by Divided into Quotient of Ratio to

One-half (one-third, one-fourth, etc.) Pay close attention to word order in subtraction and division, because these operations are not commutative!



EXAMPLES: Translating Verbal Phrases into Algebraic Expressions Translate each of the following verbal expressions into an algebraic expression. Words normally associated with addition:

A number plus 3

3 added to a number

A number increased by 3

3 more than (greater than, larger than) a number

The sum of a number and 3

Words normally associated with subtraction:

A number minus 3 3 minus a number

Take away 3 from a number Take away a number from 3

3 subtracted from a number A number subtracted from 3

A number decreased (or reduced) by 3

3 decreased (or reduced) by a number

The difference of a number and 3

The difference of 3 and a number

3 less than a number

Words normally associated with multiplication:

3 times a number

Twice a number (i.e., 2 times a number)

A number multiplied by 3

Two-thirds of a number

45% of a number

The product of 3 and a number



EXAMPLES: Translating Verbal Phrases into Algebraic Expressions (cont’d) Words normally associated with division:

A number divided by 3 3 divided by a number

3 divided into a number A number divided into 3

The quotient of a number and 3

The quotient of 3 and a number

The ratio of a number to 3 The ratio of 3 to a number

One-half of a number



EXAMPLES: Translating Verbal Phrases into Algebraic Expressions Translate each of the following verbal phrases into an algebraic expression. Try to do so without looking at the examples on the previous two pages.

1. A number increased by one ____________

2. Two less than a number ____________

3. The quotient of three and a number ____________

4. A number divided by four ____________

5. The sum of a number and five ____________

6. The difference between six and a number ____________

7. The product of a number and seven ____________

8. One-eighth of a number ____________

9. Eleven miles per hour faster than the speed limit ____________

10. Nine years younger than Renee ____________

11. Ten percent of your income ____________

12. Twelve less than the product of a number and three ____________

13. Twice the sum of a number and thirteen ____________ Match each verbal phrase on the left with an algebraic expression on the right. Some of the expressions on the right may be chosen more than once and some may never be chosen. Let x represent the unknown number.

14. ______ 7 more than 4 times a number. 15. ______ The product of 4 more than a number and 7. 16. ______ 4 more than one-seventh of the number 17. ______ 4 times the sum of a number and 7. 18. ______ 4 plus the product of the number and 7. 19. ______ The difference of 4 times the number and 7. 20. ______ The sum of 4 times a number and 7

(a) 4 7x

(b) 7 4x

(c) 47

x

(d) 4

7

x

(e) 4 7x

(f) 4 7x

(g) 7 4x

(h) 4 7x



Using Algebraic Equations to Model Data Presented in a Table When English sentences are used to describe problems, we can often translate the English sentence to an algebraic equation and solve the problem. Sometimes tables with data provide the information needed to describe a problem. Often the data in the table follow a pattern. If we can find the pattern, we can describe it in an English sentence and translate the English sentence into an equation. EXAMPLE: Movie Tickets

Number of Movie Tickets (n)

Cost in dollars (c)

1 $9.00 2 $18.00 3 $27.00 4 $36.00 5 $45.00

Describe the relation between c and n inEnglish:

Write an equation to find c, given n.

c = EXAMPLE: Classroom Handouts

Number of Students in Class (s)

Number of Handouts Printed

(h) 10 13 15 18 20 23 25 28 30 33

Describe the relation between h and s inEnglish:

Write an equation to find h, given s.

h =

MAT 083 Elementary Algebra-Part I Page 1 MAT 093 Intensive … · 2017. 10. 2. · EXAMPLES: Translating Verbal Phrases into Algebraic Expressions Translate each of the following

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