MATH+V

LICENSURE EXAMINATION FOR TEACHERS (LET)Refresher Course

Content Area: MATHEMATICSFocus: ARITHMETIC, NUMBER THEORY AND BUSINESS MATHPrepared by: Daisy de Borja-Marcelino

Competencies:

1. Simplifying expressions involving series of operations 2. Solve problems involving

a. GFC and LCMFb. prime and compositec. divisibilityd. inverse and partitive proportions

e. compound interest f. congruenceg. linear Diophantine Equation

3. Apply Euler’s function and theorems, or Fermat’s theorem in solving problem.

THE NUMBER SYSTEM

Example: Some examples of imaginary numbers are: , 3i, -7i.

Example: Simplify: 2 (3 + 2i) – 5 (4 – 6i)

Solution: 2 (3 + 2i) – 5 (4 – 6i) = ( 6 + 4i) – (20 – 30i) = 6 + 4i – 20 + 30i = -16 + 34i.

Rational numbers are numbers which can be expressed as quotient of two integers, or can be expressed as

fractions in simplest forms. Examples are 8, -3, 3.45, and .

Irrational numbers are numbers which cannot be expressed as fractions in simplest forms. Examples are , 4

, , e and .

Set of Natural/Counting numbers: {1, 2, 3, 4, … }. This set contains the numbers that we use in counting; also called natural numbers. Set of Whole Numbers: { 0 , 1, 2, 3, …}. This set is the union of the number zero and the set of counting numbers.

St. Louis Review Center, Inc-Davao Tel. no. (082) 224-2515 or 222-87332 1

Set of Integers: { … , -3, -2, -1, 0, 1, 2, 3, …}. This set is the union of the set of counting numbers, their negatives, and zero.

II. THE COUNTING NUMBERS

A. Divisibility. An integer is divisible by a certain divisor (also an integer) if it can be divided exactly by that divisor. That is, the remainder is zero after the division process is completed.To illustrate, the integer 12 is divisible by 1, 2, 3, 4, 6, and 12.

To determine whether the integer is divisible by a certain integer or not, you may use the following divisibility rules.

An integer is divisible by a) 2 if it ends with 0, 2, 4, 6, or 8. (Examples: 134 or 12 or 12,330 or 4)b) 3 if the sum of the digits is divisible by 3. (Examples: 132 or 18 or 12,330 or 45)c) 4 if the last two digits form a number which is divisible by 4. (Examples: 13,412 or 12,332)d) 5 if it ends with 0 or 5. (Examples: 135 or 10 or 12,330 or 495)e) 6 if it ends with 0, 2, 4, 6, 8 and the sum of the digits is divisible by 3.(Examples: 134 or 12)f) 7 if the difference obtained after subtracting twice the last digit from the number formed by the

remaining digits is divisible by 7. (Examples: 14 or 364)g) 8 if the last three digits form a number which is divisible by 8. (Examples: 24160 or 5328)h) 9 if the sum of the digits is divisible by 9. (Examples: 9, 432 or 18,504 or 270)i) 10 if it ends with 0. (Examples: 120 or 7, 890 or 1, 230)j) 11 if the difference between the sum of the digits on the even powers of 10 and the sum of the digits on

the odd powers of 10 is divisible by 11. (Examples: 2123 or 2816 or 94369 or 36465) k) 12 if it is both divisible by 3 and 4. (Examples: 413,412 or 112,332)l) 15 if it is both divisible by 3 and 5. (Examples: 150 or 350)

Remarks: Divisibility rules for two or more relatively prime numbers (GCF is 1) may be combined to serve as a divisibility rule for their product.

Example: The rules for 3, 4, and 5 may be combined to serve as the rule for their product which is 60 since 3, 4, and 5 are relatively prime.

Exercises: Put a check mark on the space provided for, if the integer on the first column divides exactly the integer on the top row.

456 36,720 800,112 456 36,720 800,1122 103 114 125 146 247 328 459 77

Even numbers are whole numbers which can be divided exactly by two whole numbers.

Odd numbers are whole numbers which cannot be divided exactly by two whole numbers.

Example: If n3 is odd, which of the following is true?I. n is odd II. n2 is odd III. n2 + 1 is odd

A) II only C) I onlyB) I and II only D) I and III only

Example: If x is an odd integer and y is an even integer, which of the following is an odd integer?

A. 2x-y C.

B. D.

B. Factors and Multiples. In the number sentence 2 x 3 = 6, the numbers 2 and 3 are called factors, while 6 is their product. Or we say, 2 and 3 are divisors of 6. Moreover, we say that 6 is a multiple of 2 and 3.


Example: How many factors does 42 have?A) 2 B) 4 c) 5 D) 16

Answer: (C). The factors of 42= 16 are {1, 2, 4, 8, 16}.Example: What are the multiples of 6?Answer: The multiples of 12 are {12, 24, 36, 48, …}

Exercises Fill in the blanks with either 7 or 42.a. ______ is a factor of ______.b. ______ is divisible by ______.c. ______ is a divisor of ______.d. ______ is a multiple of ______.e. ______ divides _______.

C. Prime and Composite Numbers

Prime numbers are counting numbers that have exactly two factors in the set of counting numbers: 1 and itself.

Composite numbers are counting numbers that have more than two factors in the set of counting numbers.

The numbers 0 and 1 are special numbers. They are neither prime nor composite.

Example: What is the sum of prime numbers less than 15?A) 4 B) 5 C) 6 D) 14

Answer: The number 2,3,5,7, 11 and 13 are prime number less than 15. Hence, the answer is C.

D. Prime Factorization. This is a process of expressing a number as product of prime factors.

Example: Express 24 as product of prime factors.Solution: 24= 2 x 2 x 2 x 3 = 23 x 3 or 3 x 23 .

Fundamental Theorem of Arithmetic Every composite whole numbers can be expressed as the product of primes in exactly one way (the

order of the factors is disregarded).

E. The Greatest Common Factor (GCF)

The GCF of two or more numbers is the largest possible divisor of the given numbers.

Example: Determine the GCF of 12 and 42.Solution: 24 = 2 x 2 x 3

42 = 2 x 3 x 7 GCF: 2 x 3 = 6

Example: What is the greatest integer that can divide the numbers 18, 24 and 36?Solution: 18 = 3 x 3 x 2

24 = 3 x 2 x 2 x 236 = 3 x 3 x 2 x 2

GCF: 3 x 2 = 6

F. Least Common Multiple (LCM). The LCM of two or more numbers is the smallest possible number that can be divided by the given numbers.Example: Give the LCM of 20 and 30.Solution: 20 = 2 x 2 x 5 = 22 x 5

30 = 2 x 3 x 5 LCM: 22 x 3 x 5 = 60.

Example: What is the smallest integer that can be divided by the numbers 24, 36 and 54?Solution: 24 = 2 x 2 x 2 x 3 = 23 x 3

36 = 2 x 2 x 3 x 3 = 22 x 32

54 = 2 x 3 x 3 x 3 = 2 x 33 LCM: 23 x 33 = 216

G. Relatively Prime. Two numbers are relatively prime if their GCF is 1. The numbers themselves may not be prime. The numbers 12 and 49 are relatively prime.

Example: Which of the following pairs are relatively prime to each other?St. Louis Review Center, Inc-Davao Tel. no. (082) 224-2515 or 222-87332 3

A)15 and 36 B) 23 and 51 C) 231 and 27 D) 121 and 330

III. INTEGERS

Consecutive integers are two or more integers, written in sequence, in which each integer after the first is 1 more than the preceding integer.

Examples: 1,2,3,4,5, 63, 4, 5, 6, 7, 8–4, –3, –2, –1, 0, 1, 2, 3x, x+1, x+2, x+3, x+4, x+5

The absolute value of a number x, denoted by x , is the undirected distance between x and 0 on the number line. –5 –4 –3 –2 –1 0 1 2 3 4 5

It is also defined as

x if x ≥ 0 x = – x if x < 0

Examples:Evaluate each of the following.a) │2│ = 2 c) │0│ = 0

b) │– 7│ = 7 d) – │–15│ = –15

A. Multiplication. The product of two integers with like signs is a positive while the product of two integers with unlike signs is negative.

Example: (-4) x 7 = (-28) or (-4) (7) = (-28) or (-4) 7 = (-28)Example: (-8) x (-5) = 40 or (-8) (-5) = 40 or (-8) (-5) = 40

B. Division. The quotient of two integers with like signs is a positive while the quotient of two integers with unlike signs is negative.

Example: (-72) (-8) = 9

Example: (-123) 3 = - 41

C. Addition. The sum of two integers with like signs is the sum of their absolute values with the common sign prefixed before it.

The sum of two integers with unlike signs is the difference of their absolute values with the sign of the integer with the larger absolute value prefixed before the difference.

Example: (-3) + (-23) = (-26)

Example: (-34) + 12 = (-22)

D. Subtraction. Express subtraction statements as addition statements and follow the procedure in addition. (That is, change the sign of the subtrahend to its opposite, and proceed to addition.)Example: (-12) – (-3) = (-12) + 3 = -9

Exercise: What number should a) be added to (-12) to yield 26?b) be subtracted from (-2) to yield 5?c) be multiplied by (-4) to yield (-36)?d) be divided by (-2) to yield 30?

E. P-E-MDAS. P-E-MDAS stands for “Parenthesis-Exponent-Multiplication Division Addition Subtraction.

When two or more operations are involved in a single expression, operations are performed in the order of P-E-MDAS. That is, we perform first the operation inside the parenthesis (or any grouping symbol), then


followed by determining the power of the number which is raised to a given exponent, then followed by multiplication/division, and lastly by the addition/subtraction.

Should there be multiplication and division only, perform the operation from left to right.Should there be addition and subtraction only, perform the operation from left to right.

Example: Simplify 20 + 100 ( 5 – 63 32 + 12)

Solution: 20 + 100 ( 5 – 63 32 + 12) = 20 + 100 ( 5 – 63 9 + 12) = 20 + 100 ( 5 – 7+ 12) = 20 + 100 ( (–2) + 12) = 20 + 100 10 = 20 + 10 = 30.

1. Two bells ring at 5 P.M. For the rest of the day, one bell rings every half hour whereas the other rings every 45 minutes. When is the first time, on that same day, that both bells ring at the same time again?

a. 6:30 P.M. b. 8:30 P.M. c. 8:45 P.M. d. 9:00 P.M.

2. Which is true?a. The set of prime factors of 6 is {1,2,3} c. All prime numbers are odd numbers.b. The product of irrational and rational is irrational. d. 3.14 is a rational number.

3. Which of the two-digit numbers below when inserted in the blank will make 38__09 divisible by 3? a. 98 b. 84 c. 34 d. 60

4. Which of the following number is divisible by 45?a. 300,000,000,450 b.600,000,000,045 c. 100,200,600,090 d. 400,450,000,000

5. On its anniversary, a certain store offers a free sandwich for every 4th customer and a free softdrink for every 6th customer. After 75 customers, how many had received both free sandwich and softdrink?

a. 30 b. 18 c. 12 d. 6

IV. FRACTIONS

Kinds of Fractions

As to relation between the numerator and the denominatora. Proper – the numerator is less than the denominatorb. Improper – the numerator is equal to or greater than the denominator

As to relation of the denominators of two or more fractions

a. Similar – the denominators are equal. Examples:

b. Dissimilar – the denominators are not equal. Examples:

Other classes

a. Equivalent – fractions having the same value. Examples: .

b. Mixed – composed of a whole number and a proper fraction . Examples:

Rules involving Zeroa. Zero numerator and non-zero denominator – the value is zerob. Zero denominator – no value, undefinedc. Zero value – the numerator is zero

A. Multiplication of Fractions. Multiply numerator by numerator and denominator by denominator to get the numerator and denominator respectively of the product

Example: or .


B. Division of Fractions. Multiply the supposed dividend by the reciprocal of the supposed divisor.

Example: = or .

Exercises: Evaluate the following. a) b)

c) d)

D. Changing Dissimilar Fractions to Similar Fractions. Determine the LCM of the denominators. Then with the said LCM as the denominator, express each fraction to its equivalent.

Example: Express , , to similar fractions.

Solution: = , = and =

Therefore, the similar fractions are and

E. Addition of fractions. Convert the fractions to similar fractions. Then add the numerators to obtain the numerator of the sum and copy the denominator.

Example: Evaluate + + .

Solution: The LCM is 12, so convert the addends to similar fractions with 24 as the denominator.

+ + = + or

F. Subtraction of Fractions. Convert the fractions to similar fractions. Then subtract the numerators to obtain the numerator of the difference and copy the denominator.

Example: What number should be subtracted from to obtain ?

Solution: Let the desired number be x. Then, the equation is given by

- x = x = - = = .

G. Fraction as Part of a Whole

Example: What is of 28?

Solution: Let the desired number be m. Then, the equation is given by

M = .

Example: What part of 24 is 4?Solution: Let the desired number be q. Then, the equation is given by

q 24 = 4

q = .

H. Simplifying Fractions

A fraction is in simplest form if the numerator and the denominator are relatively prime (their GCF is 1). Thus, to simplify fractions, multiply by the fraction whose numerator and denominator are the reciprocal of the GCF of the numerator and the denominator of the given fraction.

Example: The simplest form of is because .


I. Ordering Fractions

Two fractions are equivalent if their cross products are equal. Otherwise, that fraction the numerator of which was used to get the greater of the two cross products is the larger fraction.

Exercises

1. A 100-m wire is cut into two parts so that one part is ¼ of the other. How long is the shorter piece of wire?a. 120m b. 80m c. 25m d. 20m

2. Luis left ½ pan of a cake on the table. Dada ate ¾ of it. What fraction of cake was left? a. 1/8 b. 3/8 c. ¼ d. ½

3. If and are equivalent fractions, what is the value of n?

a. 13 b. 14 c. 20 d. 21

4. Mr. dela Cruz owned of a business. He sold of his share in the business at a cost of P1M. What is the

total cost of the business? a. P 6M b. P7M c. P 8M d. P 9M

5. Arrange the fractions 5/8, 4/5, 3/4 in increasing order.a. 5/8, 4/5, 3/4 c. 3/4, 4/5, 5/8b. 4/5, 3/4, 5/8 d. 4/5, 5/8, 3/4

14. Which of these fractions has the largest value?a. 3/5 b. 11/16 c. 7/10 d. 5/8

15. Mark spent his monthly salary as follows: 3/5 for food and allowances, 1/3 for his child’s education and house rental. If his monthly salary is P15, 000, how much would he left at the end of the month?

a. P 1,000 b. P2,000 c. P5,000 d. P 14,000

16. Chedy and Dada run for President for their organization. Chedy got 1/3 of the votes. If Dada got 300 votes, how many students voted for Chedy?

a. 900 b. 200 c. 150 d. 100

V. DECIMAL NUMBERS

A. The Decimal Numbers and the place value chart

The place value chart

100 000 10 000 1 000 100 10 1 0.1 0.01 0.001 0.0001 0.00001

The number 0.8 is read as “eight tenths” and .214 as “two hundred fourteen thousandths”.The number 0.8 is equal to .800.The number 0.8 is greater than 0.214. Exercise: a) Arrange the following decimal numbers in ascending order:

0.5, 0.343, 0.142, 0.5254

b) In 2.3456, what digit is in the thousandths place?

B. Addition and Subtraction of Decimal Numbers. Addition of decimals is done by writing them in a column so that their decimal points are aligned. Thus aligned, digits with the same place values would be in the same column, and the addends (or the minuend and the subtrahend) are added (or subtracted) as if they were whole numbers.


Hu

ndre

d T

hous

and

Ten

Th

ousa

nd

Tho

usa

nds

Hu

ndre

ds

Tens

One

s

Tent

hs

Tho

usa

ndth

s

Hu

ndre

dth

s

Ten

Th

ousa

ndt

hs

Hu

ndre

d T

hous

and

s

C. Multiplication of Decimal Numbers. To multiply decimals, multiply the numbers as if they were whole and so place the decimal point in the result as to have as many decimal places in it as there are in the factors combined.

D. Division of Decimal Numbers.

To divide ai. decimal by a whole number, do as in dividing whole numbers but writing the decimal point directly

above that of the dividend.ii. number by a decimal, multiply both dividend and divisor by that power of ten such that the divisor

becomes the least whole number, and then proceed as in (i) above.

VI. CONVERSION

A. Fraction to Decimal. Divide the numerator by the denominator.

Exercises: Convert the following to decimal:a) 3/5 = ______b) 5/6 = ______c) 7/8 = ______

B. Decimal to fraction

a) Terminating – multiply the number by a fraction (equal to one) whose numerator and denominator is a multiple of 10 such that the numerator of the product is a whole number.

Example: Convert 0.15 to fraction.

Solution: 0.15 =

b) Repeating decimal number

Example: Convert to fractionSolution: Let n = =

10 n = - n = -----------------------------------

9 n = 5

n =

Hence, is equal to .

Exercises

1. Jeepney fares are computed as follows: P7.50 for the four kilometers plus P0.50 for every additional kilometer thereof. How much should Au pay for a ride that covers 10 kilometers?

a. P8.00 b. P9.50 c. P10.00 d. P10.50

2. Which of the following is 0.3 of ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠?a. ♠ ♠ ♠♠ ♠ ♠ b. ♠ ♠ ♠ c. ♠ ♠ d. ♠

3. Which of the following is between 3 and 4?

a. b. -3.5 c. d.

4. Evaluate 14.8 + 3.95 + .003.a. 5433 b. 753 c. 446 d. 18.753

5. Carmen bought 4 kilograms of rice at P31.45 per kilo and 6 kilograms of salt at P22.35 per kilo. If she gave a P1000 bill to the cashier, how much change did she get?

a. P8.00 b. P9.50 c. P120.10 d. P740.10


6. Each capsule of a certain commercial vitamins contains 0.6 mg of calcium. In how many pieces of capsules can 22.2 mg of calcium be distributed?

a. 8 b. 37 c. 50 d. 105

7. Which of the following is equal to 2.4545454545… ?

a. b. 2 c. d. 2

8. The expression + + is equal to _____________.

a. 0.0051 b. 0.006 c. 0.51 d. 0.051

9. Which of these numbers is greater than ¼?a. .04 b. (1/2)2 c. 1/8 d. 1/0.04

VII. PERCENT

Per Cent – literally meaning “per hundred”, it is one way of writing fractions in which the denominator which is required to be 100 is written as “%”, and read as “per cent”.

Since 1 = 100% hence = 75%

A. CONVERSION

Percent to Decimal Number. Divide the number by 100%. Note that 100% = 1.

Example: Convert the following to decimal:a) 35% c) 8.37%

b) 6 % d) %

Solution: a) (35%) 100% = 0.35b) (6 ¼ %) 100% (6.25%) 100% = .0625c) (8.37) 100% = .0837d) (1/4%) 100% (0.25%) 100% = .0025

Decimal Numbers to Percent. Multiply the decimal number by 100%. Note that 100% = 1.

Example: Convert the following to percent.a) 0.25 b) 0.143 c) 2.03 d) .005

Solution: a) 0.25 x 100% = 25% b) 0.143 x 100% = 14.3%c) 2.03 x 100% = 203% d) .005 x 100% = 0.5%

Exercises: Fill in the blanks so that the entries in each row are equal.Fraction Decimal Percent

A 4/7B 160%C 0.95D 6/11E ½ %

B. Percentage. Percentage is a percent of a given number. The given number is called the base. The percent is called the rate.

Example: What is 50% of 140?Solution: N = 0.50 x 140 = 70.00 = 70


Formula: Percentage (P) = Base (B) x Rate (R)

and

Example: 36 is 10% of what number?

Solution: 36 = 0.10 N

Example: 22 is what percent of 88?Solution: 22 = P 88

Example: Mr. Ballaran receives a 10% increase in his salary. With the increase, he now receives P22, 000. How much is his monthly salary before the increase?Solution: present salary = previous salary + increase

22, 000 = previous salary + (10% of previous salary) 22,000= previous salary (1 + .10)

22 000 1.1 = previous salary 20,000 = previous salary

C. Discount

The discussion on DISCOUNT is very similar with the discussion on PERCENTAGE.Original Price/ Marked Price/ List price - as the BaseRate of Discount - as the RateDiscount - as the PercentageSelling Price - Original Price minus Discount

Example: A skirt with an original price of P250 is being sold at 40% discount. Find its selling price.

Solution: S.P. = Original price - Discount= 250 – (0.40 x 250) = 250 - 100 = 150

Example: An item has a selling price of P 210.00. If the selling price is 70% of the original price, what is its original price?

Solution: Selling price is 70% of the original price 210 = 0.70 O.P.

O.P. = 210 0.70 = 300.Therefore, the original price is P300.

Example: A shirt is being sold at P 199.95. If its original price is P 430, find the rate of discount.Solution: Discount = O.P. – S.P.

= 430 – 199.95 = 230.05

Rate of Discount =

D. Simple Interest Interest (I) is the amount paid for the use of money or the money earned for depositing the money. Principal (P) is the money that is borrowed or deposited. Time (t) is the number of days/months/years for which the money is being borrowed/deposited and

interest is calculated.

Example: Give the simple interest of P10,000 for three years at 5.5% per year.Solution: I = P r t

I = (10 000) (.055) (3)I = P1 100.

Example: Determine the amount of the principal if the interest at 10% per annum after 8 months is P3,600.

Solution: I = P r t = = 45,000

Compound Interest ( Final Amount = P[ 1 + r ]n )


I = Prt, P = t = r =

Compound interest is different from simple interest because after the first interest calculation, the interest is added to the principal, so interest is earned on previous interest in addition to the principal. Compound Interest rates may be given as annual (1 time a year), semiannual (2 times a year), quarterly (4 times a year), monthly (12 times a year), and daily (365 times a year). Example: If P500is invested at 8% compounded semiannually, what will the final amount be after three years? Final Amount = P[ 1 + r ]n = 500[ 1 + (8% / 2)]3 * 2 = 500[ 1 + 0.04 ]6

= 500[1.27]= 635

Exercises

1. John bought a jacket for Php 850.00. If he was given a discount of 15%, what was the original price?a. P8,500.00 b. P1,000.00 c. P900.00 d.P765.00

2. In a basket, there are 15 santol, 12 balimbing, and 3 durian. What percent of the fruits are durian?a. 10% b. 12.5% c. 12% d. 15%

3. A certain mobile phone model was sold for P4,000 in 2000. Two years later, the same mobile phone model sold for P2,800. What was the percent decrease of the price?

a. 15% b. 30% c. 20% d. 35%

4. If ♥♥♥♥ is 50% of a larger figure, which of the following is the larger figure?a. ♥ b. ♥♥ c. ♥♥♥♥ d. ♥♥♥♥♥♥♥♥

5. A senior class of 50 girls and 70 boys sponsored a dance. If 40% of the girls and 50 % of the boys attended the dance, approximately what percent attended?

a 44 b. 46 c. 42 d. 40

6. Which of the following is equal to ?

a. 2.5 b. 0.25 c. d. 0.025

7. Sarah’s earning P 9,200 a month will receive a 15% increase next month. How much will her new salary be?

a. P 10,500 b. P 10,530 c. P 10,580 d. P 10,560

8. How much is 37% of 80% of 24?a. 7.1 b. 1.92 c. 19.2 d. 71

9. According to the latest survey, 60% of the cancer patients were smokers. If there were 180 smoking cancer patients, how many cancer patients are there in all?

a. 90 b. 108 c. 240 d. 300

10. Which of the following is 70% of 50?a. 7 b. 17.5 c. 35 d. 71

11. Twenty four is 12% of what number?a. 40 b. 150 c. 200 d. 400

12. Thirty six is what percent of 90?a. 32.4% b. 40% c. 45% d. 76%

13. In a mathematics test of 40 items, Mavic got 90%. How many items did Mavic get?a. 7 b. 28 c. 36 d. 360

14. Mr. Mabini receives a 10% increase in his salary. With the increase, he now receives P13,200. How much is his monthly salary before the increase?

a. P12 000 b. P 13, 500 c. 14, 100 d. P14, 520

15. According to the latest survey, 60% of the cancer patients were smokers. If there were 180 smoking cancer patients, how many cancer patients are there in all?

a. 70 b. 150 c. 300 d. 360

VIII. RATIO AND PROPORTION

A ratio is a comparison of two or more quantities.

A proportion is a number sentence stating the equivalence of two ratios.


Note that in ratio, we are comparing quantities of the same units and that the ratio is expressed in terms of integers.Examples: a) The ratio of 12 days to 3 weeks is 12:21 or 4:7.

b) The ratio of 3 meters to 180 cm is 300:180 or 5:3.c) The ratio of 2 hours to 25 minutes is 120:25 or 24:5.d) The ratio of 1 ½ to 4 ½ is 1:3.

A. Direct Proportion. As one quantity increases, the other increases also.

Example: Find the value of x if 15:20 = 14 : x.Solution: Equate the product of the means and the product of the extremes. Then solve for x. Thus,

(15) (n) = (20) (14)

= or .

Example: A car travels at an average rate of 260 km in 5 hours. How far can it go in 8 hours, if traveling at the same rate?

Solution: 260 : 5 = x : 8 (5) x = (260) (8)

= 416.

Example: If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there?Solution: Let x be the number of teachers,

or 1 : 18 = x : 360

(18)x = (360)1 x = 360/18 = 20 teachers

B. Inverse Proportion. As one quantity increases, the other decreases.

Example: If the food is sufficient to feed 10 flood victims in 15 days, how many days would it last for 8 flood victims?

Solution: Equate the product of the terms in the first condition to the product of the terms in the second condition. Thus, we have:

(10 victims)(15 days) = (x) (8 victims)

= 18.75 days

C. Partitive Proportion. One quantity is being partitioned into different proportions.

Example: A wood 120 m long is cut in the ratio 2:3:5. Determine the measure of each part.

Solution: m

m

m

Example: A wire is cut into three equal parts. The resulting segments are then cut into 4, 6, and 8 equal parts respectively. If each of the resulting segments has an integer length, what is the minimum length of the wire?

A) 24 B) 48 C) 72 D) 96


Solution: Each third of the wire is cut into 4,6 and 8 parts respectively, and all the resulting segments have integer lengths. This means that each third of the wire has a length that is evenly divisible by 4, 6, and 8. The smallest positive integer that is divisible by 4, 6, and 8 is 24, so each third of the wire has a minimum length of 24. So, the minimum length of the whole wire is three times 24, or 72.

Exercises1. A 300 m ribbon is cut into four pieces in the ratio 1:2:3:4. Give the length of the shortest piece.2. If there are

18 boys and 45 girls in the gym, what is the ratio of the girls to the boys?a. 2:5 b. 2:3 c. 5:2 d. 3:7

2. What one number can replace x in 2: x = x: 32?a. 2 b. 6 c. 4 d. 8

3. If 5 men can do a job in 12 days, how long will it take 10 men to complete this task, assuming that they work at the same rate?

a. 20 days b. 6 days c. 2 days d. 0.06 day

4. If 3 kg of oranges cost as much as 5 kg of chicos, how many kg of oranges would cost as much as 60 kg of chicos? A. 100 B. 36 C. 7.5 D. 4

5. If 2/5 mm in a map represents 120 km, how many km will be represented by 2 mm?A. 600 km B. 300 km C. 96 km D. 24 km

6. In a Mathematics Club, the ratio of boys to girls is 3:5. If there are 240 members, how many are girls?A. 90 B. 144 C. 150 D. 450

7. A photographer wishes to enlarge a picture 18 cm long and 12 cm wide so that it will be 36 cm wide. How long will the enlarged picture be?

A. 54 cm B. 72 cm C. 24 cm D. 6 cm

8. If 8 secretaries can type 800 pages in 5 hours, how long would it take for 12 secretaries to type 800 pages at the same rate?

A. 7 1/2 hours B. 3 1/3 hours C. 10 hours D. 2 1/2 hours

THE THEORY OF CONGRUENCES

If a and b are integers, m a positive integer and m(a – b), we say that “a is congruent to b modulo m”. In symbols, we write this as a b (mod m). CONGRUENCE was invented by Karl Friedrich Gauss at the beginning of the 19th century and is a convenient statement about divisibility.

The following are equivalent and may be used interchangeably. a b (mod m). m (a – b) or (a – b) is divisible by m. a = b + mk, k Z.

Theorem: If a and b are integers and m a positive integer then a b (mod m) if and only if a and b leave the same remainder upon division by m.

Let m be a positive integer. A collection of m integers is called a complete residue system modulo m if every integer b modulo m is congruent to exactly one of the elements in the collection.

Properties of Congruence

Congruence is an equivalence relation in the set of integers; that is, congruence is reflexive, symmetric and transitive with respect to integers.In the following, let a, b, c, and d be integers and m a positive integer. If a b (mod m) then

a+c b+c (mod m). ac bc (mod m). ar br (mod m) where r is a positive integer.

If a b (mod m) and c d (mod m), then a + c (b + d) (mod m).


ac (bd) (mod m).

If ac bc (mod m), then where d = (c, m).

The following are some applications of congruence.

a) Finding the units digit (or hundreds digits) of a very large number written in exponential form; and b) Finding the remainder when a very large number is divided by another number.

Two of the most prolific mathematicians in Number Theory are Pierre de Fermat and Leonhard Euler.

FERMAT’S AND EULER’S THEOREMS

Theorem 5. (Fermat’s Little Theorem) Let p be a prime number and a Z . If p does not divide a, thenap – 1 1 (mod p) .

Theorem 6. (Fermat’s Second Theorem). Let p be a prime number and a Z . If p and a are relatively prime, then

ap a (mod p) . DEFINITION OF

Let m be a positive integer greater than 1. The number of positive integers less than and relatively prime to m is the value of Euler’s totient or function at m and is denoted by . Remarks: If p is prime, then = p – 1.

Theorem 7. Euler’s Theorem: If n is a positive integer and the greatest common divisor of a and n is 1, then .

LINEAR DIOPHANTINE EQUATIONS

An equation in one or more unknowns having integral solutions is called a Diophantine equation, in honor of Diophantus of Alexandria.

Theorem 8. Given two integers a and b where (a , b) = d. The linear Diophantine equation ax + by = c has an integral solution if and only if dc.

Theorem 9. If the equation ax + by = c has a solution x = x0 , y = y0, then any other solutions can be expressed in the form

, t Z and

, t Z.

Example: To determine the integral solution of 24x + 138y = 18, we note that since (24,138) = 6 and 618. Then we know that it has solution. We now have the following.

138 = 5 (24) + 1824 = 1 (18) + 618 = 6 (3).

Observe that,6 = 24 – 1(18)

= 24 – [138 - 5 (24)]= (-1)(138) + 6(24)

Moreover,18 = 3(6)

= 3[(-1)(138) + 6(24)] =(-3)(138) +(18)(24)

Thus, y0 = -3 and x0 = 18


Hence, the solution of the equation is of the form y = -3+23t and x = 18– 4t where t is an integer.

There are problems which can be solved using linear Diophantine equations as working equations.

The following steps may be used in solving word problems which involve linear Diophantine equations in two unknowns/variables:

Step 1. Represent the unknown values using any two variables.

Step 2. Form the equation using the condition given in the problem.

Step 3. Solve the resulting linear Diophantine equation.

Step 4. Determine the solution/s to the problem using the results in step 3.

Theorem: (Wilson’s Theorem) If p is a prime, then (p-1)! .

Exercises

1. Which of the following is true?A. B. C. D.

2. Mavic argues that . Is she correct? Why?A. Yes, because 6 divides 15 - 9. C. No, because 6 does not divide 15 + 9.B. Yes, because 6 divides 15 + 9. D. No, because .

3. Which of the following is congruent to 11 modulo 13?A. -7 B. -5 C. -2 D. 4

4. Which of the following must be the value of n if ?A. x is divisible by 7 C. x is relatively prime with 7B. x is prime D. x is any integer greater than 7

5. What is the remainder when is divided by 3?A. 1 B. 2 C. 3 D. 5

6. What is the units digit of ?A. 1 B. 3 C. 5 D. 9

7. What is ?A. 1 B.4 C.6 D. 11

PRACTICE EXERCISES

SET A Exercises. Choose the letter of the best answer.

1. What is the sum of the first four prime numbers?a. 11 b. 26 c. 17 d. 28

2. Which of the following is NOT true about the sum of two consecutive positive odd integers?

a. it is even b. it is only divisible by 12c. it is divisible by 4 d. it is always divisible by 1

3. In a sequence of starts and stops, an elevator travels from the first floor to the fourth floor and then to the second floor. From there, the elevator travels to the third floor and then to the first floor. If the floors are 3 meters apart, how far has the elevator traveled?

a. 21 m b. 24 m c. 28 m d. 32 m

4. An orange light blinks every 4 seconds. A blue one blinks every 5 seconds while a red one blinks every 6 seconds. How many times will they blink together in two hours?

a. once b. 2 times c. 10 times d. 60 times


5. If one prime factor of 84 is 3, what are the other prime factors?a. 2 and 3 b. 2 and 7 c. 3 and 5 d. 4 and 7

6. A television show reports the following temperature for 5 cities:Beijing London Chicago Philippines Moscow2 0C -6 0C 0 0C 300C -9 0CWhich city is the coldest?

a. Beijing b. Chicago c. London d. Moscow

7. If the sum of a certain number and 7 is divided by 4, the quotient is 3. What is the number?a. 5 b. 12 c. 15 d. 18

8. Which of the following numbers has the largest value?a. –22 b. –10 c. –75 d.3

9. Which of the following numbers has the least value?a. –22 b. –10 c. –75 d.3

10. What is the difference in the elevation between the top of a mountain 51 meters above sea level and a location 28 meters below sea level.

a. 23 m b. 33 m c. 79 m d. 89 m

11. A pack of P50-bills is numbered from RV628 to RV663. What is the total value of the pack of bills, in pesos?

a. 35 b. 36 c. 1750 d. 1800

12. Simplify: [ 5 81 32 – 5 3 + 2] (42 – 23)a. 15 ¼ b. 4 c. -6 d. – 15 1/4

13. If each container contains kg of flour, how many kg of flour are there in 12 container?

b. 68 kg b. 70kg c. 72 kg d. 80 kg

14. Eighteen is 2/3 of what number? a. 6 b. 12 c. 6 d. 27

15. What part of an hour has passed from 2:48 am to 3:20 am?a. 7/8 b. 1/3 c. 8/15 d. 8/25

16. Clarita spent one-sixth of her money in one store. In the next store, she spent three times as much as she spent in the first store. If she had 80 pesos left, how much money did she have from the start?

a. 240 pesos b. 252 pesos c. 300 pesos d. 360 pesos17. Philip has obtained the following grades: 1.4, 1.7, 1.8 and 2.5. What must be his fifth grade so that his average is 1.7?

a. 2.1 b. 1.9 c. 1.5 d. 1.1

18. Out of the 20 numbers, 6 were 2.5’s, 4 were 3.25’s and the rest were 2.2’s. Give the arithmetic mean of the numbers.

a. 2.5 b. 2.65 c. 10 d. 22

19. Ron bought X number of notebooks at P23.00 each, Y pad papers at 18.45 each, and Z ballpens at P8.25 each. If he gave an amount of P1000 to the cashier, how much change did he receive?

a. P 434.25 c. 1000 – [(23.00)(X) + (18.45) (Y) + (8.25) (Z)]b. P 334.25 d. none of these

20. A bag has a selling price of P60.00. If the selling price is 75% of the original price, what is its original price?a. P80 b. P120 c. P200 d. P280

21. Mr. de Borja, a store owner, advertises a polo-shirt originally sold for P200 for P170 only. What rate of discount is he giving?

a. P 30 b. P15 c. 30% d. 15%

22. Ja bought an article for P400 and sold it for P500. What rate of profit did she enjoy in that deal?a. P100 b. 100% c. 25 % d. 20%


23. The price of an item is increased by 70% and then offered at 40% discount. What happened to the original price?

a. There is an increase of 30%. c. There is an increase of 2 %.b. There is an increase of 28%. d. There is a decrease of 32%.

24. How much should Allan invest so that his money earns P2,250 deposited at 6% for 9 months?a. P 50,000 b. 37,500 c. P 135 d. P 101.25

25. Dan sells a real estate. He receives a monthly salary of P10,000 plus a commission of 1/5 % of his net sales for that month. Find his gross pay for a month during which his net sale is one million pesos.

a. P 2,000 b. P 12,000 c. P 200,000 d. P 210,000

26. There are 20 million Filipinos who are qualified voters. If 25% of the population are qualified voters, how many are not qualified voters?

a. 80 million b. 60 million c. 15 million d. 5 million

27. Three cavans of rice for a family of six members last for 5 weeks. At this rate, how many weeks will 4 cavans of rice last a family of 8 members?

A. 4 B. 5 C. 5 1/3 D. 6

28. If the assembly, ratio of boys to girls is 1:4. What percent of the assembly are the boys?A. 10% B. 20% C. 25 % D. 80%

29. What is the remainder when is divided by 31?

A. 1 B. 2 C. 10 D. 101

30. Which of the following is the remainder when is divided by 7? A. 1 B. 2 C. 3 D. 6

31. If y is the remainder when 47 is divided by 6, what is the remainder when 19 is divided by y?A. 1 B. 2 C. 4 D. 5

32. Which of the following is a value of z such that the congruence is NOT true?A. 9 B. 22 C. 30 D. 48

33. What is the remainder when 18! + 2 is divided by 19?A. 0 B. 1 C.3 D. 97

34. A certain number of sixes and nines are added to give a sum of 126. If the numbers of sixes and nines are interchanged, the new sum is 114. How many of each were there after the switch?

A. Ten sixes and 6 nines B. Four sixes and Twelve ninesC. Seven sixes and nine nines D. Six sixes and Ten nines

SET B Exercises. Choose the letter of the best answer.

1. How many prime numbers are less than 37?a. 9 b. 10 c. 11 d. 12

2. In a series of card games. Marlon starts out with P200 and wins a total of P450. If he later loses P350, wins P60 and loses P150, how much cash does Marlon have?

a. 0 b. P150 c. P210 d. P300

3. Your score in a game is -6. How many points must you earn to get a score of 10?a. -6 b 15 c. 16 D. 22

4. Arrange the fractions 5/12, 3/7, 2/5 in decreasing order. a. 2/5, 5/12, 3/7 c. 3/7, 5/12, 2/5 b. 5/12, 2/5, 3/7 d. 2/5, 3/7, 5/ 12

5. Edwin, Doms and Lon weigh 45 kg. If Edwin and Doms weigh kg and kg, respectively, what is

the weight of Lon in kilograms?

a. b. c. d.


6. Which of the following should be multiplied to so that the product is 57?

a. 6 b. 12 c. 6 d. 757. Which of the following is a value of m if ?

A. 2 B. 3 C. 23 D. 32

8. Alex works on his assignment hours a day, what part of the day does he spend doing his assignment?

a. 1/8 b. 1/7 c. 5/36 d. 15/28

9. What value of p will satisfy the equation 0.2 (2p + 1470) = p?a. 294 b. 490 c. 560 d. 1470

10. A blouse originally priced at P600 is being sold at a discount of 30%. How much would you pay if you buy that blouse?

a. P 30 b. P180 c. P 420 d. P 570

11. A pair of slippers with a selling price of P120 is sold at 40% discount. What is its original price?a. P 48 b. P72 c. P 200 d. P 300

12. An item is offered at 20% discount. Later, it is offered at 30% discount. If the new selling price is P112, what is the first original price?

a. P162 b. P200 c. P224 d. P1866.67

13. To have a 25% profit, the vendor should sell the item at P80.00. How much is his profit?a. P20.00 b. P60.00 c. P16.00 d. P64.00

14. Minda deposited P50,000 in a bank that pays a simple annual interest of %. How much

money will she have in the bank after five years.a. P 85,500 b. P35,500 c. P36,250 d. P86,250

15. The ratio of cows and carabaos in the field is 4:9. If there are 468 cows and carabaos in the field, how many are carabaos?

A. 52 B. 117 C. 144 D. 324

16. In the class, the ratio of boys to girls is 6:5. If there are 90 girls, how many persons are in the class?A. 75 B. 108 C. 165 D. 198

17. Edwin painting a wall at 9:00 a.m. and was able to finish painting 3/5 of it at 10:30 a.m. Continuing at this rate, at what time will he finish?

A. 10:45 a.m. B. 11:30 a.m. C. 11:45 a.m. D. 12:15 a.m.

18. What is the remainder when is divided by 15?A. -1 B. 1 C. 2 D. 4

19. If x is any positive integer, then 23x + 1 is _______ divisible by 8. A. Always B. Never C. Sometimes D. Equivalently

20. What is ?A. 0 B. 1 C. 2 D. 28

21. Which of the following is the remainder when is divided by 7?A. 0 B. 2 C. 4 D. 5

22. Which of the following is equivalent to the pair of congruence and ?A. B. C. D.

23. Which of the following is NOT true if a is a positive integer?A. a divides C.

B. divides . D. a and have the same units digit

24. Which of the following gives a remainder of 2 when divided by 5 and a remainder of 12 when divided by 13?


A. 22 B.38 C. 77 D. 92

25. What is the remainder upon dividing the sum by 5?A. 1 B. 2 C. 3 D. 4

26. What is the least residue if 17109 is a multiple of 6?A. 1 B. 2 C. 3 D. 5

27. Which of the following has an integral solution?A. B. C. D.

28. How many integral solutions does have? A. 2 B. 3 C. 6 D. 10

29. Which of the following has a solution if the variables are positive integers?A. 5x + 30y = 18 B. 8x + 10y = 15C. 22x + 4y = 28 D.

30. When 16! is divided by 17, the remainder is ______.A. 0 B. 1 C. 17 D. 18

31. A John’s transcript shows x number of 3-unit courses and y number of 5-unit courses for a total of 64 units. Which of the following may appear in the transcript?

A. 2 x’s and 18 y’s B. 13 x’s and 5 y’sC. 11 x’s and 3 y’s D. 9 x’s and 8 y’s

32. Which of the following is a value of x if ?A. 2 B. 5 C. 6 D. 10

33. When students in a certain college are grouped by 2’s, 3’s, 4’s, 5’s or 6’s at a time, there remain, 1,2,3,4, or 5 students respectively. When the students are grouped by 7’s, no is student left. What is the smallest possible number of students in the school?

A. 227 B.1,534 C. 1,379 D. 2,778


MATH+V

Documents