Chapter TwoFRACTIONS

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

LEARNING UNIT OBJECTIVES

2-2

LU 2-1: Types of Fractions and Conversion Procedures1. Recognize the three types of fractions.2. Convert improper fractions to whole or mixed numbers and

mixed numbers to improper fractions.3. Convert fractions to lowest and highest terms.

LU 2-3: Basic Math Functions with Fractions1. Add and subtract fractions.2. Multiply fractions.3. Divide fractions.

LU 2-2: Fraction and Decimal Conversions1. Convert decimal fractions to decimals, proper fractions to

decimals, mixed numbers to decimals, and pure and mixed decimals to decimal fractions.

TYPES OF FRACTIONS

2-3

1, 1, 1, 4, 184 2 12 7 55

Proper fractions have a value less than 1; its numerator is smaller than its denominator.

Proper Fractions

Numerator

Denominator

TYPES OF FRACTIONS

2-4

Improper FractionsImproper fractions have a value equal to or greater than 1; its numerator is equal to or greater than its denominator.

14, 7, 15, 2214 6 14 19

Denominator

Numerator

TYPES OF FRACTIONS

2-5

1, 9, 7, 5, 96 10 8 6 11

5 5 8 33

Mixed Numbers

Mixed numbers are the sum of a whole number greater than zero and a proper fraction

139

CONVERTING IMPROPER FRACTIONS TO

WHOLE OR MIXED NUMBERS

2-6

2 Steps1. Divide the numerator of the

improper fraction by the denominator

1515 = 1

2. a. If you have no remainder, the quotient is a whole number

16 1 5 5

3 R 15 16 15 1

= 3

2 b. If you have a remainder, the quotient is a mixed number. The remainder is placed over the old denominator as the proper fraction of the mixed number.

CONVERTING MIXED NUMBERS TO IMPROPER FRACTIONS

2-7

3 Steps1. Multiply the denominator of the

fraction by the whole number. (8 x 6) = 48

1 8

6

(8 x 6) = 4848 + 1 = 49

2. Add the product from Step 1 to the numerator of the old fraction.

49 8

3. Place the total from Step 2 over the denominator of the old fraction to get the improper fraction.

CONVERTING (REDUCING) FRACTIONS

TO LOWEST TERMS

Find the largest whole number that will divide into both the numerator and denominator without leaving a remainder.

24 24 / 6 430 30 / 6 5

= =

2-8

FINDING THE GREATEST COMMON DIVISOR

Step 1. Divide the numerator into the denominator.

124 30 24 6

46 24 24 0

Step 2. Divide the remainder in Step 1 into the divisor of Step 1.

24 / 6 430 / 6 5

= Step 3. Divide the

remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0. 2-9

2430

CONVERTING (RAISING) FRACTIONS TO HIGHER TERMS

Multiply the numerator and the denominator by the same whole number.

1 2 24 2 8

x =

2-10

The fractions are equivalent in value. By converting, you divided it into more parts.

47 28 28 0

RAISING FRACTIONS TO HIGHER TERMS

WHEN DENOMINATOR IS KNOWN

2 Steps1.Divide the new denominator

by the old denominator to get the common number that raises the fraction to higher terms.

4 = ? 7 28

2-11

4 x 4 = 16

1628

2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator.

CONVERTING PROPER FRACTIONS TO DECIMALS

Divide the numerator of the fraction by its denominator.

Round as necessary.

=

=

2-12

34

.75

38 .375

13

= .333

CONVERTING MIXED NUMBERS TO DECIMALS

Convert the fractional part of the mixed number to a decimal.

Add the converted fractional part to the whole number.

=

2-13

2 5

8.40

8.00+.40

8 .4025

(Step 1)

(Step 2)

CONVERTING PURE AND MIXED NUMBERS TO FRACTIONS

1. Place the digits to the right of the decimal point in the numerator of the fraction. Omit the decimal point.

2. Put a 1 in the denominator of the fraction.

2-14

3

.3 31

(Step 1) (Step 2)

3 Steps

3. Count the number of digits to the right of the decimal point. Add the same number of zeros to the denominator of the fraction.

1 310

Places

(Step 3)

ADDING AND SUBTRACTING LIKE FRACTIONS

Add or subtract the numerators and place the total over the denominator.

If the total of your numerators is the same as your original denominator, convert your answer to a whole number. If the total is larger than your original denominator, convert your answer to a mixed number.

1 4 57 7 7+ =

4 1 3 7 7 7

- =

2-15

ADDING OR SUBTRACTING UNLIKE FRACTIONS

4 Steps1. Find the LCD.

1 1 1 13 8 9 12

+++

2-16

24 9 8 6 4772 72 72 72 72

+ + + =

2. Change each fraction to a like fraction with the LCD.

3. Add or subtract the numerators and place the total over the LCD.4. If necessary, reduce the answer to lowest terms.

7

42

21

LEAST COMMON DENOMINATOR (LCD)

2-17

What is the least common denominator?

The smallest nonzero whole number into which ALL denominators will divide evenly.

3 5 7 21+

PRIME NUMBERS

A prime number is a whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number.

Examples

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43

2-18

ADDING MIXED NUMBERS

3 Steps1. Add the fractions.2. Add the whole numbers.3. Combine steps 1 & 2. Be

sure you do not have an improper fraction in your fi nal answer. If necessary, reduce the answer to lowest terms.

7 720 20

3 125 20

1 54 20

24 4 20 20

4 20

4 4

6 6

18

= 1

+ 7

+7

Step 1

Step 3

Step 2

+ 17

18 1 5

2-19

1 4 2 8

- 3 3 8 8 1 8

SUBTRACTING MIXED NUMBERS

Step 1. Subtract fractions, making sure to fi nd the LCD.

When Borrowing Is Not Necessary:

2-20

6 6

6

Step 2. Subtract whole numbers.

Step 3. Reduce the fractions to lowest

terms.

3 Steps

1 2

34

2 6 4 4

3 3 4 4

3 4

SUBTRACTING MIXED NUMBERS

Step 1. Make sure the fractions have the LCD.

When Borrowing Is Necessary:

3

1

-1 -1

2

2-21

3

-1Step 2. Borrow from the whole number.

Step 3. Subtract whole numbers and fractions.

Step 4. Reduce the fractions to lowest terms.

4 Steps

MULTIPLYING PROPER FRACTIONS

5 56

Step 1. Multiply the numerator and the denominator. 1 5

7 8

=x

2-22

Step 2. Reduce the answer to lowest terms.

2 Steps

MULTIPLYING MIXED NUMBERS

1. Convert the mixed numbers to

improper fractions.

2. Multiply the numerator and denominators.

1 1 7 3 7 13 2 3 2 2 2

2 3=X 1 X = =

1

1

2-23

3. Reduce the answer to lowest terms.

DIVIDING PROPER FRACTIONS

1. Invert (turn upside down) the divisor (the second fraction).

2. Multiply the fractions.

1 2 1 3 38 3 8 2 16

= =X..

2-24

3. Reduce the answer to lowest terms.

DIVIDING MIXED NUMBERS

1. Convert all mixed numbers to

improper fractions.

2. Invert the divisor and multiply.

3 5 35 6 105 34 6 4 17 34 34

3. Reduce the answer to lowest terms.

8 =2 X = =3

2-25

÷