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Business Math
Chapter 2: Fractions
1
2.1 Fractions
Learning Objectives

Identify types of fractions

Convert an improper fraction to a whole or mixed
number

Convert a whole or mixed number to an improper
fraction

Reduce a fraction to lowest terms

Raise a fraction to highest terms
2
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
2.1.1. Identify types of fractions

A fraction is used to
identify parts of a
whole. It describes
the relationship
between the part and
the whole.

There are four parts:
and one is shaded or
1 in 4 which is ¼.
3
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Key Terms

Denominator-the number appearing below the
fraction line.

Numerator- the number appearing above the
fraction line.

Fraction line- horizontal line dividing numerator
and denominator.

Proper fraction- a fraction has a value than is
less than “1” (⅔, for example.)
4
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at the fraction
⅔



2 is the numerator
3 is the denominator
Is it a proper fraction?
Yes, because the value of the fraction is
less than “1”.
5
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Identify the fraction
¾

What part of the area
is shaded?

The fraction is 3/7.
6
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Improper fraction
The numerator is a greater value than the
denominator, and therefore is greater than “1”.




Proper or improper?
10/4
6/7
9/8
7
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Convert an improper fraction to
a whole or mixed number

Divide the numerator or the improper fraction by
the denominator.

If the remainder is zero, the quotient is a whole
number.

If the remainder is not zero, the improper fraction
will be expressed as a mixed number.
8
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try these examples

140/10



14
260/3

86 ⅔

33 ¾
135/4
9
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Write a mixed number as an
improper fraction

Find the numerator of the improper fraction.

Multiply the denominator of the mixed number
by the whole number part.

Add the product from the previous step to the
numerator of the mixed number.

Use the denominator of the mixed number.
10
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example.
Convert 10 ¾ to an improper fraction

The numerator of the fraction is “3.”

Multiply the whole number, which is “10” by the
denominator which is “4”; the result is 40.

Add the numerator to product; 40 + 3 = 43.

Retain the same denominator.

43/4 is the improper fraction equivalent.
11
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Reduce a fraction to
lowest terms

Inspect the numerator and denominator to find
any whole number by which both can be evenly
divided.

Carry out the operation until there is no one
number that both can be evenly divided by.

Tip: Check if the denominator can be divided by
the numerator: 3/15, for example, can be
reduced to 1/5 when 3 is divided into 15.
12
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Reduce to lowest terms



18/ 30

3/5

3/7

1/7
27/63
21/147
13
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Find the greatest common
divisor of two numbers

The most direct way to reduce a fraction to
lowest terms is to use the GCD.

The GCD is the largest number by which the
denominator and the numerator can be evenly
divided.

For example, the GCD of 15 and 20 is 5. Any
number greater than 5 would result in a
quotient with a remainder.
14
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
How to find the GCD

For example: find the GCD of 42 and 28.

Divide the larger number by the smaller number:
42 divided by 28 = 1 R 14

Divide the divisor by the remainder of the
previous operation (28) by (14)
28 divided by 14 = 2 R 0.

When the R = 0, the divisor from that operation
(14, in this case) is the GCD.
15
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try these examples.



30, 36

GCD = 6

GCD = 5
30, 125
17,85

GCD =17
16
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Raise a fraction to higher terms
¾ is equal to ?

8
Look at the two denominators and divide.

“4” goes into 8 two times.

Multiply “3” by “2” to get the equivalent
numerator.

¾ = 6/8
17
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try these examples.
Determine the equivalent fraction in
higher terms:
4/5 = ?/25

20/25

35/40

36/60
7/8 = ?/40
3/5 = ?/60
18
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
2.2. Adding and subtracting
fractions
To add fractions with like denominators:

Add the numerators

The denominator remains the same

Convert an improper fraction to a mixed number,
if necessary

¼ + ¾ + ¼ = 5/4 or 1 ¼
19
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Adding fractions with
different denominators

You must first find the lowest common
denominator (LCD).

Smallest number that can be divided evenly by
each original denominator.

For example: ¾ and ⅝ [using inspection]

Convert ¾ to an equivalent fraction in eighths
and then add.
20
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Adding fractions with
different denominators

Find the LCD for 4/5, 1/2 and 1/8.

It is not as apparent which number might be the
LCD given the denominators of 5, 2 and 8.

You can use prime numbers to find the LCD

Prime number: a number greater than 1 that
can be divided evenly by only itself and 1
21
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Find the LCD
using prime numbers
Denominators
5
2
8
2
5
1
4
2
5
1
2
2
5
1
1
5
1
1
1
Prime numbers
22
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Find the LCD

Multiply the prime numbers from the first column
together (2x2x2x5) to get the LCD.

The LCD is 40.

Convert the fractions to the equivalent using 40
as the denominator.

4/5 becomes 32/40.

½ becomes 20/40.

1/8 becomes 5/40.
23
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Add the numerators

32/40 + 20/40 + 5/40 = 57/40

If the numerator is greater than the denominator,
it is an improper fraction and can be expressed
as a mixed number.

It would be 1 17/40

Inspect the fraction to determine if it is
expressed in lowest terms.
24
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Subtracting fractions
with like denominators

Subtract the smaller numerator from the greater
one.

The denominator remains the same.

5/8 – 3/8 = 2/8

Reduce to lowest terms, if necessary.

2/8 = 1/4
25
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Subtracting fractions with
different denominators

As in addition, to subtract fractions you must
have a common denominator.

Use the same methods of inspection or prime
numbers to determine the LCD.

Carry out the operation.

Reduce to lowest terms as needed.
26
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Subtracting fractions with
different denominators

5/12 -1/3 = ?

Find the LCD, which is 12.

Change 1/3 to an equivalent fraction.

1/3 = 4/12

Carry out the operation:

5/12- 4/12 = 1/12

Reduce to lowest terms, if needed.
27
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try these examples

7/8 – ½ =


2/3 – 1/5 =


3/8
7/15
4/5 -1/6 =

19/30
28
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Subtracting mixed numbers

10 ⅛ – 7 ½ =

Convert the fraction portion of each mixed
number to equivalent fractions.
10 1/8 -7 4/8 =

Borrow “1” from the whole number to carry out
the operation.
9 9/8 – 7 4/8 = 2 5/8

Reduce to lowest terms, if necessary.
29
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try these examples

Maria has 6 ⅛ cups of flour, but only needs 4 ¼
cups for her recipe. How much will she have
left?
 1⅞

Julia needs 3 ⅔ yards of tape to finish a display.
Bob brought her a 5 ⅞ yard piece from the
supply room. How much will be left?
 2 and 5/24
30
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
2.3 Multiplying and
Dividing Fractions

Multiply fractions and mixed numbers

Divide fractions and mixed numbers
1/2 divided by 1/3 = ?
3/5 x 7/8 = ?
31
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Multiply fractions
and mixed numbers

Find the numerator of the product: multiply the
numerators of the fractions.

Find the denominator of the product: multiply
the denominators of the fractions.

Reduce to lowest terms
32
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Look at this example.





⅓x⅞=
1x7=7
3 x 8 = 24
The product is 7/24.
Reduce to lowest terms, if necessary.
2/3 x 3/4 = ?
33
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Tip!

To keep things simple, if possible, reduce before
multiplying.

⅓x¾=?

The “3” in the denominator in the first fraction
and the “3” in the numerator in the second
fraction cancel each other out and become “1”.

The answer is ¼.
34
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Multiply mixed numbers
and whole numbers

Write the mixed numbers and whole numbers as
improper fractions.

Reduce numerators and denominators as
appropriate.

Multiply the fractions.

Reduce to lowest terms and / or write as a whole
number or mixed number.
35
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this example.








1⅔x3¾=?
1 2/3 becomes 5/3
3 ¾ becomes 15/ 4
5/3 x 15/4 = ?
The “3” can be reduced to “1” and the “15” to “5”
before multiplying.
Multiply: 25/4.
Convert to a mixed number.
6¼
36
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Are products always larger
than their factors?

No. When the multiplier is a proper fraction, the
product is less than the original number.
5 x 3/5 = 3

This is also true when the multiplicand is a
whole number, fraction or mixed number.
2½ x ½ = 1¼
37
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Reciprocals

The relationship between multiplying and
dividing fractions involves a concept called
reciprocals.

Two numbers are reciprocals if their product is
equal to 1.

2 is the reciprocal of ½.

What is the reciprocal of ⅓?

3
38
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Divide fractions or mixed numbers

Write numbers as fractions.

Find the reciprocal of the divisor.

Multiply the dividend by the reciprocal of the
divisor.

Reduce to lowest terms, and/or write as a whole
or mixed number.
39
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Here’s an example.

3¼ ÷ ⅔=?

To carry out this operation,
 Convert
3 ¼ to an improper fraction
 Change
⅔ to its reciprocal which is 3/2
 Change
from division to multiplication

13/4 x 3/2 = 39/8

39/8 = 4 ⅞
40
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this problem.

Madison Duke makes appliques. A customer has
ordered five appliques. Madison has ¾ yard of
fabric and each applique requires 1/6 of a yard.
Does she need more fabric?

¾ ÷ 1/6 becomes ¾ x 6

Simplify by dividing 4 and 6 by 2.

Multiply 3/2 x 3.

The answer is 4 ½; therefore she can only make
4 appliques and she needs more fabric.
41
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved
Try this problem

A home goods store is stacking decorative
boxes on shelves. If each box is 6 ⅔
inches tall, and the shelf space is 45
inches, how many boxes will fit on each
shelf?
 Six
42
Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved