Download Review of Fractions

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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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 white or 1
in 4 which is ¼.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.)
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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”.
Business Math, Eighth Edition
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07458 All Rights Reserved
Identify the fraction

Business Math, Eighth Edition
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¾

What part of the area
is shaded blue?

The fraction is 3/7.
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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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try these examples

120/10



Business Math, Eighth Edition
Cleaves/Hobbs
12
119/3

39 ⅔

33 ¾
135/4
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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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Reduce to lowest terms



Business Math, Eighth Edition
Cleaves/Hobbs
24/40

3/5

3/7

1/7
27/63
21/147
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try these examples



30, 36

GCD = 6

GCD = 5
30, 125
17, 51

Business Math, Eighth Edition
Cleaves/Hobbs
GCD =17
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try these examples



7/8 – 2/3 =
5/24

7/15
2/3 – 1/5 =
4/5 -1/6 =

Business Math, Eighth Edition
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
19/30
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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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
1/4 divided by 2/3 = ?
4/5 x 5/8 = ?
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Can this fraction be reduced further?
NO!
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved
Try this problem

Madison Duke makes appliqués. A customer has
ordered five appliqués. Madison has ¾ yard of
fabric and each appliqué 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 appliqués and she needs more fabric.
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 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
Business Math, Eighth Edition
Cleaves/Hobbs
© 2009 Pearson Education, Inc. Upper Saddle River, NJ
07458 All Rights Reserved