1.7 Fraction Review

Scott McDaniel

1.7 Fraction Review

Learning Objectives

Add and subtract fractions with like and unlike denominators.
Multiply and divide fractions.
Apply fraction operations to mixed practice problems.

Launch

Simplifying algebraic expressions often involves fractions. These fraction rules may be familiar, but reviewing them now will make algebraic steps easier later.

Addition of Fractions

Rule

If fractions have the same denominator, add the numerators and keep the denominator.
If fractions have different denominators, find a common denominator, rewrite the fractions, then add.

Example 1.7.1

Simplify: $\frac{2}{5} + \frac{1}{5}$

Solution

The denominators are the same, so add the numerators:

$\frac{2+1}{5} = \frac{3}{5}$

Example 1.7.2

Simplify: $\frac{3}{8} + \frac{1}{2}$

Solution

The denominators are different. The least common denominator (LCD) of 8 and 2 is 8. So we need to rewrite $\frac{1}{2}$ with a denominator of 8. We do this by multiplying $\frac{1}{2}$ by a “clever” form of 1. Note that I can multiply the 2 by a 4 to make it an 8. So my “clever” form of 1 would be $\frac{4}{4}$ . (See video for an easier-to-understand explanation.)

Rewrite:

$\frac{3}{8} + \frac{4}{8} = \frac{7}{8}$

Example 1.7.3

Simplify: $\frac{1}{3} + \frac{2}{9}$

Solution

The LCD of 3 and 9 is 9. Se we multiply the $\frac{1}{3}$ by $\frac{3}{3}$

Rewrite:

$\frac{3}{9} + \frac{2}{9} = \frac{5}{9}$

Subtraction of Fractions

Rule

Same denominator: subtract the numerators.
Different denominators: find the LCD, rewrite, then subtract.

Example 1.7.4

Simplify: $\frac{5}{6} - \frac{1}{6}$

Solution

Same denominator, so subtract the numerators:

$\frac{5-1}{6} = \frac{4}{6} = \frac{2}{3}$

Example 1.7.5

Simplify: $\frac{7}{8} - \frac{1}{2}$

Solution

The LCD of 8 and 2 is 8.

Rewrite:

$\frac{7}{8} - \frac{4}{8} = \frac{3}{8}$

Example 1.7.6

Simplify: $\frac{9}{14} - \frac{2}{7}$

Solution

The LCD of 14 and 7 is 14.

Rewrite:

$\frac{9}{14} - \frac{4}{14} = \frac{5}{14}$

Multiplication of Fractions

Rule

Multiply straight across:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Then simplify if possible.

Note: If you are able to simplify BEFORE multiplying, you should! This ONLY works with multiplication.

Example 1.7.7

Simplify: $\frac{2}{3} \times \frac{3}{4}$

Solution

Method I. Multiply across:

$\frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} = \frac{1}{2}$

Method II. Cancel first

$\frac{2}{3} \times \frac{3}{4}$

I can cancel the 3’s and can cancel the 2 into the 4 (see the video for a clearer explanation)

$\frac{2}{3} \times \frac{3}{4}$ = $\frac{1}{1} \times \frac{1}{2}=\frac{1}{2}$

Example 1.7.8

Simplify: $\frac{7}{8} \times \frac{4}{7}$

Solution

Method I. Multiply across:

$\frac{7 \cdot 4}{8 \cdot 7} = \frac{28}{56} = \frac{1}{2}$

Method II. Cancel first

$\frac{7}{8} \times \frac{4}{7}$

I can cancel the 7’s and 4 goes into 8, 2 times.

$\frac{7}{8} \times \frac{4}{7} = \frac{1}{2}[$

Example 1.7.9

Simplify: $\frac{4}{5} \times \frac{5}{6}$

Solution

Method I. Multiply across:

$\frac{4 \cdot 5}{5 \cdot 6} = \frac{20}{30} = \frac{2}{3}$

Method II. Cancel first

$\frac{4}{5} \times \frac{5}{6}$

The 5’s cancel and 2 goes into 4 twice and 2 goes into 6 three times.

$\frac{4}{5} \times \frac{5}{6} =\frac{2}{3}$

Division of Fractions

Rule

To divide fractions, multiply by the reciprocal:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

(This is often remembered as “Keep, Change, Flip.”)

We multiply by the reciprocal because dividing by a fraction means asking what number undoes multiplying by that fraction—and the reciprocal is exactly what reverses it.