Chapter 02 of 10

Fraction Operations

How do fraction operations connect to whole-number reasoning?

Fractions are numbers, not just pieces of shapes. Addition and subtraction require equal-sized parts, while multiplication finds a part of a part. Division asks how many groups of one fractional size fit into another quantity.

The big ideas

Understand these before you worry about speed.

Use common denominators to add or subtract

The denominator names the size of each part. Rewrite fractions with a common denominator before combining their numerators, then simplify.

Example

\displaystyle \frac{3}{4}+\frac{1}{6}=\frac{9}{12}+\frac{2}{12}=\frac{11}{12}

Multiply straight across

Multiply numerators and denominators, then simplify. Canceling common factors before multiplying can keep the numbers smaller.

Example

\displaystyle \frac{2}{3}\cdot\frac{9}{10}=\frac{18}{30}=\frac{3}{5}

Divide by multiplying by the reciprocal

Dividing by a fraction is equivalent to multiplying by its reciprocal. Keep the first fraction, change division to multiplication, and flip only the divisor.

Example

\displaystyle \frac{5}{6}\div\frac{2}{3}=\frac{5}{6}\cdot\frac{3}{2}=\frac{5}{4}

Convert mixed numbers before multiplying or dividing

An improper fraction makes every part of the quantity available to the operation. Convert the result back to a mixed number when that form is easier to interpret.

Example

\displaystyle 2\frac{1}{3}=\frac{7}{3}

Worked example

Follow the reason for each step—not just the symbols.

Problem

Evaluate $\displaystyle 3\frac{1}{2}\div\frac{3}{4}$ .

Convert the mixed number: $\displaystyle 3\frac{1}{2}=\frac{7}{2}$ .
Multiply by the reciprocal of $\displaystyle \frac{3}{4}$ : $\displaystyle \frac{7}{2}\cdot\frac{4}{3}$ .
Simplify before multiplying: $\displaystyle \frac{7}{1}\cdot\frac{2}{3}=\frac{14}{3}$ .
Convert back to a mixed number: $\displaystyle \frac{14}{3}=4\frac{2}{3}$ .

Answer $\displaystyle 4\frac{2}{3}$

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

Compute $\displaystyle \frac{5}{8}+\frac{1}{6}$ .

Check answer

$\displaystyle \frac{19}{24}$

Compute $\displaystyle \frac{7}{9}-\frac{1}{3}$ .

Check answer

$\displaystyle \frac{4}{9}$

Compute $\displaystyle \frac{4}{5}\cdot\frac{15}{16}$ .

Check answer

$\displaystyle \frac{3}{4}$

Compute $\displaystyle 2\frac{1}{4}\div\frac{3}{8}$ .

Check answer

$\displaystyle 6$

Practice and tools

Use the resource that matches what you need next.

Calculator

Fraction Operations

The big ideas

Use common denominators to add or subtract

Multiply straight across

Divide by multiplying by the reciprocal

Convert mixed numbers before multiplying or dividing

Worked example

Evaluate $\displaystyle 3\frac{1}{2}\div\frac{3}{4}$ .

Try it yourself

Practice and tools

Fraction Calculator

Adding and Subtracting Fractions

Multiplying and Simplifying Fractions

Dividing Fractions