Topic 1 of 9

Adding and subtracting fractions with different denominators

Step 7 · Fraction and decimal arithmetic topics

Add and subtract fractions with different denominators by rewriting them with a common denominator, including mixed numbers, and check answers with benchmarks.

Finding a common denominator

$\frac{1}{2}$ + $\frac{1}{3}$ cannot be done by counting pieces, because halves and thirds are different sizes. First cut both into pieces of the same size.

1/2

3/6

1/3

2/6

Halves and thirds both cut into sixths: 1/2 = 3/6 and 1/3 = 2/6.

A common denominator is a number that both denominators go into. Multiplying the two denominators always works; the smallest common multiple keeps the numbers small.

Example: A common denominator

Find a common denominator for $\frac{3}{4}$ and $\frac{1}{6}$ .

Multiples of 6: 6, 12, 18.
12 is also a multiple of 4.
$\frac{3}{4}$ = $\frac{9}{12}$ and $\frac{1}{6}$ = $\frac{2}{12}$ .

Answer12

Adding and subtracting

Once the denominators match, add or subtract the numerators as before.

Example: Adding

What is $\frac{1}{2}$ + $\frac{1}{3}$ ?

$\frac{1}{2}$ = $\frac{3}{6}$ and $\frac{1}{3}$ = $\frac{2}{6}$ .
$\frac{3}{6}$ + $\frac{2}{6}$ = $\frac{5}{6}$ .

Answer $\frac{5}{6}$

Example: Subtracting

What is $\frac{3}{4}$ − $\frac{1}{6}$ ?

In twelfths: $\frac{9}{12}$ − $\frac{2}{12}$ .
9 − 2 = 7.

Answer $\frac{7}{12}$

Mixed numbers

Add the whole numbers and the fractions separately, then tidy up if the fraction part comes to more than 1.

Example: Adding mixed numbers

What is $2\frac{3}{4}$ + $1\frac{1}{2}$ ?

Wholes: 2 + 1 = 3.
Fractions: $\frac{3}{4}$ + $\frac{2}{4}$ = $\frac{5}{4}$ = $1\frac{1}{4}$ .
3 + $1\frac{1}{4}$ = $4\frac{1}{4}$ .

Answer $4\frac{1}{4}$

Example: Subtracting mixed numbers

What is $3\frac{1}{3}$ − $1\frac{1}{2}$ ?

In sixths: $3\frac{2}{6}$ − $1\frac{3}{6}$ .
$\frac{2}{6}$ is less than $\frac{3}{6}$ , so trade 1 whole: $3\frac{2}{6}$ = $2\frac{8}{6}$ .
$2\frac{8}{6}$ − $1\frac{3}{6}$ = $1\frac{5}{6}$ .

Answer $1\frac{5}{6}$

Try it yourself

Work each one on paper first, then open the answer to check.

What is $\frac{1}{4}$ + $\frac{1}{3}$ ?
Show answer
$\frac{7}{12}$
What is $\frac{5}{6}$ − $\frac{1}{2}$ ?
Show answer
$\frac{2}{6}$ , which is $\frac{1}{3}$ .
What is $\frac{2}{5}$ + $\frac{3}{10}$ ?
Show answer
$\frac{7}{10}$
What is $1\frac{1}{2}$ + $2\frac{2}{3}$ ?
Show answer
$4\frac{1}{6}$
Is $\frac{1}{2}$ + $\frac{2}{5}$ more or less than 1?
Show answer
Less. $\frac{2}{5}$ is less than $\frac{1}{2}$ , so the sum is less than 1. It is $\frac{9}{10}$ .