Add and Subtract Fractions with Unlike Denominators | Grade 5 Math | LevelKite

You already know how to add fractions with the same denominator. In $\frac{2}{7} + \frac{3}{7}$ , every part is a seventh, so you count the parts: $2$ sevenths and $3$ sevenths make $5$ sevenths. The denominator (the bottom number) tells the size of each part. The numerator (the top number) counts the parts.

Now try $\frac{1}{2} + \frac{1}{3}$ . These fractions have unlike denominators. That means the denominators are different, so the parts are different sizes. Can you still count the parts?

Two bars can help you decide. Both bars stand for the same whole. The first bar has $2$ equal parts, and $1$ part is shaded. It shows $\frac{1}{2}$ .

\frac{1}{2}

: one of the two equal parts is shaded.

\frac{1}{3}

: one of the three equal parts is shaded.

The second bar has $3$ equal parts, and $1$ part is shaded. It shows $\frac{1}{3}$ .

Each bar has $1$ shaded part. But the parts are not the same size. A half of the whole is bigger than a third of it.

So you can't just count $1$ part plus $1$ part and call it $2$ parts. Two parts of what size? Halves? Thirds? You can count parts together only when they are the same size.

So make the parts the same size.

Picture the first bar again. Cut each half into $3$ equal pieces. Now the whole has $6$ equal parts, and the shaded half is $3$ of them: $\frac{1}{2} = \frac{3}{6}$ .

Now picture the second bar. Cut each third into $2$ equal pieces. This whole has $6$ equal parts too, and the shaded third is $2$ of them: $\frac{1}{3} = \frac{2}{6}$ .

You just renamed both fractions. To rename a fraction is to write an equivalent fraction. The amount stays the same. Only the size and the number of the parts change.

Both fractions are in sixths now, so you can count the parts: $3$ sixths and $2$ sixths make $5$ sixths. The next bar shows this sum, the total you get when you add.

\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

: five of the six equal parts are shaded.

This bar is the same whole cut into $6$ equal parts, and $5$ of them are shaded. Three of the shaded sixths come from $\frac{1}{2}$ . The other two come from $\frac{1}{3}$ . Here is the number sentence:

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

Sixths help you subtract too. Think of $\frac{1}{2}$ as $3$ sixths. Take away $\frac{1}{3}$ , which is $2$ sixths. Then $1$ sixth is left:

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

Key concept

Same-size parts first

You can add or subtract fractions only when their parts are the same size. Fractions with unlike denominators have parts of different sizes. So rename them first with a common denominator. A common denominator is one denominator that both fractions can use, like the $6$ in $\frac{3}{6}$ and $\frac{2}{6}$ . Then the parts match, and you add or subtract the way you already know: add or subtract the numerators and keep the denominator.

You don't need to draw bars every time. Here is how to do the same thing with numbers.

Find a common denominator. It must be a multiple of both denominators. The multiples of a number are the numbers you say when you count by it. List multiples of each denominator. Use the first number that is on both lists.
Rename each fraction. Multiply the numerator and the denominator by the same number to reach the common denominator.
Add or subtract the numerators. Keep the denominator, because the size of the parts stays the same.
Check for lowest terms. A fraction is in lowest terms when $1$ is the only whole number that divides both the numerator and the denominator evenly. If a bigger number divides both, divide the numerator and the denominator by that number, then check again.

Try step 1 with halves and thirds. Multiples of $2$ : $2$ , $4$ , $6$ . Multiples of $3$ : $3$ , $6$ . The first number on both lists is $6$ . That is why sixths worked.

Worked example

Add with unlike denominators

Add $\frac{1}{4} + \frac{2}{3}$ .

Fourths and thirds are different sizes, so find a common denominator first. Multiples of 4: 4, 8, 12. Multiples of 3: 3, 6, 9, 12. The first number on both lists is 12, so rename both fractions as twelfths.
Rename one fourth. 4 times 3 is 12, so multiply the numerator and the denominator by 3. This cuts each fourth into 3 equal pieces, so the amount stays the same.
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Rename two thirds the same way. 3 times 4 is 12, so multiply the numerator and the denominator by 4. This cuts each third into 4 equal pieces.
$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
Now every part is a twelfth, so you can count the parts together. Add the numerators and keep the denominator.
$\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$
Check for lowest terms. The only whole number that divides both 11 and 12 evenly is 1, so the sum is already in lowest terms.

Answer

The sum of $\frac{1}{4}$ and $\frac{2}{3}$ is $\frac{11}{12}$ .

Worked example

Add, then write the sum in lowest terms

Add $\frac{1}{6} + \frac{3}{10}$ . Write the sum in lowest terms.

Find a common denominator. Multiples of 6: 6, 12, 18, 24, 30. Multiples of 10: 10, 20, 30. The first number on both lists is 30, so rename both fractions as thirtieths.
Rename both fractions. 6 times 5 is 30, so multiply the numerator and the denominator of one sixth by 5. 10 times 3 is 30, so multiply the numerator and the denominator of three tenths by 3.
$\frac{1 \times 5}{6 \times 5} = \frac{5}{30} \quad \text{and} \quad \frac{3 \times 3}{10 \times 3} = \frac{9}{30}$
Both fractions are in thirtieths now, so you can count the parts together. Add the numerators and keep the denominator.
$\frac{5}{30} + \frac{9}{30} = \frac{14}{30}$
Write the sum in lowest terms. 2 divides both 14 and 30 evenly, so divide the numerator and the denominator by 2. This joins the thirtieths in pairs to make fifteenths, so the amount stays the same. Now 1 is the only whole number that divides both 7 and 15 evenly, so you are done.
$\frac{14}{30} = \frac{14 \div 2}{30 \div 2} = \frac{7}{15}$

Answer

In lowest terms, the sum of $\frac{1}{6}$ and $\frac{3}{10}$ is $\frac{7}{15}$ .

Worked example

Subtract with unlike denominators

Subtract $\frac{5}{6} - \frac{3}{8}$ .

Sixths and eighths are different sizes, so find a common denominator before you subtract. Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. The first number on both lists is 24.
Rename both fractions as twenty-fourths. 6 times 4 is 24, so multiply the numerator and the denominator of five sixths by 4. 8 times 3 is 24, so multiply the numerator and the denominator of three eighths by 3.
$\frac{5 \times 4}{6 \times 4} = \frac{20}{24} \quad \text{and} \quad \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$
Both fractions are in twenty-fourths now, so you can take 9 of them away from 20 of them. Subtract the numerators and keep the denominator.
$\frac{20}{24} - \frac{9}{24} = \frac{11}{24}$
Check the answer. The only whole number that divides both 11 and 24 evenly is 1, so it is already in lowest terms.

Answer

When you subtract $\frac{3}{8}$ from $\frac{5}{6}$ , you get $\frac{11}{24}$ .

Remember the big idea: you can add or subtract fractions only when their parts are the same size. If the denominators are unlike, rename the fractions with a common denominator first. Then add or subtract the numerators, keep the denominator, and check for lowest terms.

Next, you will use this idea to subtract a fraction from a whole number, add three or more fractions, and find a missing fraction in a number sentence.