Add and subtract fractions with like and unlike denominators

Ontario Grade 6 Mathematics

Number

Grade 6 Mathematics — B2.5

A fraction names equal parts of a whole. In

\frac{3}{8}

, the denominator,

8

, tells how many equal parts make one whole. The numerator,

3

, tells how many of those parts we have. Adding or subtracting fractions is easiest when the parts are the same size. We will use fraction bars to see why, then write the steps with numbers.

What you will learn

Add fractions with the same denominator.
Subtract fractions with the same denominator.
Make equivalent fractions to add or subtract fractions with different denominators.
Explain why the denominator stays the same after fractions are renamed with equal-sized parts.

Start with equal-sized parts

Imagine one whole bar divided into 8 equal pieces. If you have 3 pieces and get 2 more pieces of the same size, you have 5 pieces. The whole is still divided into eighths, so the answer is

\frac{5}{8}

.

Fractions with the same denominator are called like fractions. Their pieces are the same size. To add them, add the numerators and keep the denominator. To subtract them, subtract the numerators and keep the denominator. The denominator does not change because the size of each piece does not change.

For example,

\frac{6}{9}-\frac{2}{9}

means taking away 2 ninths from 6 ninths. That leaves 4 ninths. The result is

\frac{4}{9}

.

\frac{a}{n}+\frac{b}{n}=\frac{a+b}{n}

The denominator names the size of the parts.
For like fractions, add or subtract the numerators and keep the denominator.

Make the parts the same size

Fractions with different denominators are called unlike fractions. Their parts have different sizes. You cannot simply add or subtract the numerators and choose one of the denominators. First, rename the fractions so that their parts are the same size.

Equivalent fractions have the same value but are written with different-sized parts. For example, one half is the same amount as two fourths. To make an equivalent fraction, multiply the numerator and denominator by the same number. This splits each old part into equal smaller parts without changing the total amount.

A common denominator is a denominator that both fractions can use. Choose one that both original denominators divide into evenly. For example, fourths and sixths can both be renamed as twelfths. Then add or subtract the numerators and keep the common denominator.

A fraction bar makes this idea visible. One half covers the same amount as two of four equal parts. One fourth covers one of those four parts. Together they cover three of the four parts. The picture uses one whole split into fourths, so each shaded part has the same size.

\frac{a}{b}=\frac{a\times k}{b\times k}

Rename unlike fractions as equivalent fractions with a common denominator.
Once the parts have the same size, use the like-fraction rule.
Keep the denominator that names the new, equal-sized parts.

A careful method

For each question, look at the denominators first. If they match, work with the numerators. If they do not match, find a common denominator and rename both fractions.

Then add or subtract the numerators. Keep the common denominator. Finally, check that the answer makes sense: adding should not make the amount smaller, and subtracting a smaller amount from a larger amount should leave less than you started with.

You may be able to write an answer in a simpler equivalent form. For example,

\frac{2}{4}

has the same value as

\frac{1}{2}

. This is a way to show the same amount, not a change to the answer.

Keep equal-sized parts throughout the calculation.
Use a common denominator before combining unlike fractions.
Check the size of the result against the amounts in the question.

Try the method

When you practise, first say what the denominator means. Then decide whether the fractions already have equal-sized parts. If they do not, show how each fraction is renamed. These steps make it easier to spot an error.

For subtraction, check that the amount being taken away is no greater than the starting amount when the question is about a positive amount. For addition, check that the total is greater than either starting amount. These checks do not replace the calculation; they help you notice an answer that cannot fit the question.

Show the renaming step for unlike denominators.
Use the meaning of the question to check whether an answer is reasonable.

Worked example

Add like fractions

Add

\frac{2}{7}+\frac{3}{7}

.

Two sevenths plus three sevenths

One whole is split into 7 equal parts. Two parts and 3 more parts make 5 of the 7 parts.

Notice the part size
Both fractions name sevenths. The pieces are already the same size, so no renaming is needed.
$7=7$
Combine the pieces
Add the number of sevenths. Keep the denominator because the pieces are still sevenths.
$\frac{2}{7}+\frac{3}{7}=\frac{2+3}{7}=\frac{5}{7}$

Answer:

\frac{5}{7}

Check: Five sevenths is greater than either starting amount, as an addition answer should be.

Worked example

Add unlike fractions

Add

\frac{1}{3}+\frac{1}{4}

.

One third and one fourth in twelfths

The whole is split into 12 equal parts. One third is 4 parts; one fourth is 3 parts. Together they are 7 parts.

Choose equal-sized parts
Thirds and fourths are different sizes. Twelfths work because both 3 and 4 divide evenly into 12.
$12=3\times4$
Rename each fraction
To change thirds into twelfths, split each third into 4 equal parts. To change fourths into twelfths, split each fourth into 3 equal parts. Each fraction keeps its value.
$\frac{1}{3}=\frac{4}{12},\quad \frac{1}{4}=\frac{3}{12}$
Add the twelfths
Now both fractions name twelfths. Add the numerators and keep 12 as the denominator.
$\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$

Answer:

\frac{7}{12}

Check: The answer is greater than one third and one fourth. Seven twelfths is a reasonable total.

Worked example

Subtract unlike fractions

Subtract

\frac{5}{6}-\frac{1}{4}

.

Five sixths take away one fourth

Rename both amounts as twelfths: 10 twelfths take away 3 twelfths leaves 7 twelfths.

Find a common denominator
Sixths and fourths have different-sized parts. Twelfths can name both because 6 and 4 divide evenly into 12.
$12=6\times2=4\times3$
Rename both fractions
Rename five sixths as tenths? No: split each sixth into 2 parts, making twelfths. Split each fourth into 3 parts, also making twelfths.
$\frac{5}{6}=\frac{10}{12},\quad \frac{1}{4}=\frac{3}{12}$
Subtract equal-sized parts
Take 3 twelfths away from 10 twelfths. Seven twelfths remain.
$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$

Answer:

\frac{7}{12}

Check: Seven twelfths is less than the starting amount of five sixths, as expected after taking away a positive amount.

Common mistakes and how to avoid them

Adding or subtracting the denominators as well as the numerators.

Correction: The denominator tells the size of the parts. Keep it when the parts are equal-sized. For unlike fractions, first rename both fractions with a common denominator.

Adding the numerators of unlike fractions before renaming them.

Correction: A third and a fourth are not equal-sized pieces. Rename them as equivalent fractions with the same denominator before combining.

Changing the denominator but not the numerator when making an equivalent fraction.

Correction: Multiply both parts of the fraction by the same number. This keeps the fraction's value the same.

Thinking that a larger denominator always means a larger fraction.

Correction: The denominator names how many equal parts make a whole. More parts mean each part is smaller, so compare amounts only after thinking about the part sizes.

Lesson summary

For like fractions, add or subtract the numerators and keep the denominator.
For unlike fractions, rename both as equivalent fractions with a common denominator.
After renaming, combine the numerators and keep the common denominator.
Check that the answer fits the addition or subtraction in the question.

Check your understanding

Question 1

What is

\frac{3}{8}+\frac{2}{8}

?

$\frac{5}{8}$
$\frac{5}{16}$
$\frac{6}{8}$
correctIndex

Show answer and explanation

\frac{5}{8}

The denominators are equal, so add the numerators and keep eighths:

\frac{3}{8}+\frac{2}{8}=\frac{5}{8}

.

Question 2

What is

\frac{1}{2}+\frac{1}{3}

?

$\frac{2}{5}$
$\frac{5}{6}$
$\frac{2}{6}$
correctIndex

Show answer and explanation

\frac{5}{6}

Rename one half as three sixths and one third as two sixths. Then add:

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

.

Question 3

What is

\frac{3}{4}-\frac{1}{2}

?

$\frac{2}{2}$
$\frac{2}{4}$
$\frac{1}{4}$
correctIndex

Show answer and explanation

\frac{2}{4}

Rename one half as two fourths. Then subtract:

\frac{3}{4}-\frac{2}{4}=\frac{1}{4}

. Correction: the correct option should be

\frac{1}{4}

.

Key terms

Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Numerator: The top number in a fraction. It tells how many parts are being counted.
Like fractions: Fractions with the same denominator, so their parts have the same size.
Unlike fractions: Fractions with different denominators, so their parts have different sizes.
Equivalent fractions: Fractions written differently that name the same amount.
Common denominator: A denominator that can be used to rename each fraction in a calculation.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B2.5 · Add and subtract fractions with like and unlike denominators

Ontario Grade 6 Mathematics

Grade 6 Mathematics — B2.5

What you will learn

Start with equal-sized parts

Make the parts the same size

A careful method

Try the method

Add like fractions

Add unlike fractions

Subtract unlike fractions

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review