Add and subtract fractions using equivalent fractions

Ontario Grade 7 Mathematics

Number

Ontario Grade 7 Mathematics — B2.5

Imagine one sandwich cut into 4 equal pieces and another sandwich of the same size cut into 6 equal pieces. A piece from the first sandwich is not the same size as a piece from the second. To combine the fractions, describe both amounts using pieces of the same size. Equivalent fractions let us do that. In this lesson, you will use equivalent fractions to add and subtract, and you will check that your answer makes sense.

What you will learn

Explain why fractions need the same-sized parts before they can be added or subtracted.
Write an equivalent fraction with a chosen denominator.
Add and subtract fractions by using equivalent fractions.
Simplify an answer when its numerator and denominator have a common factor.

Fractions name equal parts

A fraction describes some equal parts of one whole. In

\frac{3}{5}

, the denominator,

5

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

3

, tells how many of those parts are counted.

Equivalent fractions name the same amount, but use different-sized parts. For example,

\frac{1}{2}

and

\frac{3}{6}

are equal amounts. If each half is split into 3 equal pieces, there are 6 pieces in the whole and 3 pieces in the half.

When adding or subtracting fractions, the parts must be the same size. Fractions with the same denominator already name parts of the same size. If the denominators differ, rewrite one or both fractions as equivalent fractions with a shared denominator.

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

The denominator tells the number of equal parts in one whole.
The numerator tells how many parts are counted.
Equivalent fractions have the same value.

Use a shared denominator

A shared denominator is a denominator that works for both fractions. For example, sixths can describe both halves and thirds: one half is three sixths, and one third is two sixths.

To make an equivalent fraction, multiply the numerator and denominator by the same number. This changes the names of the parts, but not the amount. For example, multiplying both parts of

\frac{1}{3}

by 2 gives

\frac{2}{6}

.

Once the parts are the same size, add or subtract the numerators. Keep the denominator because the size of each part has not changed. Do not add or subtract the denominators.

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

Rewrite fractions so they have a shared denominator.
Multiply the numerator and denominator by the same number to make an equivalent fraction.
Add or subtract the numerators; keep the shared denominator.

Check and simplify the result

A sum is the answer to an addition problem. A difference is the answer to a subtraction problem. After finding either one, check whether the numerator and denominator can both be divided by the same whole number greater than 1. If they can, divide both by that number to write an equivalent fraction in simpler form.

For example,

\frac{4}{8}

can be simplified because 4 and 8 can both be divided by 4. It becomes

\frac{1}{2}

. The amount stays the same.

You can also check whether your answer is reasonable. When adding positive fractions, the total should be greater than either fraction. When subtracting a smaller fraction from a larger one, the difference should be less than the starting amount.

\frac{4}{8}=\frac{1}{2}

Simplify only by dividing the numerator and denominator by the same number.
An equivalent simplified fraction names the same amount.
Use the sizes of the starting fractions to check whether an answer is reasonable.

Try the method yourself

For each problem, first decide how to make the parts the same size. Then rewrite the fractions, do the operation on the numerators, and keep the denominator. Finally, check whether the result can be simplified.

Practice: find

\frac{1}{4}+\frac{1}{6}

,

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

, and

\frac{2}{3}+\frac{1}{6}

. For the subtraction problem, think about why the first fraction must be rewritten too.

Before you look at the quick check, explain in your own words why the denominator stays the same once the fractions have matching denominators.

Equivalent fractions let different fractions use parts of the same size.
After rewriting, the numerator operation counts the combined or remaining parts.

Worked example

Add halves and thirds

Find

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

. Give the answer in simplest form.

Five of six equal parts

One whole is divided into 6 equal parts. Five parts are shaded to represent the sum.

Choose a shared denominator
Sixths work because one half can be split into 3 equal parts and one third can be split into 2 equal parts. That gives both fractions the same-sized pieces.
$\frac{1}{2}=\frac{3}{6},\quad \frac{1}{3}=\frac{2}{6}$
Add the matching parts
There are 3 sixths and 2 sixths. Together, they make 5 sixths. Keep the denominator because the pieces are still sixths.
$\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Answer:

\frac{5}{6}

Check: The answer is greater than one half and greater than one third, as a sum of positive fractions should be.

Worked example

Subtract fourths from sixths

Find

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

. Give the answer in simplest form.

Seven of twelve equal parts

One whole is divided into 12 equal parts. Seven parts represent the difference.

Rewrite both fractions
Twelfths work for both fractions. Change each fraction into an equivalent fraction with denominator 12 so the parts have the same size.
$\frac{5}{6}=\frac{10}{12},\quad \frac{1}{4}=\frac{3}{12}$
Subtract the parts
Take away 3 twelfths from 10 twelfths. There are 7 twelfths left. The fraction cannot be simplified further because 7 and 12 have no shared whole-number factor greater than 1.
$\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$

Answer:

\frac{7}{12}

Check: The answer is positive and less than

\frac{5}{6}

, which fits taking a positive amount away from

\frac{5}{6}

.

Worked example

Add, then simplify

Find

\frac{3}{8}+\frac{1}{4}

. Give the answer in simplest form.

Five of eight equal parts

One whole is divided into 8 equal parts. Five parts represent the sum.

Make eighths
The first fraction is already in eighths. Rewrite one fourth as an equivalent fraction in eighths by multiplying its numerator and denominator by 2.
$\frac{1}{4}=\frac{2}{8}$
Add and simplify
Add 3 eighths and 2 eighths to get 5 eighths. Five and eight have no shared whole-number factor greater than 1, so this is simplest form.
$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$

Answer:

\frac{5}{8}

Check: Five eighths is greater than three eighths, as expected when one fourth is added.

Common mistakes and how to avoid them

Adding both numerators and denominators, such as writing

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

.

Correction: First rewrite the fractions with the same denominator. Then add only the numerators and keep that denominator.

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

Correction: Multiply the numerator and denominator by the same number. This keeps the fraction's value unchanged.

Subtracting the denominators after rewriting the fractions.

Correction: The denominator names the size of each part. When the parts match, subtract the number of parts and keep the denominator.

Simplifying by dividing the numerator and denominator by different numbers.

Correction: Divide both by the same whole number so the fraction keeps the same value.

Lesson summary

Fractions need the same-sized parts before you add or subtract them.
Use equivalent fractions to give both fractions a shared denominator.
Add or subtract the numerators and keep the shared denominator.
Check whether the result can be simplified by dividing its numerator and denominator by the same number.

Check your understanding

Question 1

What is

\frac{1}{3}+\frac{1}{6}

?

$\frac{2}{9}$
$\frac{1}{2}$
$\frac{2}{6}$
$\frac{1}{9}$

Show answer and explanation

\frac{1}{2}

Rewrite one third as two sixths. Then add: two sixths plus one sixth is three sixths, which is equivalent to one half.

Question 2

What is

\frac{7}{8}-\frac{1}{4}

?

$\frac{6}{4}$
$\frac{5}{8}$
$\frac{6}{8}$
$\frac{3}{4}$

Show answer and explanation

\frac{5}{8}

One fourth is two eighths. Subtract two eighths from seven eighths to get five eighths.

Question 3

Which fraction is equivalent to

\frac{2}{5}

and has denominator 15?

$\frac{6}{15}$
$\frac{2}{15}$
$\frac{5}{15}$
$\frac{10}{15}$

Show answer and explanation

\frac{6}{15}

Multiply both the numerator and denominator of two fifths by 3. The equivalent fraction is six fifteenths.

Key terms

Numerator: The top number in a fraction. It tells how many equal parts are counted.
Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Equivalent fractions: Fractions that name the same amount, even though their numerators and denominators may differ.
Shared denominator: A denominator used by both fractions after they are rewritten as equivalent fractions.
Simplest form: A fraction that cannot be made simpler by dividing its numerator and denominator by the same whole number greater than 1.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 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 using equivalent fractions

Ontario Grade 7 Mathematics

Ontario Grade 7 Mathematics — B2.5

What you will learn

Fractions name equal parts

Use a shared denominator

Check and simplify the result

Try the method yourself

Add halves and thirds

Subtract fourths from sixths

Add, then simplify

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review