Simplify fractions using equivalent fractions

Ontario Grade 7 Mathematics

Number

Keep the amount the same while making the numbers smaller

A fraction names equal parts of one whole. If a strip is split into 8 equal parts and 6 are shaded, the shaded amount is six eighths. Pairing the small parts makes 4 larger equal parts, 3 of them shaded. The amount has not changed. It can be named as six eighths or three fourths. Simplifying uses this idea to write the same amount with smaller numbers.

What you will learn

Explain how equivalent fractions name the same amount.
Simplify a fraction by dividing its numerator and denominator by the same number.
Check that a fraction is in simplest form.

See the amount before the symbols

The numerator is the top number in a fraction. It tells how many parts are counted. The denominator is the bottom number. It tells how many equal parts make one whole.

Imagine a strip split into 8 equal parts, with 6 shaded. Pair the small parts. The whole strip now has 4 pairs, and the shaded area has 3 pairs. The shaded amount is still the same. Only the way we count the parts has changed.

Fractions that name the same amount are called equivalent fractions. For example, six eighths and three fourths are equivalent. A fraction is in simplest form when its numerator and denominator have no shared whole-number divisor greater than 1.

The numerator counts parts; the denominator counts equal parts in the whole.
Equivalent fractions name the same amount.
Simplifying changes the fraction numbers, not the amount.

Make equivalent fractions

To simplify, divide the numerator and denominator by the same whole number. Choose a number that divides both evenly, with no remainder. This number is a shared divisor.

Why do both numbers change together? If we group small parts into pairs, the number of parts in the whole is cut in half. The number of counted parts is also cut in half. The shaded amount stays the same.

For example, dividing both numbers in six eighths by 2 gives three fourths. Dividing only one number would change the amount. When simplifying, use the same shared divisor for the numerator and denominator.

You can also make an equivalent fraction by multiplying both numbers by the same whole number. This creates more, smaller parts without changing the amount. Simplifying goes in the other direction: it combines equal-sized parts into fewer, larger groups.

\frac{6}{8}=\frac{6\div2}{8\div2}=\frac{3}{4}

Divide both numbers by the same shared divisor.
Both divisions must give whole-number answers.
Multiplying both numbers by the same whole number also keeps a fraction equivalent.

Keep simplifying until you can stop

After one division, look at the new numerator and denominator. They might still share a divisor greater than 1. If they do, divide both again by that same divisor.

For example, eighteen twenty-fourths can first become nine twelfths. The new numbers still divide evenly by 3, so the fraction can become three fourths. At that point, no whole number greater than 1 divides both 3 and 4 evenly.

A divisor is a whole number that divides another whole number evenly. For example, 3 is a divisor of 12 because 12 divided by 3 is 4 with no remainder. A shared divisor works evenly for both numbers in the fraction.

You may simplify in one step or in several steps. Either way, check the final numbers. Do not stop just because they are smaller. Stop when no shared divisor greater than 1 remains.

Check the new fraction after every division.
A fraction in simplest form has no shared divisor greater than 1.
The same fraction can be simplified in different numbers of steps.

Explain and check your work

A good explanation names the shared divisor and says that both fraction numbers were divided by it. This shows why the new fraction is equivalent to the original.

You can check by reversing your divisions. Multiply the final numerator and denominator by the same numbers you divided by. You should get the original numerator and denominator. You can also compare shaded models: the shaded amount should not change.

Try this routine: find a shared divisor, divide both numbers by it, and check the new fraction. Repeat if needed. When no shared divisor greater than 1 remains, the fraction is in simplest form.

Name the shared divisor you used.
Check that the amount stayed the same.
Check for another shared divisor before stopping.

Worked example

Regroup shaded parts

A paper strip is divided into 8 equal parts, and 6 parts are shaded. Simplify the fraction of the strip that is shaded.

Six eighths equals three fourths

The shaded amount is 6 of 8 equal parts. Pairing the parts shows the same amount as 3 of 4.

Name the shaded fraction
There are 6 shaded parts out of 8 equal parts in the whole strip.
$\frac{6}{8}$
Divide by a shared divisor
Both 6 and 8 divide evenly by 2. Dividing both numbers by 2 groups the small parts into pairs without changing the shaded amount.
$\frac{6\div2}{8\div2}=\frac{3}{4}$
Check the result
No whole number greater than 1 divides both 3 and 4 evenly. The fraction is in simplest form.
$\frac{6}{8}=\frac{3}{4}$

Answer: The shaded fraction in simplest form is

\frac{3}{4}

.

Check: Multiplying the numerator and denominator of

\frac{3}{4}

by 2 gives

\frac{6}{8}

, confirming that the fractions are equivalent.

Worked example

Simplify in two steps

Simplify

\frac{18}{24}

.

Divide by a shared divisor
Both 18 and 24 divide evenly by 2. Divide the numerator and denominator by 2 to keep an equivalent fraction.
$\frac{18}{24}=\frac{9}{12}$
Check the new fraction
The new numbers, 9 and 12, both divide evenly by 3. Divide both by 3. This keeps the amount the same and makes the numbers smaller.
$\frac{9}{12}=\frac{3}{4}$
Confirm simplest form
No whole number greater than 1 divides both 3 and 4 evenly, so the process stops.
$\frac{18}{24}=\frac{9}{12}=\frac{3}{4}$

Answer: The simplified fraction is

\frac{3}{4}

.

Check: The divisions used 2 and then 3. Multiplying 3 and 4 by 6 gives 18 and 24, so

\frac{3}{4}

is equivalent to

\frac{18}{24}

.

Worked example

Simplify a fraction with larger numbers

A container is full to

\frac{35}{49}

of its capacity. Simplify this fraction.

Find a shared divisor
Both 35 and 49 divide evenly by 7. Dividing both numbers by 7 keeps the fraction equivalent.
$\frac{35\div7}{49\div7}=\frac{5}{7}$
Check whether to continue
No whole number greater than 1 divides both 5 and 7 evenly. The fraction is in simplest form.
$\frac{35}{49}=\frac{5}{7}$

Answer: The simplified fraction is

\frac{5}{7}

.

Check: Multiplying 5 and 7 by 7 gives 35 and 49, confirming that the fractions name the same amount.

Common mistakes and how to avoid them

Dividing only the denominator to make it smaller.

Correction: Divide the numerator and denominator by the same number. Otherwise, the fraction may name a different amount.

Using different divisors for the numerator and denominator in one step.

Correction: Use one shared divisor for both numbers in that step. This keeps the fraction equivalent.

Stopping after one step even though both new numbers still share a divisor greater than 1.

Correction: Check the new numerator and denominator again. Continue until no shared divisor greater than 1 remains.

Calling a fraction simplified just because its numbers are smaller.

Correction: Check whether a whole number greater than 1 divides both numbers evenly. If one does, simplify again.

Lesson summary

Equivalent fractions name the same amount.
To simplify, divide the numerator and denominator by the same shared divisor.
Keep simplifying until no whole number greater than 1 divides both numbers evenly.
Check that your final fraction names the same amount as the original.

Check your understanding

Question 1

Which fraction is the simplified form of

\frac{14}{21}

?

$\frac{2}{3}$
$\frac{7}{21}$
$\frac{14}{3}$
$\frac{3}{2}$

Show answer and explanation

\frac{2}{3}

Divide 14 and 21 by 7. This gives

\frac{2}{3}

, and 2 and 3 have no shared divisor greater than 1.

Question 2

Which change keeps

\frac{3}{5}

equivalent?

Multiply the numerator by 2 and leave the denominator unchanged.
Multiply the numerator and denominator by 2.
Add 2 to the numerator and denominator.
Divide the numerator by 3 and the denominator by 2.

Show answer and explanation

Multiply the numerator and denominator by 2.

Multiplying both numbers by the same whole number makes an equivalent fraction:

\frac{3}{5}=\frac{6}{10}

.

Question 3

Is

\frac{4}{9}

in simplest form?

Yes, because no whole number greater than 1 divides both 4 and 9 evenly.
No, divide both numbers by 2.
No, divide both numbers by 3.
Yes, because the denominator is larger than the numerator.

Show answer and explanation

Yes, because no whole number greater than 1 divides both 4 and 9 evenly.

Neither 2 nor 3 divides both 4 and 9 evenly, so there is no shared divisor greater than 1.

Key terms

Numerator: The top number in a fraction. It tells how many 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 numbers may be different.
Divisor: A whole number that divides another whole number evenly.
Simplest form: An equivalent fraction whose numerator and denominator have no shared whole-number divisor 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 B1.4. 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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B1.4 · Simplify fractions using equivalent fractions

Ontario Grade 7 Mathematics

Keep the amount the same while making the numbers smaller

What you will learn

See the amount before the symbols

Make equivalent fractions

Keep simplifying until you can stop

Explain and check your work

Regroup shaded parts

Simplify in two steps

Simplify a fraction with larger numbers

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