Grade by Grade
B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Simplifying Fractions

B1.4About 40 minutes
Learning goals
  • I can use equivalent fractions to write a fraction in simplest form.
  • I can use a common factor, or the greatest common factor, to simplify.
  • I can simplify negative fractions and improper fractions.
  • I can decide when a simplified fraction is helpful and when the original fraction tells more.

Words to know

equivalent fractions
Fractions that name the same amount, like 6/8 and 3/4.
simplify
Write an equivalent fraction with a smaller numerator and denominator by dividing both by a common factor.
simplest form
A fraction whose numerator and denominator have no common factor other than 1, like 3/4.
common factor
A number that divides into two or more numbers with no remainder. 2 is a common factor of 6 and 8.
greatest common factor (GCF)
The largest factor two numbers share. The GCF of 24 and 36 is 12.

Remember

In Grade 5 you made equivalent fractions by multiplying or dividing the numerator and denominator by the same number. In Grade 6 you wrote decimals as fractions in simplest form, like 0.75 = 75/100 = 3/4, and learned divisibility rules. Now you will simplify all kinds of fractions and decide when simplifying is a good idea.

Two answers, same amount

Liam and Zoe each ate part of a tray of maple fudge.

  • Liam says, "I ate 6/8 of a row."
  • Zoe says, "Liam ate 3/4 of a row."

They are both right! 6/8 and 3/4 are equivalent fractions. 3/4 uses smaller numbers, so it is in simplest form.

A baking tray of light brown maple fudge cut into square pieces in rows.
Fudge cut into equal pieces is a good model for fractions. Eating 6 of 8 pieces in a row is the same amount as 3 of 4. AI-generated image
6/83/4
6/8 and 3/4 cover the same part of the same-sized strip. They are equivalent.

How to simplify a fraction

Divide the numerator and the denominator by the same common factor. This makes bigger pieces, but the same amount.

  • 6/8: both 6 and 8 can be divided by 2. 6 ÷ 2 = 3 and 8 ÷ 2 = 4. So 6/8 = 3/4.
  • Keep going until the only common factor is 1. Then the fraction is in simplest form.

Fast way: divide by the greatest common factor (GCF) and you are done in one step. (You will practise finding the GCF in the Operations unit. For now, you can list the factors of both numbers and pick the largest one they share.)

Worked example

Example 1: Simplify in steps

Write 24/36 in simplest form.

  1. Both are even. Divide by 2: 24/36 = 12/18.
  2. Both are still even. Divide by 2: 12/18 = 6/9.
  3. Both 6 and 9 can be divided by 3: 6/9 = 2/3.
  4. 2 and 3 have no common factor except 1. Stop.

24/36 = 2/3

Worked example

Example 2: Simplify in one step

Write 24/36 in simplest form using the GCF.

  1. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  2. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
  3. The GCF is 12. 24 ÷ 12 = 2 and 36 ÷ 12 = 3.

24/36 = 2/3, the same answer in one step.

243612186923÷ 2÷ 2÷ 3÷ 12 (the GCF): one step2 × 2 × 3 = 12
Dividing by 2, then 2, then 3 does the same job as dividing by the GCF, 12, all at once. Both paths land on 2/3.
Worked example

Example 3: Negative and improper fractions

Simplify −45/60 and 18/12.

  1. −45/60: the GCF of 45 and 60 is 15. 45 ÷ 15 = 3 and 60 ÷ 15 = 4. Keep the negative sign: −3/4.
  2. 18/12: the GCF of 18 and 12 is 6. 18 ÷ 6 = 3 and 12 ÷ 6 = 2: 3/2, which is 1 1/2.

−45/60 = −3/4 and 18/12 = 3/2 = 1 1/2

Worked example

Example 4: When is simplifying appropriate?

On a quiz, Sara got 18 out of 24 questions right. Should she write her score as 18/24 or 3/4?

  1. 18/24 = 3/4 (divide both by 6). They are the same amount.
  2. 3/4 is easy to compare with other scores. It quickly tells you she got three-quarters right.
  3. 18/24 tells you something 3/4 doesn't: there were 24 questions and she got 18 right.

Both are correct. Simplify when it makes the fraction easier to understand or compare. Keep the original when the actual numbers matter, like a quiz score or a recipe measured with a 1/8 cup scoop.

Divisibility rules help

Look for common factors with the rules you learned in Grade 6. Both even? Divide by 2. Both digit sums divisible by 3? Divide by 3. Both end in 0 or 5? Divide by 5. A fraction is in simplest form when the numerator and denominator have no common factor except 1.

Hands-on activity

Egg carton fractions

You need: an empty 12-cup egg carton and 12 small objects (like dried pasta, coins or balls of paper).

  1. An egg carton has 2 rows of 6 cups. Fill the cups two at a time, across the carton (one cup from each row), starting at one end, until 8 cups are full. What fraction is filled? (8/12)
  2. Now think of each pair of cups across the carton as one group. How many groups are there? How many are full? Write the fraction (4/6).
  3. Now make groups of 4 cups (two pairs side by side). How many groups? How many full? (2/3)
  4. Try thinking of the carton as 4 groups of 3 cups. Can every group be completely full or completely empty? (No: 8 full cups can't be split into groups of 3, because 3 is not a factor of 8.)
  5. Which fraction is in simplest form? Why can't you make bigger groups?

What to look for: 8/12 = 4/6 = 2/3. Groups of 4 are the biggest groups that work because 4 is the greatest common factor of 8 and 12.

Just for fun

6/8 said to 3/4, "Hey, stop copying me!"

3/4 replied, "I'm not copying you. I'm just you in simplest form."

Your turn

1

Simplify 4/10.

2

Simplify 9/12.

3

Simplify 15/20.

4

Simplify 16/24.

5

Simplify −10/25.

6

Simplify 14/21.

7

Is 7/12 in simplest form? Explain.

More practice: warm-up

1

Simplify 2/4.

2

Simplify 5/10.

3

Simplify 3/9.

4

Simplify 6/8.

5

Simplify 8/10.

More practice: on your own

1

Simplify 36/48.

2

Simplify 45/75.

3

Simplify −28/35.

4

Simplify 30/12 and write it as a mixed number.

5

In a class of 30 students, 18 walk to school. What fraction walk? Write it in simplest form.

Challenge

1

Write three fractions equivalent to 2/3 with denominators between 20 and 40.

2

Liam simplified 26/39 to 2/3 in one step. What number did he divide by? How do you know it's right?

3

A hockey team won 32 of 48 games. The coach's report says "we won 2/3 of our games". A newspaper says "they won 32 of 48 games". Are both correct? Which tells you more?

Check your understanding

Quiz

1.

What is 12/18 in simplest form?

2.

Which fraction is already in simplest form?

3.

To simplify 20/32 to simplest form in one step, divide both by...

4.

What is −35/50 in simplest form?

5.

Sam says 16/20 = 6/10 because he subtracted 10 from both. What is 16/20 in simplest form?

6.

When would you keep a quiz score as 18/24 instead of writing 3/4?

For parents and teachers

Big idea: Grade 7 expectation B1.4 asks students to use equivalent fractions to simplify fractions, when appropriate, in various contexts. Finding the greatest common factor is its own expectation (B2.6) in the Operations unit; here we use common factors and the GCF as a tool.

Common difficulties:

  • Subtracting instead of dividing (16/20 → 6/10). Equivalent fractions come from multiplying or dividing both parts by the same number.
  • Stopping too early (12/18 → 6/9). Ask: "Is there another common factor besides 1?"
  • Dropping the negative sign.
  • Thinking a simplified fraction is "more correct". It is the same amount. Talk about when the original numbers matter.

How to help: Do the egg carton activity. In recipes, sports stats and test scores, ask "Could we simplify this? Would it help?"