Simplifying Fractions

Simplifying does not change what a fraction is worth. It only changes how it is written, using the smallest numbers that still say the same thing.

Grades 5 to 7Number8 min read7 practice questions

What simplifying means

Simplifying a fraction means rewriting it with the smallest possible numbers, without changing its value.

Look at these two bars. The top one is cut into eighteenths with twelve shaded. The bottom one is cut into thirds with two shaded. The shading reaches exactly the same point.

121812 eighteenths232 thirds
Same amount, fewer cuts. Simplifying is just choosing the tidier way to describe the shaded part.

So 1218 and 23 are the same number. Teachers ask for the second one because it is easier to picture, easier to compare and easier to work with.

How to do it

The method is the equivalent fraction rule running backwards. Instead of multiplying top and bottom, you divide them.

The rule

Find a number that divides exactly into both the top and the bottom. Divide both by it. Repeat until nothing divides into both except 1.

121869=23=÷ 2÷ 2÷ 3÷ 3
Two small steps get you there. Halve both numbers, then divide both by 3. Each step keeps the value and shrinks the digits.

You do not have to find the perfect divisor first time. Dividing by 2 twice is just as correct as dividing by 4 once, and it is far easier to spot.

The one-step shortcut

If you can find the highest common factor, the largest number that divides into both, you get there in a single step.

For 1218:

  • Factors of 12 are 1, 2, 3, 4, 6, 12.
  • Factors of 18 are 1, 2, 3, 6, 9, 18.
  • They share 1, 2, 3 and 6. The highest is 6.
  • 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so the answer is 23 straight away.

Worth knowing, but not worth stressing about. Under exam pressure, dividing by 2 and 3 repeatedly is usually faster than hunting for the highest common factor.

Knowing when to stop

This is where marks actually get lost. A child divides once, gets smaller numbers and thinks the job is done.

The finished check

A fraction is in its lowest terms when the only number that divides into both the top and the bottom is 1. Run through 2, 3, 5 and 7 in your head. If none of them go into both, you are done.

Quick habit: if both numbers are even, you are definitely not finished.

Worked examples

1

Simplify in stages

Write 2436 in its lowest terms.
  1. Both numbers are even, so halve them. 24 ÷ 2 = 12, 36 ÷ 2 = 18, giving 1218.
  2. Still both even, so halve again. 12 ÷ 2 = 6, 18 ÷ 2 = 9, giving 69.
  3. Now 6 and 9 are not both even, but 3 goes into both. 6 ÷ 3 = 2, 9 ÷ 3 = 3, giving 23.
  4. Check: does 2, 3, 5 or 7 divide into both 2 and 3? No. Finished.
2436 = 23
2

One step with the highest common factor

Simplify 4560.
  1. Both end in 5 or 0, so 5 divides into both. That is a good place to start.
  2. 45 ÷ 5 = 9 and 60 ÷ 5 = 12, giving 912.
  3. Now 3 divides into both. 9 ÷ 3 = 3 and 12 ÷ 3 = 4, giving 34.
  4. The highest common factor was 15, so 45 ÷ 15 = 3 and 60 ÷ 15 = 4 would have done it in one move. Same answer either way.
4560 = 34
3

A fraction that will not simplify

Simplify 715.
  1. 7 is prime, so the only numbers that divide into it are 1 and 7.
  2. Does 7 divide into 15? No, 15 ÷ 7 does not give a whole number.
  3. So the only shared factor is 1, which means there is nothing to cancel.
  4. Writing "cannot be simplified" is the full and correct answer here. It is not a trick question.
715 is already in its lowest terms

Common mistakes

1. Stopping too early

Turning 2436 into 1218 and putting the pen down. It is a correct equivalent fraction, but it is not simplified, and in most mark schemes that loses the mark. Always run the finished check.

2. Subtracting instead of dividing

Turning 69 into 36 by taking 3 off each number. Subtracting changes the value completely: 69 is two thirds, while 36 is a half. Only dividing keeps the amount the same.

3. Dividing only one of the two numbers

Turning 1015 into 215 by dividing the top by 5 and leaving the bottom. Both numbers move or neither does. This is the same rule as equivalent fractions, and it does not bend.

Practice questions

Seven questions. One attempt each, and the explanation appears once the answer is in.

Check yourself

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For parents

If your child gets the right idea but keeps stopping one step early, the problem is a missing habit rather than missing knowledge. Get them to write the finished check under every answer for a week: does 2, 3, 5 or 7 go into both? It takes five seconds and it turns a recurring lost mark into a solved problem.

What to learn next

Equivalent fractionsMath · Grades 4 to 6Comparing fractionsMath · Grades 4 to 6Adding fractions, different denominatorsMath · Grades 5 to 7Fractions to decimalsMath · Grades 5 to 7

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