Simplifying Fractions

Simplifying changes nothing about what a fraction is worth. It changes how it looks, and most of the marks lost here go to children who stopped one step too early.

Fractions · topic 3 of 9Grades 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 the two bars below. One has been cut into eighteen pieces with twelve of them shaded, the other into three pieces with two shaded, and the shading reaches the same point on both.

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 version not out of fussiness but because nobody can picture twelve eighteenths, while everybody can picture two thirds, and the smaller numbers make every later step easier to carry out.

How to do it

There is no new method to learn here. It is the equivalent fraction rule pointed the other way: instead of multiplying the 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, and children who think they do tend to freeze. Halving twice gets you to the same place as dividing by four once, and spotting that both numbers are even is far easier under pressure than working out a highest common factor.

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.

Useful when it comes quickly, not worth hunting for when it does not. In an exam, halving and dividing by three until nothing goes in beats staring at a fraction trying to spot the perfect number.

Knowing when to stop

Here is where the marks actually go. A child divides once, sees numbers that are visibly smaller than the ones they started with, and quite reasonably concludes that they have simplified the fraction. They have. They have just not finished.

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.

One habit catches most of it: if both numbers are still even, you are not finished, no matter how tidy the fraction looks.

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

2436 becomes 1218 and the pen goes down. Nothing about that is wrong, which is what makes it so frustrating to mark: the value is right, the working is right, and the answer still scores zero because the question asked for lowest terms. The finished check takes five seconds and it is the difference.

2. Subtracting instead of dividing

Three gets taken off each number, so 69 turns into 36. Both numbers were treated identically, which is why it feels legitimate, but the amount has quietly changed from two thirds to a half. Dividing preserves value and subtracting does not, and that difference is worth being explicit about rather than assuming a child will infer it.

3. Dividing only one of the two numbers

The top gets divided by 5 and the bottom is left alone, giving 215. This is the same rule that governs equivalent fractions and it does not bend for simplifying: both numbers move, or neither does.

Practice questions

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

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For parents

When a child understands this perfectly and still keeps stopping early, the problem is a missing habit, not missing knowledge, and habits are fixed by repetition rather than explanation. Have them write the finished check underneath every single answer for a week, in full: does 2, 3, 5 or 7 divide into both of these? After about twenty of those it becomes automatic and the lost marks stop.

Keep going

Maths CheckTwelve questions, five minutes, then a printable report naming what to work on next. Free, no sign up.OpenShow me the workingLong division, long multiplication and column methods, walked step by step on your own numbers.Open

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