Finding a Fraction of a Number

Three quarters of 20, two fifths of 45. Two steps in a fixed order, and the reason for the order is the thing worth teaching, because a child who understands it will never divide by the wrong number.

Fractions · topic 7 of 9Grades 4 to 6Number7 min read7 practice questions

The picture

Suppose there are 20 counters and you want three quarters of them.

Quarters means four equal groups, so share the 20 counters into 4 groups and then take 3 of them.

555520 shared into 4 equal groups
20 shared into 4 equal groups gives 5 in each. Take three of the groups and you have 15.

The method is nothing more than that picture written down. The bottom number told you how many groups to make and the top number told you how many of them to keep, which is the same reading order children were taught for what a fraction means in the first place.

Divide then multiply

The rule

Divide by the bottom number. Multiply by the top number.

For 34 of 20: 20 ÷ 4 = 5, then 5 × 3 = 15.

Read the fraction from the bottom upwards and the two steps come out in the right order without anybody having to remember a rule: cut into 4, take 3.

Why that order

Multiplying first and dividing second gets you to the same place. 20 × 3 = 60, then 60 ÷ 4 = 15, and there is nothing wrong with that route.

Dividing first is still worth preferring, for two reasons that matter under exam conditions. The numbers stay small enough to handle in your head, and the steps match the picture, so a child who has forgotten the rule entirely can draw four groups and work it out from scratch.

When dividing first does not work neatly

For 23 of 10, dividing first gives 10 ÷ 3, which is not a whole number. Multiply first instead: 10 × 2 = 20, then 20 ÷ 3, which is 203 or 6 and two thirds. Same answer, less mess.

Worked examples

1

A unit fraction

Find 15 of 45.
45 counters to start withThe bottom number is 5, so share them into 5 equal groups of 9The top number is 1, so keep 1 of the 5 groups1/5 of 45 is 9
Follow it downwards. The counters never change size, because sharing them out does not change how many there are.
  1. The bottom number is 5, so share 45 into 5 equal groups.
  2. 45 ÷ 5 = 9, so each group has 9.
  3. The top number is 1, so take just one group.
  4. Multiplying by 1 changes nothing, so the answer is 9.
15 of 45 is 9
2

More than one part

Find 25 of 45.
45 counters to start withThe bottom number is 5, so share them into 5 equal groups of 9The top number is 2, so keep 2 of the 5 groups2/5 of 45 is 18
Same 45, same five groups. Only the number of groups you keep has changed.
  1. Same first step. 45 ÷ 5 = 9 in each group.
  2. This time the top number is 2, so take two groups.
  3. 9 × 2 = 18.
  4. Sense check: two fifths is a bit less than half, and 18 is a bit less than half of 45. Good.
25 of 45 is 18
3

A money question

A jacket costs 60 dollars. It is reduced by 13. What is the new price?
  1. First find the discount. 60 ÷ 3 = 20, so a third of 60 is 20 dollars off.
  2. The question asks for the new price, not the discount, so keep going.
  3. 60 − 20 = 40.
  4. Faster route: taking a third off leaves two thirds. 20 × 2 = 40 gets there in one step.
The jacket now costs 40 dollars

Where this shows up

Unlike a good deal of school maths, this one gets used constantly in ordinary life, and saying so out loud changes how willingly a child learns it.

  • Sales. A third off, a quarter off, half price.
  • Recipes. Making three quarters of a recipe that serves 8.
  • Sharing. Splitting a bill or a bag of sweets unevenly but fairly.
  • Percentages later on. 25 percent is just 14, and 75 percent is 34. A child solid on this topic finds percentages easy.

Common mistakes

1. Dividing by the top number

For 34 of 20 the child works out 20 ÷ 3, usually because 3 is the number they read first. The bottom number is the one that does the sharing, since it is the one that says how many equal groups the whole gets cut into, and reading the fraction from the bottom upwards puts the steps in the right order automatically.

2. Stopping after the division

20 ÷ 4 = 5 gets written down as the answer to "three quarters of 20", which is one quarter of it. The division felt like the hard part, so it feels like the work is done, and unless the top number happens to be 1 there is always a second step waiting.

3. Answering the wrong question in word problems

The discount gets calculated correctly and written down, when the question asked what you actually pay. Across a whole test this costs more marks than every arithmetic slip combined, and the cure is to read the final line of the question again immediately before writing anything.

Practice questions

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

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For parents

Practise this in shops rather than on paper. Point at a sign offering a third off and ask what the new price will be before you get to the till, where the answer is about to be checked in public by a machine. Real money and immediate feedback teach this faster than any worksheet, and ten of those conversations will do more than a page of exercises.

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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