Two kinds and what is left

Primary 5 Four Operations of Fractions study guide: the whole chapter, step by step

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Rewrite each part on the same whole before using the difference between two parts to find one unit.

Lesson video, with the working

Two kinds and what is left

2 min 45 s

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Today's question

Wei Jie had a book of used and unused stamps. He gave away 2/3 of them. 1/2 of the stamps he gave away were used. 1/4 of the stamps left were used. He gave away 21 more used stamps than the used stamps left. How many stamps were in the book at first?

  1. 12 units = 12 × 7 = 84 stamps

Ans: 84

How to solve it
  1. The total is unknown, so use equal units. Make the whole book 12 units, so thirds, halves and quarters all split evenly.
  2. Two thirds of 12 units is 8 units given away. That leaves 12 minus 8, which is 4 units in the book.
  3. Find the used stamps: one half of 8 units is 4 units given away, and one quarter of 4 units is 1 unit left.
  4. Compare: 4 units minus 1 unit is 3 units, which is 21 stamps. So 1 unit is 21 divided by 3, which is 7.
  5. The book is 12 units, so 12 times 7 is 84 stamps at first.

One half refers to the stamps given away. One quarter refers to the stamps left. These fractions refer to different parts of the book. Equal-sized units let us compare the two counts.

Practice questions with worked solutions

Question 1 is free to try. The other 6 open in the free trial, each with its full working.

Basic

1Free2

Intermediate

34

Challenge

567

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6 more questions, each with its full working.Answers checked, progress kept, wrong ones come back in revision. No card needed.

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Question 1 of 7 · Basic · free to try1/3 of the books on a shelf are storybooks. 3/4 of the remainder are comics…1/3 of the books on a shelf are…

1/3 of the books on a shelf are storybooks. 3/4 of the remainder are comics and the rest are magazines. There are 24 magazines. How many books are on the shelf?

storybooksthe restcomicsmagazines
books

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Video solution · 1 min 23 s

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Working

  1. lowest common multiple of 3 and 4 = 12
  2. the books = 12 units
  3. storybooks = 1/3 × 12 units = 4 units
  4. remainder = 12 units − 4 units = 8 units
  5. comics = 3/4 × 8 units = 6 units
  6. magazines = 8 units − 6 units = 2 units
  7. 2 units = 24 books
  8. 1 unit = 24 ÷ 2 = 12 books
  9. 12 units = 12 × 12 = 144 books

Answer: 144 books

For parents

How do you solve P5 fraction remainder questions with three parts?

These word problems split a whole into three kinds: a fraction of the whole, a fraction of the remainder, and the rest.

What your child learns here

These word problems split a whole into three kinds: a fraction of the whole, a fraction of the remainder, and the rest. This page shows how to rewrite all three as units of one whole, applying the Primary 5 outcomes on multiplying fractions to multi-step word problems.

  • Choose a number of units for the whole that both fractions split evenly.
  • Take the first fraction of the whole, then the second fraction of what is left, not of the whole.
  • Write each of the three kinds as a number of units of the same whole.
  • Match the given count, or the given difference between two kinds, to its units to find one unit.
  • Build the answer the question asks for: the whole, one kind or the rest.
Does the answer make sense?

When the question asks for the whole, the answer must be bigger than every count given, because each of the three kinds is only part of it. When it asks for one kind, that kind must be smaller than the whole your child found along the way.

How to practise these questions

Basic questions train drawing that two-stage model and working from the count of the last group, or of the middle group, back to the whole. Intermediate questions give the first group's count instead and ask for the last, so the model is read forwards. In Challenge questions the number given is a difference between two groups, so units are compared before they are valued. When the remainder model gets tangled, watch that question's solution film, then attempt it again from Revision later.

Questions parents ask

Why do we choose 12 units for the whole? The number of units has to split evenly at both steps. For one third and then one quarter of the rest, 12 works because one third of 12 is 4 and one quarter of the 8 left is 2.

Does my child have to use the unit method? The unit method keeps all three kinds on one whole, which makes them easy to compare. The lesson on this page uses it with a bar, and the working follows the bar line by line.

Is this in the Primary 5 syllabus? It applies the Primary 5 outcomes on multiplying fractions to multi-step word problems. The syllabus lists the operations; this page shows how they combine in one question.