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PSLE Mathematics · Primary 5 and 6

Ratio

A ratio compares two or three quantities by showing how many equal parts each one has. In Primary 5 you write ratios in their simplest form and share quantities in a given ratio. In Primary 6 you link ratio to fractions and solve problems where the quantities change. Ratio thinking, built on units, is one of the most useful tools in PSLE word problems and links directly to fractions, percentage and model drawing. This guide covers each idea and ends with before-and-after practice.

What the syllabus expects

  • Write a ratio of two or three quantities and express it in its simplest form
  • Find equivalent ratios and a missing term in a pair of equivalent ratios
  • Find the ratio of two or three quantities given their values, using the same units
  • Find a quantity given the total, a part or a difference and the ratio
  • Relate a ratio to fractions, for example A : B = 2 : 3 means A is 2/5 of the total
  • Solve before-and-after problems where one quantity, the total or the difference stays the same

Writing ratios in their simplest form

A ratio compares quantities of the same kind. If a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18. Dividing both terms by their highest common factor, 6, gives the simplest form 2 : 3. The order matters, so the ratio of girls to boys is 3 : 2. A ratio has no units, but the quantities must be in the same unit before you compare them: 50 cm to 2 m is 50 : 200 = 1 : 4. Three-term ratios work the same way, with every term divided by a common factor, so 6 : 9 : 15 = 2 : 3 : 5. A ratio in its simplest form uses whole numbers only, so 1.5 : 2 becomes 3 : 4 after multiplying both terms by 2.

Ratio and fractions

A ratio tells you how many units each quantity has. If A : B = 2 : 3, then A has 2 units, B has 3 units and the total is 5 units. From this, A is 2/5 of the total, B is 3/5 of the total, A is 2/3 of B, and B is 3/2 = 1 1/2 times of A. Going the other way, if 3/7 of the beads in a box are red and the rest are blue, then red is 3 units and blue is 7 − 3 = 4 units, so red : blue = 3 : 4. Switching between fractions and ratios is often the first step in a PSLE problem, because it turns words into units you can draw in a model.

Sharing and combining ratios with the unitary method

When you know one amount, find the value of 1 unit, then multiply. To share $360 between Ann and Bala in the ratio 4 : 5, the total is 9 units, so 1 unit = $360 ÷ 9 = $40, Ann gets 4 × $40 = $160 and Bala gets 5 × $40 = $200. If instead you are told that Bala has $40 more than Ann, the difference is 5 − 4 = 1 unit, so 1 unit = $40 and the same answers follow. To combine two ratios that share a quantity, make that quantity the same number of units. If A : B = 2 : 3 and B : C = 4 : 5, multiply the first ratio by 4 and the second by 3 to get A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.

Before-and-after problems

In these problems the ratio changes because something is added, removed or transferred. Find what stays the same, then make it the same number of units in both ratios. There are three common cases. If only one person's amount changes, the other person's amount is unchanged. If one person gives some to the other, the total is unchanged. If both receive or lose the same amount, the difference is unchanged. For example, Ali and Ben shared money in the ratio 3 : 5. Ben gave Ali $20 and they then had the same amount. The total is unchanged at 8 units, so after the transfer each has 4 units. Ali went from 3 units to 4 units, so 1 unit = $20 and the total is 8 × $20 = $160. Check: Ali had $60 and Ben had $100, and after the transfer each has $80.

Worked example

The ratio of the number of John's marbles to Kumar's marbles was 5 : 3. After John gave away 77 marbles, the ratio became 3 : 4. How many marbles did John have at first?

  1. Only John's marbles changed, so Kumar's number of marbles stayed the same.
  2. Kumar has 3 units before and 4 units after. Make these equal using the lowest common multiple of 3 and 4, which is 12.
  3. Before: 5 : 3 = 20 : 12 (multiply both terms by 4). After: 3 : 4 = 9 : 12 (multiply both terms by 3).
  4. John went from 20 units to 9 units, so he gave away 20 − 9 = 11 units = 77 marbles.
  5. 1 unit = 77 ÷ 11 = 7 marbles.
  6. John at first = 20 units = 20 × 7 = 140 marbles.
  7. Check: Kumar has 12 × 7 = 84. Before, 140 : 84 = 5 : 3. After, 63 : 84 = 3 : 4 (divide both by 21).

Answer: John had 140 marbles at first.

Common mistakes

  • Comparing units from two different ratios without first making the unchanged quantity the same number of units. In the worked example, Kumar's 3 units and 4 units must both become 12 units.
  • Thinking that adding the same number to both terms keeps the ratio. 2 : 3 becomes 3 : 4 when 1 is added to each, which is a different ratio. What stays the same is the difference.
  • Writing the terms in the wrong order. 'The ratio of girls to boys' must start with girls.
  • Forgetting to change to the same units, such as writing 45 min : 2 h as 45 : 2. Use 45 : 120 = 3 : 8.
  • Confusing a part with the whole. If A : B = 3 : 5, A is 3/8 of the total, not 3/5 of it. 3/5 is A as a fraction of B.
  • Leaving fractions or decimals inside a ratio, such as 1/2 : 3/4. Multiply both terms by 4 to get 2 : 3.

Exam tips

  • For every before-and-after problem, write the two ratios one above the other and label which quantity, total or difference stays the same before you start.
  • Use the lowest common multiple when making units equal, so the numbers stay small.
  • Always substitute your final values back into both the before and after ratios. It takes a few seconds and catches most unit errors.

Practice questions

1. Easy: Express the ratio 45 minutes : 2 hours in its simplest form.

Show answer

3 : 8. Working: 2 hours = 120 minutes, so the ratio is 45 : 120. Dividing both terms by 15 gives 3 : 8.

2. Medium: $540 is shared among Amy, Bala and Chen in the ratio 2 : 3 : 4. How much does Chen receive?

Show answer

$240. Working: total = 2 + 3 + 4 = 9 units = $540, so 1 unit = $60 and Chen gets 4 × $60 = $240.

3. Harder: The ratio of Mei's stickers to Siti's stickers was 4 : 7. Each girl then bought 15 more stickers, and the ratio became 3 : 5. How many stickers did Mei have at first?

Show answer

120 stickers. Working: both bought the same number, so the difference is unchanged. The differences are 7 − 4 = 3 units and 5 − 3 = 2 units, so make both 6: before 8 : 14, after 9 : 15. Mei went from 8 to 9 units, so 1 unit = 15 and Mei had 8 × 15 = 120. Check: 120 : 210 = 4 : 7 and 135 : 225 = 3 : 5.

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