Ratios and equal shares

Keep the named order

A box has 6 red and 9 blue counters. The ratio red : blue is 6:9 = 2:3. Divide both terms by the same non-zero factor. The ratio blue : red would be 3:2. Name the quantities in order before writing their numbers, and compare amounts in the same unit.

There are five equal ratio units altogether: two red and three blue. Each unit represents three counters, so the fifteen counters can be represented by five boxes. A box represents a ratio unit, not one counter.

Red counters are 2/5 of all counters; 2/3 instead compares red with blue. Part-to-part and part-to-whole comparisons have different denominators.

Keep every term in an equivalent ratio

For 6 red, 9 blue and 15 white counters, the ratio in that order is 6:9:15 = 2:3:5. All three terms use the same scale. Dividing only the first two terms would change the comparison with white.

A simplest ratio has no whole-number factor greater than 1 common to every term. For 12:18:30, divide every term by 6 to get 2:3:5. An equivalent ratio need not already be simplest: 4:6:10 names the same proportions.

Check your understanding 1 One scale for all three terms

Complete 2:3:5 = 8:12: ?. Then decide whether any third term could make 2:3:5 = 8:11: ? true.

Your missing term and explanation

Show answer

The scale factor from 2 to 8 is 4. It also changes 3 to 12, so the missing term is 5 × 4 = 20. In the second statement, the first two terms already disagree: 3 × 4 = 12, not 11. No third term can repair that disagreement.

Share a total

Share $84 in the ratio 2:5. Seven units represent $84, so one unit is $12. The shares are $24 and $60. Check both their sum and the ratio 24:60 = 2:5. The sum alone is not enough: $30 and $54 also total $84 but have a different ratio.

You can also use fractions of the total: the first share is 2/7 × 84 = 24 dollars and the second is 5/7 × 84 = 60 dollars. Both methods use the same seven-unit whole.

For three shares in ratio 2:3:4 totalling 72, the nine units are worth 8 each. The shares are 16, 24 and 32. Add all three terms before dividing the total; the second share is 3/9 = 1/3 of the total.

Sometimes a known amount is one part

If red : blue is 5:2 and there are 14 blue counters, the given 14 represents two units. It is not the seven-unit whole. One unit is 14÷2 = 7, so there are 5 × 7 = 35 red counters.

For three groups in ratio 3:4:5, a first group of 24 represents three units. One unit is 8, so the other groups contain 32 and 40. Use the term for the named given group, even when that group is not last in the ratio.

Check your understanding 2 What does the given amount represent?

A class has boys : girls = 4 : 5. There are 20 girls. Find the boys and the fraction of the class who are boys.

Your answer or drawing on paper

Show answer

Five units are 20, so one unit is 4 and four units are 16 boys. There are 36 pupils; the fraction boys is 16/36 = 4/9. The given 20 represents the girls, not the whole class.

Sometimes the known amount is a difference

Two amounts are in ratio 3:7 and differ by 28. Their difference is four units, so one unit is 7. The amounts are 21 and 49, total 70. Dividing 28 by ten would wrongly treat a difference as a total. Check 49-21 = 28 as well as 21:49 = 3:7.

Before dividing, identify whether the given amount represents all the units, one named part, or the difference between two parts.

Check your understanding 3 A difference in a three-part ratio

Three groups have counters in the ratio 2:3:5. The third group has 18 more counters than the first. Find every group’s count in the named order.

Your three counts and a check

Show answer

The difference is 5-2 = 3 units, so one unit is 18÷3 = 6 counters. The counts are 12, 18 and 30. Check 30-12 = 18 and 12:18:30 = 2:3:5. The given difference is not the total of the three groups.

Turn a fraction into a part-to-rest ratio

If 3/5 of a collection is red and the rest is blue, five units form the whole. Three are red and 5-3 = 2 are blue. Red : blue is 3:2, not 3:5; 3:5 would compare red with the whole.

The reverse works too. From red : blue = 3:2, the red fraction of the whole is 3/(3 + 2) = 3/5.

Check your understanding 4 Name the denominator

Two fifths of the beads in a bag are red; the rest are blue. Write red : blue in simplest form. Separately, a mix has red : blue : white = 2:3:4. What fraction of that mix is blue?

Your part-to-rest ratio and fraction of the mix

Show answer

The bag has two red units and three blue units, so red : blue is 2:3. In the mix, blue occupies three of the nine units: 3/9 = 1/3. Blue : all other colours would be 3:(2 + 4) = 1:2, a different comparison from the original three-colour ratio. A single fraction does not tell you how two other colours are split, so it cannot determine all three ratio terms by itself.

Additive and multiplicative change differ

A ratio is 2:3. If both actual quantities double, it becomes 4:6, an equivalent ratio. If two are added to each actual quantity starting from 2 and 3, it becomes 4:5, which is not equivalent. Explain using fractions of the totals.

Your comparison of the two changes