If both shares grow by the same amount then the gap between them is fixed
When both change by the same amount
Two amounts are 3 and 7, so the difference between them is 7 − 3 = 4. Now add 2 to both: they become 5 and 9, and the difference is 9 − 5 = 4 again. Adding the same amount to both numbers moves both ends of the gap by the same distance, so the gap does not change. Taking the same amount away from both works the same way.
This happens in many problems. Two people's ages both grow by the same number of years, because the same time passes for both of them. Two friends who each spend the same amount of money both go down by that amount. In every case the difference between the two stays the same.
Both bars grow by 2, so the gap between them is 4 before and 4 after.
Make the differences match first
In a ratio problem, the difference is the anchor, because it is the same real amount before and after. To use it, both ratios must give the difference the same number of parts.
The ages of two brothers are in the ratio 3 : 5. In 6 years' time, their ages will be in the ratio 3 : 4. In 3 : 5 the difference is 5 − 3 = 2 parts, but in 3 : 4 it is 4 − 3 = 1 part. These parts are different sizes, so they cannot be compared yet. Multiply both numbers of the second ratio by 2: 3 : 4 = 6 : 8, and now the difference is 8 − 6 = 2 parts in both ratios. One part is the same number of years in both.
Now compare. The younger brother goes from 3 parts to 6 parts, a gain of 3 parts, and so does the older brother, from 5 parts to 8 parts. Those 3 parts are the 6 years that pass, so one part is 6 ÷ 3 = 2 years. Now the brothers are 3 × 2 = 6 and 5 × 2 = 10 years old. Check: in 6 years they will be 12 and 16, and 12 : 16 = 3 : 4.
The two ratios with the difference matched, measured in parts of 2 years each. Each brother gains 3 parts, and the gap stays 2 parts.
The usual mistakes
Comparing parts before the differences match. From 3 : 5 to 3 : 4, the younger brother seems to stay at 3 parts, which would mean he did not get older. The parts in the two ratios are different sizes until the difference has the same number of parts in both.
Adding the change to the gap. If 3 and 7 both gain 2, the gap is still 4, not 4 + 2 = 6. Each number gained its own 2, and the two gains cancel.
Changing only one of the amounts. When both people spend $18, take $18 from both of them, not only from one.
Counting the time wrongly. From 5 years ago to 10 years from now is 5 + 10 = 15 years, not 10 − 5 = 5.
When the differences already match
Sometimes the two ratios already give the difference the same number of parts, and you can compare them straight away. In the next problem, 2 : 5 and 1 : 4 both have a difference of 3 parts.
Fractions · Two Children Spend the Same Amount, and Their Fraction Changes
QuestionEthan had 25 as much pocket money as Fiona. Then each of them spent $18 on stationery. After that, Ethan had 14 as much money as Fiona. (a) How much money did Fiona have at first? (b) How much money did Fiona have left?
1.At first Ethan had 25 as much as Fiona: Ethan : Fiona = 2 : 5, and the difference is 5 − 2 = 3 units.
Ethan : Fiona = 2 : 5, and the difference is 5 − 2 = 3 units. 2.Afterward Ethan had 14 as much as Fiona: Ethan : Fiona = 1 : 4, and the difference is 4 − 1 = 3 units.
Both spend the same $18. Move it: the difference between the bars stays the same. 3.They spent the same amount, so the difference in dollars did not change. It is 3 units both times, so a unit before is the same size as a unit after.
In the end Ethan : Fiona = 1 : 4, and the difference is again 3 units, so the units are the same size. 4.Ethan went from 2 units to 1 unit, so he spent 1 unit. He spent $18, so 1 unit is $18.
Ethan went from 2 units to 1 unit, so he spent 1 unit: 1 unit is $18. 5.(a) Fiona had 5 × $18 = $90 at first.
(a) Fiona had 5 × $18 = $90 at first. 6.(b) Fiona had 4 × $18 = $72 left. Check: Ethan had 2 × $18 = $36 and was left with $36 − $18 = $18, which is 14 of $72.
(b) Fiona had 4 × $18 = $72 left.
Answer: (a) $90; (b) $72
Watch out for
- Taking 25 as Ethan's share of all the money, so that Ethan : Fiona = 2 : 3. 'Ethan had 25 as much as Fiona' compares Ethan with Fiona, so Ethan is 2 units when Fiona is 5 units.
- Comparing Ethan's 2 units with his 1 unit without checking the differences first. The units are the same size only because both ratios give a difference of 3 units; when the differences are not the same, scale the ratios until they are.
Ages, when the differences must be matched
In the next problem, the difference is 7 − 2 = 5 parts in the first ratio and 2 − 1 = 1 part in the second, so the second ratio is multiplied by 5 before the two are compared.
Ratio and proportion · Mr. Wong and His Son: Their Ages Five Years Ago and in Ten Years' Time
QuestionFive years ago, the ratio of Mr. Wong's age to his son's age was 7 : 2. In 10 years' time, the ratio of Mr. Wong's age to his son's age will be 2 : 1. (a) How old is Mr. Wong's son now? (b) What is the ratio of Mr. Wong's age to his son's age now, in its simplest form?
1.Five years ago, Mr. Wong was 7 units and his son was 2 units, so the difference was 7 − 2 = 5 units. In 10 years' time, the ratio is 2 : 1, so the difference is 2 − 1 = 1 part.
Five years ago, Mr. Wong was 7 units and his son 2, a difference of 5 units. In 10 years' time the ratio is 2 : 1, a difference of 1 part. 2.The difference between their ages never changes, so make it 5 units in both. Multiply the second ratio by 5: 2 : 1 = 10 : 5.
The difference in their ages never changes, so make it 5 units in both: 2 : 1 = 10 : 5. 3.The son grows from 2 units to 5 units, and Mr. Wong grows from 7 units to 10 units. Each of them gains 3 units.
Mr. Wong goes from 7 units to 10, and his son from 2 units to 5: each gains 3 units. 4.From 5 years ago to 10 years from now is 5 + 10 = 15 years. So 3 units are 15 years, and 1 unit is 15 ÷ 3 = 5 years.
From 5 years ago to 10 years from now is 5 + 10 = 15 years, so 3 units are 15 years and 1 unit is 5 years. 5.(a) Five years ago the son was 2 × 5 = 10 years old, so now he is 10 + 5 = 15 years old.
(a) The son was 2 × 5 = 10 five years ago, so now he is 10 + 5 = 15. 6.(b) Five years ago Mr. Wong was 7 × 5 = 35, so now he is 35 + 5 = 40. The ratio now is 40 : 15, and dividing both by 5 gives 8 : 3. Check: in 10 years' time they will be 50 and 25, and 50 : 25 = 2 : 1.
(b) Now Mr. Wong is 35 + 5 = 40 and his son is 15: 40 : 15 = 8 : 3.
Answer: (a) 15 years old; (b) 8 : 3
Watch out for
- Counting the years between the two ratios as 10 − 5 = 5. One ratio is 5 years before now and the other is 10 years after now, so they are 5 + 10 = 15 years apart.
- Giving the son's age five years ago, 10, as his age now. The units give the ages five years ago, so add 5 years to find each age now: the son is 15 and Mr. Wong is 40.
Your turn
1.5 and 8 each grow by 2. What is the gap now?