The ratio of the ages of A and B, 3 years ago, was 5 : 7. The ratio of their ages, 13 years from now, will be 9 : 11. If the present age of C is 11 years less than that of B, then the present age of C, in years, is:
- (a)31
- (b)20
- (c)23
- (d)36
Answer
Why
Correct — B. Put the earlier ratio in parts, age both people forward by the same amount, then match the later ratio.
3 years ago: A = 5k, B = 7k.
Present: A = 5k + 3, B = 7k + 3.
13 years from now: A = 5k + 16, B = 7k + 16.
(5k + 16) : (7k + 16) = 9 : 11
11(5k + 16) = 9(7k + 16)
55k + 176 = 63k + 144
8k = 32, so k = 4.
Present ages: A = 23, B = 31.
C = B − 11 = 31 − 11 = 20 → option (b).
Why the others are wrong
- (a)31 — 31 is B's present age, one step short of the answer. C is defined as 11 years younger than B, so 31 is the number you subtract from, not the number you report.
- (c)23 — 23 is A's present age. C is tied to B by the stem, not to A, so A's age never enters the final line.
- (d)36 — 36 is A's age 13 years from now, 23 + 13. It is one of the figures the second ratio 9 : 11 is built on, not a present age.
Concept
Two-ratio age problems are a single linear equation in disguise.
Give the earlier ratio one unknown — 5k and 7k — rather than two, because the ratio already ties the two ages together.
Then move both ages forward by the same number of years. From 'three years ago' to 'thirteen years hence' is a jump of 16, so both expressions get +16.
Cross-multiply the second ratio, solve for k, and finish by answering the question actually asked — here the age of a third person.
The gap between the two people never changes: B − A = 2k = 8 years at every date in this question.
Checking that your answer preserves that gap is the fastest way to catch an arithmetic slip.
Key facts
- Three years ago the ages were 20 and 28, which is the ratio 5 : 7.
- Thirteen years from now they will be 36 and 44, which is the ratio 9 : 11.
- The present ages are A = 23 and B = 31, so C is 31 − 11 = 20.
- An age difference is constant over time, so it must be the same in every ratio the question gives.
Study next
Common traps
- Adding 13 to the past ages instead of 16, which drops the three-year offset.
- Answering B's age because B is the last person named before C.
- Cross-multiplying 9 : 11 the wrong way round, as 9(5k + 16) = 11(7k + 16).
Age items in this corpus run from one-date to two-date set-ups.
23 Sep 2024, 12:30, Quant Q.12 asks how many years ago 20 and 30 stood in the ratio 3 : 5, and 24 Sep 2024, 16:00, Quant Q.6 pairs 'one year ago' with 'after 9 years' before asking for the sum of the present ages.
Related PYQs
No directly related past PYQ was found.