Problems on Ages • Topic 2 of 3

Ratio of Ages

When ages are given as a ratio m:n, write them as mk and nk so a single multiplier k carries the unknown. Because the actual difference of their ages equals (n − m)k and that difference is fixed for all time, one extra fact — a known difference or a sum — instantly gives k and hence both ages. The signature CAT trap lives here: the ratio of two ages does NOT stay the same as years pass. If two ages are 3:1 today, after some years the gap is unchanged but the ratio shrinks toward 1:1, because the same constant t is added to both numbers. So "their ages are 3:1 now and will be 2:1 later" is a perfectly consistent statement — set mk + t and nk + t to the second ratio and solve. Always anchor on the invariant difference, not the moving ratio.

✅ Solved examples

1. The ages of A and B are in the ratio 5:3 and A is 8 years older than B. Find A’s age.
A = 5k, B = 3k, difference 2k = 8 ⇒ k = 4. A = 20, B = 12.
2. Two ages are in the ratio 4:3 and their sum is 35. Find the older age.
Take 4k and 3k. Sum 7k = 35 ⇒ k = 5. Older = 4k = 20.
3. The present ages of X and Y are in the ratio 7:5. Four years later the ratio becomes 4:3. Find X’s present age.
X = 7k, Y = 5k. (7k + 4)/(5k + 4) = 4/3 ⇒ 21k + 12 = 20k + 16 ⇒ k = 4. X = 28.
4. Six years ago the ages of P and Q were in the ratio 3:2. Now they are 4:3. Find Q’s present age.
Six years ago P = 3k, Q = 2k. Now (3k + 6)/(2k + 6) = 4/3 ⇒ 9k + 18 = 8k + 24 ⇒ k = 6. Q now = 2k + 6 = 18.

✏️ Practice — try these, take hints as needed

1. The ages of M and N are 9:4 and M is 25 years older than N. Find N’s age.
M = 9k, N = 4k.
Difference 5k = 25.
k = 5, N = 4k.
20 years
2. Two ages are 5:7 with sum 60. Find the younger age.
5k and 7k.
Sum 12k = 60.
k = 5, younger = 5k.
25 years
3. Present ages of A and B are 3:4. After 6 years the ratio is 5:6. Find B’s present age.
A = 3k, B = 4k.
(3k + 6)/(4k + 6) = 5/6.
18k + 36 = 20k + 30.
12 years
4. The ratio of ages of a father and son is 7:2; the father is 35 years older. Find the father’s age.
7k and 2k.
Difference 5k = 35.
k = 7, father = 7k.
49 years
5. Four years ago the ages of two sisters were 2:3; now they are 5:7. Find the present age of the younger sister.
Then 2k and 3k.
(2k + 4)/(3k + 4) = 5/7.
14k + 28 = 15k + 20 ⇒ k = 8.
20 years

📝 Topic test — 8 questions

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