Problems on Ages Questions and Answers

Answer: A

P's present age is 50 years.

Let the present ages be \(5x\) and \(7x\). Eighteen years ago they were in the ratio 8:13.

\[ \begin{aligned} \frac{5x - 18}{7x - 18} &= \frac{8}{13} \\ 65x - 234 &= 56x - 144 \\ 9x &= 90,\ x = 10 \end{aligned} \]

So P is \(5 \times 10 = 50\) and Q is 70. Check: 18 years ago they were 32 and 52, and \(32:52 = 8:13\). 70 years is Q's age, not P's.

Answer: A

The father's present age is 35 years.

Let the present ages be \(7k\) and \(2k\). After 10 years:

\[ \begin{aligned} \frac{7k + 10}{2k + 10} &= \frac{9}{4} \\ 28k + 40 &= 18k + 90 \\ k &= 5 \end{aligned} \]

So the father is 35 and the son 10. Check: in 10 years they will be 45 and 20, and 45 : 20 = 9 : 4. 40, 30 and 25 are not multiples of 7, so none of them can be the father's share of a 7 : 2 ratio in whole years.

Ratio questions on ages are solved by writing the present ages as multiples of one unknown and using the fact that every age changes by the same number of years.

Answer: A

The boy was 13 five years ago.

The father's age when the son was born equals the son's present age, \(s\). The father is that much older than the son, so his present age is \(s + s = 2s\):

\[ \begin{aligned} 2s &= 36 \\ s &= 18 \end{aligned} \]

Five years ago the boy was \(18 - 5 = 13\). 18 would be his present age, which is not what is asked, and 15, 17 and 20 do not follow.

Check: the son is 18 and the father 36, and the father was 18 when the son was born, which matches what he says.

Answer: A

Seema's present age is 72 years.

Work from Prem's age, the only fixed number:

\[ \begin{aligned} \text{Prem now} &= 5 + 6 = 11 \\ \text{Priti now} &= 3 \times 11 = 33 \\ \text{Seema} + 6 &= 2 \times (33 + 6) = 78 \\ \text{Seema now} &= 72 \end{aligned} \]

Both ages in the 'six years hence' statement must be taken six years on. Doubling Priti's present age, 66, and then subtracting 6 gives 60, the trap among the options.

Answer: A

The son's present age is 10 years.

Let the son's age be s, so the father's is 3s. In 10 years the father will be twice the son's age:

\[ \begin{aligned} 3s + 10 &= 2(s + 10) \\ 3s + 10 &= 2s + 20 \\ s &= 10 \end{aligned} \]

Check: the son is 10 and the father 30; in 10 years they will be 20 and 40, and 40 is twice 20.

With a son of 15 the father would be 45, and in 10 years 55 against 25, which is not double; 20 and 25 fail in the same way.

Answer: C

B's age is 20 years.

A is twice as old as B, so A = 2B, and together they total 60:

\[ \begin{aligned} 2B + B &= 60 \\ 3B &= 60 \\ B &= 20 \end{aligned} \]

So B is 20 and A is 40. Check: 40 + 20 = 60, and 40 is twice 20.

With B at 15, A would be 30 and the total only 45; with 18 the total is 54, and with 25 it is 75. Only 20 gives 60.

Answer: A

Amit's present age is 30 years.

Let the son's age be s, so Amit is 6s. In four years both are 4 years older:

\[ \begin{aligned} (6s + 4) + (s + 4) &= 43 \\ 7s + 8 &= 43 \\ s = 5, \ 6s &= 30 \end{aligned} \]

Check: Amit is 30 and his son 5; in four years they will be 34 and 9, which add to 43.

32, 34 and 28 are not multiples of 6, so none can be six times a whole-number age; 34 is Amit's age four years from now, a common slip.

Answer: B

The mother is 40 years old.

Let my present age be x, so my mother's is 4x. Five years ago our ages added to 40.

\[ \begin{aligned} (4x - 5) + (x - 5) &= 40 \\ 5x &= 50 \Rightarrow x = 10 \end{aligned} \]

So my mother is 40 and I am 10. Check: five years ago we were 35 and 5, which add to 40. 32 and 60 would break the 4 : 1 ratio or the sum, and 26 is not a multiple of 4.

Answer: C

The son is 16 years old.

Let the son be s years old, so the father is 3s. Eight years ago the father was five times as old.

\[ \begin{aligned} 3s - 8 &= 5(s - 8) \\ 3s - 8 &= 5s - 40 \Rightarrow s = 16 \end{aligned} \]

Check: the father is 48; eight years ago they were 40 and 8, and 40 = 5 x 8. 12, 14 and 20 fail this check.

Answer: B

Write the condition algebraically.

\[ A + B = (B + C) + 12 \]

The \( B \) cancels from both sides:

\[ A - C = 12 \]

So C is 12 years younger than A. B's age never matters - which is why it is not given.