Problems on Ages Questions and Answers

Answer: C

The son's present age is 15 years.

Let the son be S, so the father is 60 - S. Five years ago the father was four times the son:

\[ \begin{aligned} 60 - S - 5 &= 4(S - 5) \\ 55 - S &= 4S - 20 \\ 75 &= 5S \Rightarrow S = 15 \end{aligned} \]

Check: the father is 45; five years ago they were 40 and 10, and 40 is four times 10. 5 and 10 are ages five or ten years earlier, and 20 fails the check.

Answer: A

Sachin is 24.5 years old.

With the ratio 7 : 9, the ages are 7k and 9k, and their difference 2k equals 7 years:

\[ \begin{aligned} 9k - 7k &= 7 \Rightarrow k = 3.5 \\ \text{Sachin} &= 7 \times 3.5 = 24.5 \end{aligned} \]

Rahul is then 31.5, and 24.5 : 31.5 = 7 : 9. The other options give differences that do not match the ratio; 27.5, for example, would make Rahul 34.5, a ratio of about 0.8.

Answer: C

A's present age is 10 years.

Let the present ages be 5x and 4x. Two years ago they were 5x - 2 and 4x - 2, in the ratio 4 : 3:

\[ \begin{aligned} 3(5x - 2) &= 4(4x - 2) \\ 15x - 6 &= 16x - 8 \\ x &= 2 \end{aligned} \]

So A is 5 × 2 = 10 years old and B is 8. Two years ago they were 8 and 6, which is 4 : 3. The option of 8 years is B's age, not A's, and 12 years does not satisfy the second ratio.

Answer: A

B is 10 years old.

Let C be \(x\). Then B is \(2x\) and A is \(2x + 2\):

\[ \begin{aligned} (2x + 2) + 2x + x &= 27 \\ 5x &= 25, \quad x = 5 \end{aligned} \]

So C is 5, B is 10 and A is 12. 12 is A's age, the usual slip of answering for the wrong person, and 8 and 14 do not satisfy the total.

Answer: A

The ratio is 11 : 53.

Let Sudha be \(S\) and her grandfather \(G\) years old now.

\[ \begin{aligned} G - 16 &= 8(S - 16) \Rightarrow G = 8S - 112 \\ G + 8 &= 3(S + 8) \Rightarrow G = 3S + 16 \\ 8S - 112 &= 3S + 16 \Rightarrow S = 25.6 \end{aligned} \]

Then \(G = 92.8\). Eight years ago they were 17.6 and 84.8:

\[ \frac{17.6}{84.8} = \frac{176}{848} = \frac{11}{53} \]

The ages are not whole numbers, but the ratio is exact. The neater ratios 1 : 5 and 3 : 8 do not satisfy both conditions.

Answer: A

After 10 years the ratio will be 3:2.

Let the father's age be \(F\) and the daughter's \(D\), so \(F + D = 80\). Ten years ago their ages were in the ratio 2:1:

\[ \begin{aligned} F - 10 &= 2(D - 10) \\ (F - 10) + (D - 10) &= 60 \\ 3(D - 10) &= 60 \\ D &= 30,\ F = 50 \end{aligned} \]

After 10 years they will be 60 and 40, a ratio of 60:40 = 3:2.

5:3 is their present ratio (50:30), the usual slip of forgetting to add the 10 years.

Answer: D

Let C's age be \(x\) years. Then, B's age = \(2x\) years. A's age \(=(2x+2)\) years.

\(\therefore (2x + 2) + 2x + x = 27\)

\(\Rightarrow 5x = 25\)

\(\Rightarrow x=5\)

Hence, B's age \(=2x=10\) years.

Answer: B

Let varun's age = \(x\) years.

From the given conditions,

\(\Rightarrow x+1 = 2(x-12)\)

\(\Rightarrow x=25\) years. 
 

Answer: C

Let Rajan's present age be '\(x \)' years.

Then his daughter's present age is =  \(\frac{x}{3}\) years.

His mother's present age = \(\frac{13x}{9}\) years.

Now, according to the question,
\(\therefore x+\frac{x}{3}+\frac{13x}{9}=125\)

\(\Rightarrow x=\)\(\frac{125*9}{25}\)\(=45\)

Therefore, required difference = \(\frac{13x}{9}-\)\(\frac{x}{3}=13x -\)\(\frac{3x}{9}=\)\(\frac{10x}{9}\)

\(\Rightarrow \frac{10*45}{9}\)= 50 years 

Answer: C

Ayush's present age = 10 years.

His mother's present age = (10 + 20) years = 30 years.
Ayush's father's present age = (30 + 5) years = 35 years.
Ayush's father's age at the time of Ayush's birth = (35 - 10) years = 25 years.

Therefore Ayush's father's age at the time of marriage = (25 - 2) years = 23 years.