The ratio of the ages of A and B 25 years ago was 3: 8. The ratio of their present ages is 8 : 13. What will be the present age (in years) of B?
(A) 15 (B) 40 (C) 65 (D) 50
Let’s denote the present ages of A and B as \(A_p\) and \(B_p\), respectively.
According to the given information:
1. The ratio of their ages 25 years ago was 3:8.
2. The ratio of their present ages is 8:13.
Let’s first find their ages 25 years ago:
\[ \text{25 years ago:} \]
\[ A – 25 \]
\[ B – 25 \]
Now, set up the ratio according to the first statement:
\[\frac{A – 25}{B – 25} = \frac{3}{8}\]
Similarly, set up the ratio according to the second statement for their present ages:
\[\frac{A_p}{B_p} = \frac{8}{13}\]
Now, we can set up a system of equations to solve for \(A_p\) and \(B_p\):
\[A – 25 = 3k\]
\[B – 25 = 8k\]
\[A_p = 8m\]
\[B_p = 13m\]
where \(k\) and \(m\) are common factors.
Let’s solve these equations:
\[A – 25 = 3k\]
\[B – 25 = 8k\]
Subtract the first equation from the second:
\[B – A = 5k\]
Now substitute \(B = A + 5k\) into the first equation:
\[A – 25 = 3k\]
\[A – 25 = 3k\]
\[A = 3k + 25\]
Now substitute this expression for \(A\) back into the expression for \(B\):
\[B = A + 5k\]
\[B = (3k + 25) + 5k\]
\[B = 8k + 25\]
Now, we have the present ages in terms of \(k\). Since the present age ratio is 8:13, set up the equation:
\[\frac{A_p}{B_p} = \frac{8}{13}\]
\[\frac{(3k + 25)}{(8k + 25)} = \frac{8}{13}\]
Cross-multiply:
\[13(3k + 25) = 8(8k + 25)\]
\[39k + 325 = 64k + 200\]
\[25k = 125\]
\[k = 5\]
Now that we have the value of \(k\), we can find \(B_p\) (the present age of B):
\[B_p = 8k + 25\]
\[B_p = 8(5) + 25\]
\[B_p = 40 + 25\]
\[B_p = 65\]
Therefore, the present age of B is 65 years.
So, the correct answer is (C) 65.
