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?

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.

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