The ratio of the present ages of A and B is $3:4$. $X$ years ago, the ratio of their ages was $5:7$. After $X$ years, the sum of their ages will be $48$. What is the present age of B?

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**Solution:**

1. Let the present ages of A and B be $3k$ and $4k$ respectively, based on the given ratio $3:4$.

2. $X$ years ago, their ages were $(3k - X)$ and $(4k - X)$, and the ratio was $5:7$. So,
   $$\frac{3k - X}{4k - X} = \frac{5}{7}$$

3. Cross-multiplying,
   $$7(3k - X) = 5(4k - X)$$
   $$21k - 7X = 20k - 5X$$

4. Rearranging,
   $$21k - 20k = -5X + 7X$$
   $$k = 2X$$

5. After $X$ years, their ages will be $(3k + X)$ and $(4k + X)$, and their sum is given as $48$:
   $$(3k + X) + (4k + X) = 48$$
   $$7k + 2X = 48$$

6. Substitute $k = 2X$ from step 4:
   $$7(2X) + 2X = 48$$
   $$14X + 2X = 48$$
   $$16X = 48$$
   $$X = 3$$

7. Now, find $k$:
   $$k = 2X = 2 	imes 3 = 6$$

8. Present age of B is $4k = 4 	imes 6 = 24$ years.

**Answer:** $24$ years

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