The ratio of the ages of two friends is $4 : 3$. After $4$ years, this ratio becomes $6 : 5$. What will be the sum of their ages (in years) after $22$ years from now?

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**Solution:**

1. Let the present ages of the two friends be $4x$ and $3x$ respectively, based on the given ratio $4 : 3$.

2. After $4$ years, their ages will be $4x + 4$ and $3x + 4$.

3. According to the problem, the ratio of their ages after $4$ years is $6 : 5$. So, we set up the equation:

$$\frac{4x + 4}{3x + 4} = \frac{6}{5}$$

4. Cross-multiplying gives:

$$5(4x + 4) = 6(3x + 4)$$

$$20x + 20 = 18x + 24$$

5. Simplify the equation:

$$20x - 18x = 24 - 20$$

$$2x = 4$$

$$x = 2$$

6. Therefore, the present ages are:

- First friend: $4x = 4 	imes 2 = 8$ years
   - Second friend: $3x = 3 	imes 2 = 6$ years

7. After $22$ years from now, their ages will be:

- First friend: $8 + 22 = 30$ years
   - Second friend: $6 + 22 = 28$ years

8. The sum of their ages after $22$ years will be:

$$30 + 28 = 58 	ext{ years}$$

**Answer:** $58$

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