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

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**Solution:**

1. Let the current ages of the two friends be $4x$ and $3x$ respectively.

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:

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

4. Cross-multiply:

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

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

5. Simplify to find $x$:

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

$$2x = 4$$

$$x = 2$$

6. Calculate current ages:

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

7. After 22 years, their ages will be:

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

8. Sum of their ages after 22 years:

$$30 + 28 = 58$$

**Answer:** $58$

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