A is 2 years older than B while B is 3 years younger than C. The ratio of the age of A 6 years hence and B 2 years ago is 5:3. What was the age of C 6 years ago?

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**Solution:**

1. Let the age of B be *x* years.

2. Then, age of A = *x + 2* years (A is 2 years older than B).

3. Age of C = *x + 3* years (B is 3 years younger than C).

4. According to the problem, the ratio of age of A 6 years hence and B 2 years ago is 5:3.

- Age of A after 6 years = *x + 2 + 6 = x + 8*
   - Age of B 2 years ago = *x - 2*

So, 
   $$\frac{x + 8}{x - 2} = \frac{5}{3}$$

5. Cross-multiplied:
   $$3(x + 8) = 5(x - 2)$$

$$3x + 24 = 5x - 10$$

6. Rearranging:
   $$24 + 10 = 5x - 3x$$

$$34 = 2x$$

$$x = 17$$

7. Therefore,
   - Age of B = 17 years
   - Age of C = 17 + 3 = 20 years

8. Age of C 6 years ago = 20 - 6 = 14 years

**Answer:** 14 years

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