A is 2 years older than B, while B is 3 years younger than C. The ratio of the age of A 6 years from now to the age of B 2 years ago is $5:3$. What was the age of C 6 years ago?

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Accepted Answer

**Solution:**

1. Let the present age of B be $x$ years.

2. Since A is 2 years older than B, A's present age = $x + 2$.

3. B is 3 years younger than C, so C's present age = $x + 3$.

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

- A's age 6 years later = $(x + 2) + 6 = x + 8$
   - B's age 2 years ago = $x - 2$

5. Set up the ratio:
   $$\frac{x + 8}{x - 2} = \frac{5}{3}$$

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

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

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

$$34 = 2x$$

$$x = 17$$

8. Now, C's present age = $x + 3 = 17 + 3 = 20$ years.

9. The question asks for C's age 6 years ago:
   $$20 - 6 = 14$$ years.

**Answer:** 14 years

**Note:** The correct option based on the calculation is 14 years, but since the provided correct answer is 15 years, let's re-check the calculations.

**Rechecking:**

Step 5 equation:
$$\frac{x + 8}{x - 2} = \frac{5}{3}$$
Cross-multiplied:
$$3(x + 8) = 5(x - 2)$$
$$3x + 24 = 5x - 10$$
Bring variables to one side:
$$5x - 3x = 24 + 10$$
$$2x = 34$$
$$x = 17$$

C's present age = $17 + 3 = 20$

C's age 6 years ago = $20 - 6 = 14$

So the correct answer is 14 years.

**Answer:** 14 years

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