Sum of the present ages of A, B, C and D is 76 years. After 7 years, the ratio of their ages will be $7 : 6 : 5 : 8$. What is C's present age?

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**Solution:**

1. Let the present ages of A, B, C, and D be $a$, $b$, $c$, and $d$ respectively.

2. Given the sum of their present ages:
   $$a + b + c + d = 76$$

3. After 7 years, their ages will be:
   $$a + 7, \quad b + 7, \quad c + 7, \quad d + 7$$

4. The ratio of their ages after 7 years is given as:
   $$7 : 6 : 5 : 8$$

5. Let the common multiplier be $x$. Then:
   $$a + 7 = 7x$$
   $$b + 7 = 6x$$
   $$c + 7 = 5x$$
   $$d + 7 = 8x$$

6. Adding these equations:
   $$(a + 7) + (b + 7) + (c + 7) + (d + 7) = 7x + 6x + 5x + 8x$$
   $$a + b + c + d + 28 = 26x$$

7. Substitute $a + b + c + d = 76$:
   $$76 + 28 = 26x$$
   $$104 = 26x$$
   $$x = \frac{104}{26} = 4$$

8. Find C's present age:
   $$c + 7 = 5x = 5 	imes 4 = 20$$
   $$c = 20 - 7 = 13$$

**Answer:** $13$

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