The ratio of the present age of P to the age of Q four years ago is $2 : 1$. The present age of R is equal to the average of the present ages of P and Q. Sixteen years ago, R was 10 years old. What is the sum of the present ages of P, Q, and R (in years)?

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**Solution:**

1. Let the present age of P be $2x$ years.

2. Since the ratio of P's present age to Q's age four years ago is $2 : 1$, Q's age four years ago is $x$ years. Therefore, Q's present age is $x + 4$ years.

3. The present age of R is given as the average of the present ages of P and Q:
   $$R = \frac{2x + (x + 4)}{2} = \frac{3x + 4}{2}$$

4. Sixteen years ago, R's age was 10 years, so:
   $$R - 16 = 10 \implies R = 26$$

5. Substitute $R = 26$ into the expression:
   $$\frac{3x + 4}{2} = 26 \implies 3x + 4 = 52 \implies 3x = 48 \implies x = 16$$

6. Calculate present ages:
   - P's age = $2x = 2 	imes 16 = 32$ years
   - Q's age = $x + 4 = 16 + 4 = 20$ years
   - R's age = 26 years

7. Sum of their present ages:
   $$32 + 20 + 26 = 78 	ext{ years}$$

**Answer:** 78 years

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