The ratio of the present ages of P and Q is 9 : 4, and the difference between their present ages is 20 years. What will be the sum of their ages, in years, after 10 years from now?

Introduction / Context:
This problem is a straightforward application of age ratios and differences, followed by a simple time shift. It is commonly used in aptitude tests to check basic comfort with ratio and proportion and with the idea that all people age by the same number of years over any given interval.

Given Data / Assumptions:

• The present age ratio of P and Q is 9 : 4.
• The difference between their present ages is 20 years.
• We must find the sum of their ages after 10 years.
• Ages are in years and positive.

Concept / Approach:
We first express the present ages of P and Q as 9x and 4x respectively, using the given ratio. The difference between these expressions is then set equal to 20 years, which allows us to solve for x. Once we know x, we can compute the actual present ages of P and Q. To find the sum of their ages after 10 years, we add 10 years to each age and then add them together. This method uses the ideas of ratio based representation and constant age differences over time.

Step-by-Step Solution:
Step 1: Let the present age of P be 9x years and the present age of Q be 4x years. Step 2: The difference between their present ages is 9x - 4x = 5x. Step 3: According to the question, this difference is 20 years, so 5x = 20. Step 4: Divide both sides by 5 to obtain x = 4. Step 5: Therefore, P present age is 9x = 9 * 4 = 36 years. Step 6: Q present age is 4x = 4 * 4 = 16 years. Step 7: After 10 years, P age will be 36 + 10 = 46 years and Q age will be 16 + 10 = 26 years. Step 8: The sum of their ages after 10 years is 46 + 26 = 72 years. Step 9: Hence, the required sum is 72 years.

Verification / Alternative check:
We can verify by checking the ratio and difference at the present time. With ages 36 and 16, the ratio 36 : 16 simplifies by dividing both numbers by 4 to get 9 : 4, which matches the given ratio. The difference 36 - 16 equals 20 years, which also matches the statement. Since both conditions hold at the present, and adding the same 10 years to each age does not change the difference, the computed future sum of 72 years is reliable.

Why Other Options Are Wrong:
Option A (62 years) would correspond to significantly smaller ages after 10 years and does not match the given ratio and difference when traced back to the present.
Option B (66 years) also fails to align with the correct present age ratio and difference when checked.
Option D (76 years) is too large for the sum after 10 years and would imply present ages that do not satisfy both the 9 : 4 ratio and the 20 year difference.

Common Pitfalls:
A frequent mistake is to add 10 years to the age difference as well, but the difference between two ages remains constant when the same number is added to both. Another pitfall is incorrectly setting up the equation for the difference, such as writing 9x + 4x instead of 9x - 4x. Keeping a clear picture of ratio based representation and the concept of age difference helps avoid these errors.

Final Answer:
The sum of their ages after 10 years will be 72 years.