Mary is 12 years younger than Pinky. After 3 years, the ratio of their ages will be 5 : 8. If Sam's present age is equal to the sum of the present ages of Mary and Pinky, what was Sam's age 15 years ago?

Introduction / Context:
This question combines age differences, future age ratios, and the sum of ages to determine the past age of a third person, Sam. We are told how Mary and Pinky's ages relate to each other now and in the future, and that Sam's present age equals their combined present ages. From this, we must work backwards to find Sam's age 15 years ago.

Given Data / Assumptions:

- Mary is 12 years younger than Pinky at present.
- After 3 years, the ratio of Mary's age to Pinky's age will be 5 : 8.
- Sam's present age is equal to the sum of the present ages of Mary and Pinky.
- We must find Sam's age 15 years ago.

Concept / Approach:
We first express the current ages of Mary and Pinky using a variable, using the age difference and the future ratio to form an equation. Once their ages are found, we add them to get Sam's current age. Finally, we subtract 15 years from Sam's present age to determine how old he was 15 years ago. This problem uses both linear equations and the concept of age sums.

Step-by-Step Solution:
Step 1: Let Pinky's present age be P years. Then Mary's present age is P − 12 years. Step 2: After 3 years, Mary's age will be (P − 12) + 3 = P − 9 years. Step 3: After 3 years, Pinky's age will be P + 3 years. Step 4: The ratio after 3 years is (P − 9) : (P + 3) = 5 : 8. Step 5: Form the equation (P − 9) / (P + 3) = 5 / 8. Step 6: Cross-multiply: 8(P − 9) = 5(P + 3). Step 7: Expand: 8P − 72 = 5P + 15 ⇒ 3P = 87 ⇒ P = 29 years. Step 8: Mary's present age = P − 12 = 29 − 12 = 17 years. Step 9: Sam's present age is the sum of Mary's and Pinky's present ages: 17 + 29 = 46 years. Step 10: Sam's age 15 years ago = 46 − 15 = 31 years.

Verification / Alternative check:
Check the future ratio: After 3 years, Mary will be 17 + 3 = 20 years old and Pinky will be 29 + 3 = 32 years old. The ratio 20 : 32 simplifies to 5 : 8, which matches the given condition. Sam's age is correctly set as 46, the sum of 17 and 29. Subtracting 15 gives Sam's age 15 years ago as 31 years, confirming the result.

Why Other Options Are Wrong:
Values such as 29, 33, 37, or 46 years do not match Sam's age 15 years ago when all relationships are checked. Some of them correspond to Sam's present age or other unrelated quantities, but only 31 years is consistent with all the given conditions.

Common Pitfalls:
Common mistakes include reversing the age difference (taking Mary older than Pinky), misusing the future ratio, or forgetting to subtract 15 years at the final step. It is important to clearly define the variables and carefully handle the time shift and age difference together.

Final Answer:
Sam's age 15 years ago was 31 years.