The average age of a husband, wife and their child 3 years ago was 27 years. The average age of the wife and the child 5 years ago was 20 years. What is the present age of the husband?

Introduction / Context:
This question tests your ability to handle age problems using averages and time shifts. Information is given about average ages some years in the past, and you must use it to deduce present ages. The key technique is to translate statements about averages into equations involving sums of ages, then adjust those sums for the passing of time.

Given Data / Assumptions:

• Three people: husband (H), wife (W) and child (C).
• Three years ago, the average age of H, W and C was 27 years.
• Five years ago, the average age of W and C was 20 years.
• We must find the present age of the husband H.
• All ages are in years and increase linearly with time.

Concept / Approach:
Average age is sum of ages divided by number of people. If we know an average at a certain time in the past, we can convert it into a sum. Then we adjust that sum to the present by adding the appropriate number of years for each person. We are given two pieces of information: one about three years ago for the whole family, and one about five years ago for just the wife and child. Converting each to current sums allows us to solve for the husband's current age.

Step-by-Step Solution:
Step 1: Let H, W and C be the present ages of husband, wife and child respectively. Step 2: Three years ago, their ages were H - 3, W - 3 and C - 3. Step 3: The average age three years ago was 27, so the sum then was 3 * 27 = 81. Step 4: Therefore, (H - 3) + (W - 3) + (C - 3) = 81. Step 5: Simplify: H + W + C - 9 = 81 ⇒ H + W + C = 90. Step 6: Five years ago, the ages of wife and child were W - 5 and C - 5. Step 7: Their average then was 20, so the sum was 2 * 20 = 40. Step 8: Therefore, (W - 5) + (C - 5) = 40 ⇒ W + C - 10 = 40 ⇒ W + C = 50. Step 9: Substitute W + C = 50 into H + W + C = 90 to find H. Step 10: H + 50 = 90 ⇒ H = 90 - 50 = 40. Step 11: Therefore, the present age of the husband is 40 years.

Verification / Alternative check:
Check the conditions using these values. If H = 40 now, W + C = 50 now. Three years ago, their ages were 37, W - 3 and C - 3. The sum was 40 + 50 - 9 = 81, and the average for 3 people is 81 / 3 = 27, satisfying the first condition. Five years ago, wife and child had ages W - 5 and C - 5, with sum 50 - 10 = 40, giving an average of 40 / 2 = 20, satisfying the second condition. So all constraints hold.

Why Other Options Are Wrong:
If H were 30, then H + W + C would be 30 + 50 = 80, contradicting the earlier result that H + W + C = 90. Values 50 or 60 similarly give wrong total sums or fail to satisfy one of the time shifted average conditions. The value 35 also does not satisfy both equations at once. Only 40 years for the husband fits both given averages properly.

Common Pitfalls:
Many students attempt to guess ages or treat the averages as current rather than time shifted, which leads to inconsistent equations. Others forget to add the appropriate number of years when converting from past sums to present sums. Always write an equation for each time point and carefully adjust all ages by the same time interval when shifting between past and present.

Final Answer:
The present age of the husband is 40 years.