The average age of 8 men increases by 4 years when one man aged 30 years is replaced by a new man. What is the age of the new man in years?

Introduction / Context:
This is a standard average and age replacement problem from aptitude. We have a group of men with a known average age, and when one member of the group leaves and is replaced by another, the average changes. From this change, we can deduce the age of the new person. Such questions are frequently used to test understanding of averages and how they relate to totals.

Given Data / Assumptions:
- There are 8 men in the group.
- The average age increases by 4 years after the replacement.
- The man who leaves is 30 years old.
- The new man has an unknown age that we must determine.
- All ages are in complete years.

Concept / Approach:
The average age of a group is equal to the total of the ages divided by the number of people. When one person leaves and another joins, the number of people remains the same, but the total age changes. The increase in average multiplied by the number of people gives the increase in total age. Knowing how the total age changed when a 30 year old man left makes it possible to find the age of the new man by simple algebra.

Step-by-Step Solution:
Step 1: Let the original average age of the 8 men be A years.Step 2: The original total age of the 8 men is 8 * A.Step 3: After the 30 year old man is replaced, the new average becomes A + 4 years.Step 4: The new total age is 8 * (A + 4) = 8A + 32.Step 5: Let the age of the new man be N years. The new total age can also be written as original total minus 30 plus N, that is 8A - 30 + N.Step 6: Equate the two expressions for the new total age: 8A - 30 + N = 8A + 32.Step 7: Cancel 8A from both sides to obtain -30 + N = 32. Therefore N = 32 + 30 = 62 years.

Verification / Alternative check:
We can check the logic with a simple interpretation. Because there are 8 men and the average age increases by 4 years, the total age of the group must have increased by 8 * 4 = 32 years. When the 30 year old man leaves and the new man joins, the net increase in total age is the age of the new man minus 30. This increase must equal 32. So new age minus 30 equals 32, leading to new age equal to 62 years, which confirms the algebraic solution.

Why Other Options Are Wrong:
If the new man were 55, 42, 69, or 38 years old, the change in total age would not be exactly 32 years. For example, if he were 55 years old, the increase in total age would be 55 - 30 = 25 years, which would only raise the average by 25 / 8, not by 4 years. Similar checks show that 42, 69, and 38 years do not give the required average increase.

Common Pitfalls:
Some learners mistakenly add 4 directly to the age of the man who left or divide the change in total age by the wrong number of people. Others confuse the original and new averages. The key is to remember that average times number of people equals total age, and that the increase in average multiplied by the number in the group gives the change in total age.

Final Answer:
The age of the new man is 62 years.