In a village, the average age of n people is 42 years. Later it is discovered that the age of one person had been recorded as 20 years less than the actual age. After correcting this mistake, the average age increases by 1 year. What is the value of n?

Introduction / Context:
This question involves the concept of average and how it changes when an error in one of the data points is corrected. It is a classic example of error correction problems in averages where a single incorrect entry affects the overall mean. Understanding how total sum and average are related is the key to solving such problems efficiently.

Given Data / Assumptions:

• There are n people in the village.
• The original average age is 42 years.
• One person's age was recorded as 20 years less than the actual age.
• After correcting this person's age, the new average increases by 1 year, becoming 43 years.
• We must find the value of n.

Concept / Approach:
The average age is the total sum of ages divided by the number of people. If the average is 42, then the total recorded sum of ages is 42n. But one person's age was understated by 20 years, so the true total should be 20 higher than the recorded total. When we add this missing 20 years and divide by n, the average becomes 43. Setting up this relationship as an equation allows us to solve for n.

Step-by-Step Solution:
Step 1: Let the recorded total sum of ages be S. Since the recorded average is 42, we have S = 42n. Step 2: One person's true age is 20 years more than what was recorded. So the correct total sum is S + 20. Step 3: After correction, the new average is 43 years, so (S + 20) / n = 43. Step 4: Substitute S = 42n into this equation: (42n + 20) / n = 43. Step 5: Simplify: 42 + 20 / n = 43. Step 6: Subtract 42 from both sides: 20 / n = 1. Step 7: Multiply both sides by n: 20 = n. Step 8: Therefore, the number of people n is 20.

Verification / Alternative check:
Check numerically. If n = 20 and the average age is 42, the recorded total is 42 * 20 = 840. One age is 20 years too low, so the actual total should be 840 + 20 = 860. The corrected average is 860 / 20 = 43, which matches the description that the average increased by 1 year. This confirms that n = 20 is consistent with all the conditions in the problem.

Why Other Options Are Wrong:
If n = 21, then the original total would be 42 * 21 and adding 20 and dividing by 21 would not give exactly 43. Similarly, n = 22 or n = 19 would lead to fractional increases in the average that are not exactly 1 year. The option 'none of these' is incorrect because we have found a specific value of n that satisfies the conditions. Only n = 20 makes the increase in average exactly 1 year when 20 is added to the total.

Common Pitfalls:
A common mistake is to try to work directly with averages rather than converting to total sums. Some students also misinterpret the phrase '20 years less than the actual age' and add or subtract 20 incorrectly. Another error is to forget that the number of people n remains the same before and after correction. Keeping track of total sums, the number of entries and the definition of average helps avoid these mistakes.

Final Answer:
The total number of people in the village is 20.