The average number of visitors to a library on the first four days of a week is 58. The average number of visitors on the second, third, fourth, and fifth days is 60. The number of visitors on the first and fifth days are in the ratio 8 : 9. How many visitors came to the library on the fifth day?

Introduction:
This question tests using averages to form equations about sums, and then combining that with a ratio condition. Averages are just total sum divided by number of days, so two different averages over overlapping day ranges allow you to isolate the difference between two specific days. After finding the difference between day 5 and day 1, the ratio 8:9 converts that difference into exact values. This is a classic “overlapping averages + ratio” aptitude pattern.

Given Data / Assumptions:

• Average visitors on days 1 to 4 = 58
• Average visitors on days 2 to 5 = 60
• Days 1 and 5 are in ratio 8 : 9
• Let visitors on day 1 = D1 and on day 5 = D5

Concept / Approach:
Convert averages to sums: Sum(1..4) = 58*4 and Sum(2..5) = 60*4. Subtract the sums to eliminate common days 2,3,4 and obtain D5 - D1. Then apply the ratio D1:D5 = 8:9 to solve for D5.

Step-by-Step Solution:
Sum of days 1 to 4 = 58 * 4 = 232 Sum of days 2 to 5 = 60 * 4 = 240 Subtract: (days 2 to 5) - (days 1 to 4) = D5 - D1 So D5 - D1 = 240 - 232 = 8 Given D1 : D5 = 8 : 9, let D1 = 8k and D5 = 9k Then D5 - D1 = k = 8 So k = 8 and D5 = 9k = 72

Verification / Alternative check:
D1 = 64 and D5 = 72 satisfy the ratio. Also D5 - D1 = 8 matches the difference derived from sums, so the result is consistent.

Why Other Options Are Wrong:
Any value other than 72 breaks either the required difference D5 - D1 = 8 or the ratio D1:D5 = 8:9 (or both).

Common Pitfalls:
Using 5 days in the average (instead of 4), subtracting in the wrong order (getting D1 - D5), or applying the ratio incorrectly as 9:8 instead of 8:9.

Final Answer:
72 visitors came to the library on the fifth day.