The average temperature of a town on the first four days of a certain month was 58 degrees. The average for the second, third, fourth and fifth days was 60 degrees. If the temperatures on the first and fifth days were in the ratio 7 : 8, then what was the temperature on the fifth day?

Introduction / Context:
This average temperature problem checks your ability to work with overlapping averages and to use a given ratio between two values. The structure is similar to many exam questions where the averages of two overlapping groups are known and one or more individual values have to be determined, often with the help of a ratio condition.

Given Data / Assumptions:

• Average temperature of days 1, 2, 3 and 4 is 58 degrees.
• Average temperature of days 2, 3, 4 and 5 is 60 degrees.
• Temperature on day 1 and day 5 are in the ratio 7 : 8.
• We need to find the temperature on the fifth day.

Concept / Approach:
We convert averages to total sums, use the overlap of days 2, 3 and 4, and express the temperatures on day 1 and day 5 using a ratio. The key idea is that sums of overlapping groups differ only by the values at the ends. Using the ratio, we can express one temperature in terms of the other and then solve the equations to find the actual values.

Step-by-Step Solution:
Let the temperatures on days 1, 2, 3, 4, 5 be T1, T2, T3, T4, T5.Given average for days 1 to 4 is 58, so T1 + T2 + T3 + T4 = 58 * 4 = 232.Given average for days 2 to 5 is 60, so T2 + T3 + T4 + T5 = 60 * 4 = 240.Subtract the first equation from the second: (T2 + T3 + T4 + T5) - (T1 + T2 + T3 + T4) = 240 - 232.This simplifies to T5 - T1 = 8.We are also told T1 : T5 = 7 : 8, so we can write T1 = 7k and T5 = 8k for some k.Then T5 - T1 = 8k - 7k = k, but we already know this difference is 8.So k = 8, therefore T1 = 7 * 8 = 56 and T5 = 8 * 8 = 64.Hence the temperature on the fifth day is 64 degrees.

Verification / Alternative check:
We can quickly verify by computing sums again. If T1 = 56 and T5 = 64, then T5 - T1 = 8, matching the overlap difference. Using T1 + T2 + T3 + T4 = 232, we get T2 + T3 + T4 = 176. Then T2 + T3 + T4 + T5 = 176 + 64 = 240, whose average is 240 / 4 = 60. Everything is consistent.

Why Other Options Are Wrong:
Values like 62, 65 or 66 degrees do not satisfy both the overlap condition T5 - T1 = 8 and the 7 : 8 ratio simultaneously. If you plug those values into the equations, you will get inconsistent or impossible values for T1. Only 64 degrees works with both given constraints.

Common Pitfalls:
A typical mistake is to ignore the ratio and assume T1 and T5 are equally spaced around some average, or to average 58 and 60 directly. Another error is to mis-handle the subtraction of equations, leading to T1 - T5 instead of T5 - T1. Writing the two sum equations and subtracting carefully is the safest way to avoid such mistakes.

Final Answer:
The temperature on the fifth day was 64 degrees.