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

Introduction / Context:
This temperature problem combines average calculations with a ratio relation. The averages are given for two overlapping sets of four days each. In addition, you know the ratio of temperatures on the first and fifth days. Using this information, you need to determine the temperature on the fifth day. It is a typical exam style question that checks algebraic manipulation and understanding of averages.

Given Data / Assumptions:
- Let the temperatures on days 1, 2, 3, 4 and 5 be T1, T2, T3, T4 and T5 respectively.
- Average for days 1 to 4 is 58 degrees.
- Average for days 2 to 5 is 60 degrees.
- Temperatures T1 and T5 are in the ratio 7 : 8, so T1 : T5 = 7 : 8.
- We need to find T5, the temperature on the fifth day.

Concept / Approach:
The average of a group of days can be written as the sum of those day temperatures divided by the number of days. The given averages give two equations for the sums of T1 to T4 and T2 to T5. Subtracting these equations relates T1 and T5. The ratio condition between T1 and T5 provides a second relation. Solving these two simple equations gives the values of T1 and T5.

Step-by-Step Solution:
Step 1: From the first average, (T1 + T2 + T3 + T4) / 4 = 58, so T1 + T2 + T3 + T4 = 4 * 58 = 232.Step 2: From the second average, (T2 + T3 + T4 + T5) / 4 = 60, so T2 + T3 + T4 + T5 = 4 * 60 = 240.Step 3: Subtract the first sum from the second: (T2 + T3 + T4 + T5) - (T1 + T2 + T3 + T4) = 240 - 232.Step 4: This simplifies to T5 - T1 = 8.Step 5: From the ratio T1 : T5 = 7 : 8, we have T5 = (8 / 7) * T1.Step 6: Substitute into T5 - T1 = 8 to get (8 / 7) * T1 - T1 = 8.Step 7: Factor T1: T1 * (8 / 7 - 1) = 8, so T1 * (1 / 7) = 8.Step 8: Therefore T1 = 8 * 7 = 56 degrees.Step 9: Then T5 = (8 / 7) * 56 = 64 degrees.

Verification / Alternative Check:
We can quickly verify: T5 - T1 = 64 - 56 = 8, which matches the difference derived from the averages. Also, T1 : T5 = 56 : 64 simplifies to 7 : 8, which agrees with the given ratio. So T5 is consistent with all conditions in the problem.

Why Other Options Are Wrong:
Temperatures such as 62, 65 or 66 degrees fail to satisfy both relations simultaneously. For each of those values, either the difference T5 - T1 does not equal 8 when combined with the ratio, or the ratio T1 : T5 is not exactly 7 : 8. Only 64 degrees fits both the difference and the ratio conditions.

Common Pitfalls:
Many learners mistakenly divide by 5 instead of 4 when converting averages to sums, or they misinterpret the ratio and use T5 : T1 = 7 : 8 instead of T1 : T5. Another common error is not subtracting the equations correctly, which leads to an incorrect relation between T1 and T5. Careful algebra avoids these problems.

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