The mean temperature from Monday to Wednesday is 37 degrees Celsius and the mean temperature from Tuesday to Thursday is 34 degrees Celsius. If the temperature on Thursday is four fifths of the temperature on Monday, what is the temperature on Thursday?

Introduction / Context:
This problem involves average temperatures over overlapping groups of days and a ratio relation between two specific days. It is a classic example of how averages and simple algebra combine to solve practical questions about weather data. Students need to translate average conditions into equations for the individual day temperatures.

Given Data / Assumptions:
- Average temperature for Monday, Tuesday and Wednesday = 37 degrees Celsius.
- Average temperature for Tuesday, Wednesday and Thursday = 34 degrees Celsius.
- Temperature on Thursday is four fifths of the temperature on Monday.
- We need to find the temperature on Thursday in degrees Celsius.

Concept / Approach:
Let the temperatures on Monday, Tuesday, Wednesday and Thursday be M, T, W and Th. The given averages can be converted to equations involving sums of these variables. Subtracting these equations allows us to relate Monday and Thursday directly. The ratio condition Th = (4 / 5) * M gives a second equation. Solving these two equations together yields the actual temperatures.

Step-by-Step Solution:
Step 1: From the first average, (M + T + W) / 3 = 37 so M + T + W = 3 * 37 = 111.Step 2: From the second average, (T + W + Th) / 3 = 34 so T + W + Th = 3 * 34 = 102.Step 3: Subtract the second equation from the first: (M + T + W) - (T + W + Th) = 111 - 102.Step 4: This simplifies to M - Th = 9.Step 5: We are also given Th = (4 / 5) * M.Step 6: Substitute into M - Th = 9 to get M - (4 / 5) * M = 9.Step 7: This becomes (1 / 5) * M = 9, so M = 9 * 5 = 45 degrees Celsius.Step 8: Then Th = (4 / 5) * 45 = 36 degrees Celsius.

Verification / Alternative Check:
We can check quickly. If Monday is 45 and Thursday is 36, compute sums: M + T + W = 111 and T + W + Th = 102 from the equations. Their difference gives M - Th = 9 which matches 45 - 36. Also Th is four fifths of M because (4 / 5) * 45 = 36. Thus all conditions are satisfied, confirming the answer.

Why Other Options Are Wrong:
Values such as 38, 39 or 40 degrees Celsius do not satisfy both the difference equation M - Th = 9 and the ratio Th = (4 / 5) * M at the same time. If we assume any of those as Thursday temperature and back calculate Monday, one or both conditions fail. Only 36 degrees Celsius fits all given relationships.

Common Pitfalls:
Some students confuse which average corresponds to which group of days, leading to incorrect equations. Others may misapply the ratio, using 5 / 4 instead of 4 / 5, or they forget to multiply the averages by the number of days to get total sums. The safe method is always to convert each average into a sum equation, subtract them carefully, and then use the given ratio to solve for the unknowns.

Final Answer:
The temperature on Thursday is 36 degrees Celsius.