Recovering day-wise temperatures from grouped averages: Over four days, the average temperature is 48°C. The average of the 2nd and 3rd days is 34°C, and the temperatures of the 1st and 4th days are in the ratio 9 : 11. Find the temperatures on the 1st and 4th days (in °C).

Introduction / Context:
This is a system-of-equations problem from averages and ratios. Using the four-day mean, the two-day mean (days 2 and 3), and a ratio between day 1 and day 4, we can solve for both unknown temperatures.

Given Data / Assumptions:

• Average of 4 days = 48°C → total of 4 days = 4 * 48 = 192
• Average of days 2 and 3 = 34°C → T2 + T3 = 68
• T1 : T4 = 9 : 11 → let T1 = 9k, T4 = 11k

Concept / Approach:
From the total: T1 + T2 + T3 + T4 = 192. Substitute T2 + T3 = 68 and T1, T4 in terms of k to solve for k, then compute T1 and T4.

Step-by-Step Solution:
9k + 68 + 11k = 19220k = 124 → k = 6.2T1 = 9 * 6.2 = 55.8°CT4 = 11 * 6.2 = 68.2°C

Verification / Alternative check:
T1 + T4 = 124 and T2 + T3 = 68 sum to 192, matching the four-day total; the ratio 9 : 11 is preserved exactly.

Why Other Options Are Wrong:
Other pairs do not satisfy both the total and the 9 : 11 ratio simultaneously.

Common Pitfalls:
Forgetting that the 34°C is an average for two days (hence 68 total), or mixing up the ratio assignment between day 1 and day 4.

Final Answer:
55.8°C, 68.2°C