The average temperature for the first four days of a week is 40.2°C and that of the last four days is 41.3°C. If the average temperature for the whole week is 40.6°C, then the temperature on the fourth day is

### Concept & Strategy A week has $7$ days. The "first four days" and the "last four days" sum up to $8$ days. This means the 4th day is counted in both groups. We can find the temperature of this overlapping day by finding the sum of both groups and subtracting the sum of the entire week. $$\text{Overlapping Day} = (\text{Sum of Group 1} + \text{Sum of Group 2}) - \text{Sum of Whole}$$ ### Step-by-Step Solution * **Given:** * Total days in a week = $7$, Average temp = $40.6^\circ\text{C}$. * First $4$ days, Average temp = $40.2^\circ\text{C}$. * Last $4$ days, Average temp = $41.3^\circ\text{C}$. * **Calculation:** 1. Calculate total sum of temperatures for the whole week ($7$ days): $7 \times 40.6 = 284.2^\circ\text{C}$. 2. Calculate sum of temperatures for the first $4$ days: $4 \times 40.2 = 160.8^\circ\text{C}$. 3. Calculate sum of temperatures for the last $4$ days: $4 \times 41.3 = 165.2^\circ\text{C}$. 4. Summing the two groups gives the total for all $7$ days plus the 4th day counted a second time: $160.8 + 165.2 = 326.0^\circ\text{C}$. 5. Temperature on 4th day = $326.0 - 284.2 = 41.8^\circ\text{C}$. ### Exam Strategy & Shortcut **Deviation Method:** Check the difference from the weekly average ($40.6^\circ\text{C}$). First $4$ days deviation: $40.2 - 40.6 = -0.4^\circ\text{C}$. Total = $4 \times (-0.4) = -1.6$. Last $4$ days deviation: $41.3 - 40.6 = +0.7^\circ\text{C}$. Total = $4 \times (+0.7) = +2.8$. Net deviation = $-1.6 + 2.8 = +1.2$. Because the 4th day overlaps, we add this net deviation directly to the overall average: $40.6 + 1.2 = 41.8^\circ\text{C}$. This avoids multiplying decimals like $7 \times 40.6$. ### Common Pitfall Simple arithmetic mistakes when multiplying decimals. Using the deviation method completely bypasses the need to calculate $284.2$ and $326$, which are prime spots for calculation errors under time pressure. ### Final Answer **Therefore, the correct answer is 41.8°C.**