Understanding the Average Temperature Problem

This problem involves calculating a specific day's temperature based on average temperatures provided for different periods within a week. We are given:

• The average temperature for the first 4 days of the week.
• The average temperature for the last 4 days of the week.
• The average temperature for the entire 7-day week.

Our objective is to determine the precise temperature on the fourth day. Note that the fourth day is part of both the 'first four days' group and the 'last four days' group.

Solving for the Fourth Day Temperature

To solve this, we first need to calculate the total sum of temperatures for each given period. The formula for the sum is: Sum = Average × Number of items.

1. Calculate the sum for the first four days:
Average temperature for the first 4 days = 25°C.
Sum of temperatures for the first 4 days ($ \text{Sum}_{\text{1-4}} $) is: $ \text{Sum}_{\text{1-4}} = 25^\circ\text{C} \times 4 = 100^\circ\text{C} $ This sum represents $ T_1 + T_2 + T_3 + T_4 $.

2. Calculate the sum for the last four days:
Average temperature for the last 4 days = 30°C.
Sum of temperatures for the last 4 days ($ \text{Sum}_{\text{4-7}} $) is: $ \text{Sum}_{\text{4-7}} = 30^\circ\text{C} \times 4 = 120^\circ\text{C} $ This sum represents $ T_4 + T_5 + T_6 + T_7 $.

3. Calculate the sum for the entire week:
Average temperature for the entire 7 days = 27°C.
Sum of temperatures for the entire week ($ \text{Sum}_{\text{Total}} $) is: $ \text{Sum}_{\text{Total}} = 27^\circ\text{C} \times 7 = 189^\circ\text{C} $ This sum represents $ T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 $.

4. Determine the temperature on the fourth day ($T_4$):
If we add the sum of the first four days and the sum of the last four days, the temperature of the fourth day ($T_4$) gets counted twice. $ \text{Sum}_{\text{1-4}} + \text{Sum}_{\text{4-7}} = (T_1 + T_2 + T_3 + T_4) + (T_4 + T_5 + T_6 + T_7) $ $ 100^\circ\text{C} + 120^\circ\text{C} = T_1 + T_2 + T_3 + T_4 + T_4 + T_5 + T_6 + T_7 $ $ 220^\circ\text{C} = (T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7) + T_4 $ We can see that the expression in the parenthesis is the total sum for the week ($ \text{Sum}_{\text{Total}} $). So, the equation becomes: $ 220^\circ\text{C} = \text{Sum}_{\text{Total}} + T_4 $ Now substitute the value of $ \text{Sum}_{\text{Total}} $: $ 220^\circ\text{C} = 189^\circ\text{C} + T_4 $ To find $T_4$, subtract the total sum from the combined sum: $ T_4 = 220^\circ\text{C} - 189^\circ\text{C} $ $ T_4 = 31^\circ\text{C} $

Thus, the temperature on the fourth day is 31°C.