The mean high temperature of the first, four days of a week is 25°C whereas the mean of the last four days is 25.5°C. If the mean temperature of the whole week is 25.2°C then the temperature on the 4th day is:

Options

A. `25.5^(@)C`
B. `25^(@)c`
C. `25.2^(@)C`
D. `25.6^(@)C`

Answer

To solve the problem step by step, we need to set up equations based on the information given about the mean temperatures.

### Step 1: Define Variables
Let:
- \( T_1, T_2, T_3, T_4, T_5, T_6, T_7 \) be the temperatures from Monday to Sunday respectively.

### Step 2: Set Up the First Equation
The mean temperature of the first four days (Monday to Thursday) is given as 25°C. Therefore, we can write the equation:
\[ \frac{T_1 + T_2 + T_3 + T_4}{4} = 25 \]
Multiplying both sides by 4 gives:
\[ T_1 + T_2 + T_3 + T_4 = 100 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The mean temperature of the last four days (Thursday to Sunday) is given as 25.5°C. Therefore, we can write the equation:
\[ \frac{T_4 + T_5 + T_6 + T_7}{4} = 25.5 \]
Multiplying both sides by 4 gives:
\[ T_4 + T_5 + T_6 + T_7 = 102 \quad \text{(Equation 2)} \]

### Step 4: Set Up the Third Equation
The mean temperature of the whole week is given as 25.2°C. Therefore, we can write the equation:
\[ \frac{T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7}{7} = 25.2 \]
Multiplying both sides by 7 gives:
\[ T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 = 176.4 \quad \text{(Equation 3)} \]

### Step 5: Substitute and Solve
Now we have three equations:
1. \( T_1 + T_2 + T_3 + T_4 = 100 \)
2. \( T_4 + T_5 + T_6 + T_7 = 102 \)
3. \( T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 = 176.4 \)

From Equation 3, we can express \( T_1 + T_2 + T_3 \) in terms of \( T_4 \):
\[ T_1 + T_2 + T_3 = 176.4 - (T_4 + T_5 + T_6 + T_7) \]
Substituting Equation 2 into this gives:
\[ T_1 + T_2 + T_3 = 176.4 - 102 = 74.4 \]

### Step 6: Substitute into Equation 1
Now substitute \( T_1 + T_2 + T_3 = 74.4 \) into Equation 1:
\[ 74.4 + T_4 = 100 \]
Solving for \( T_4 \):
\[ T_4 = 100 - 74.4 = 25.6 \]

### Conclusion
The temperature on the 4th day (Thursday) is:
\[ \boxed{25.6°C} \]