The average of four numbers A, B, C, and D is 40. The average of four numbers A, B, E and F is also 40 (A, B are common). Which of the following must be true?

Options

A. (A +B) `ne` (C+D)
B. (C+D) = (E+F)
C. Either C = E or F and D = F or E
D. C = E and D = F

Answer

To solve the problem, we need to analyze the information given about the averages of the numbers A, B, C, D, E, and F.

### Step-by-Step Solution:

1. **Understanding the Average of A, B, C, D:**
- The average of four numbers A, B, C, and D is given as 40.
- The formula for average is:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of observations}} \]
- Therefore, we can write:
\[ \frac{A + B + C + D}{4} = 40 \]
- Multiplying both sides by 4 gives:
\[ A + B + C + D = 160 \quad \text{(Equation 1)} \]

2. **Understanding the Average of A, B, E, F:**
- The average of four numbers A, B, E, and F is also given as 40.
- Using the same average formula:
\[ \frac{A + B + E + F}{4} = 40 \]
- Multiplying both sides by 4 gives:
\[ A + B + E + F = 160 \quad \text{(Equation 2)} \]

3. **Setting Up the Equations:**
- We now have two equations:
- From Equation 1: \( A + B + C + D = 160 \)
- From Equation 2: \( A + B + E + F = 160 \)

4. **Eliminating Common Terms:**
- Since \( A + B \) is common in both equations, we can subtract Equation 1 from Equation 2:
\[ (A + B + E + F) - (A + B + C + D) = 160 - 160 \]
- This simplifies to:
\[ E + F - C - D = 0 \]
- Rearranging gives:
\[ C + D = E + F \quad \text{(Equation 3)} \]

5. **Conclusion:**
- From Equation 3, we conclude that the sum of C and D is equal to the sum of E and F. Therefore, the statement "C + D = E + F" must be true.

### Final Answer:
The statement that must be true is:
\[ C + D = E + F \]