There are seven numbers. Average of first four numbers is 4. Average of last four numbers is also 4. Average of all seven numbers is 3. Find the fourth number

Question

Accepted Answer

To solve the problem and justify why Option 3 is the correct answer, let's break down the information given and use logical reasoning.

1. **Understanding the Averages:**
   - The average of the first four numbers is 4. This means the sum of the first four numbers is $4 \times 4 = 16$.
   - The average of the last four numbers is also 4. This means the sum of the last four numbers is $4 \times 4 = 16$.
   - The average of all seven numbers is 3. This means the sum of all seven numbers is $3 \times 7 = 21$.

2. **Setting Up the Equations:**
   - Let the first four numbers be $a, b, c, d$.
   - Let the last four numbers be $d, e, f, g$.
   - From the information given, we have:
     - $a + b + c + d = 16$
     - $d + e + f + g = 16$
     - $a + b + c + d + e + f + g = 21$

3. **Finding the Fourth Number:**
   - We know that $a + b + c + d + e + f + g = 21$.
   - We also know that $a + b + c + d = 16$ and $d + e + f + g = 16$.
   - Adding these two equations gives us $a + b + c + d + d + e + f + g = 32$.
   - However, since $a + b + c + d + e + f + g = 21$, we can equate and solve for $d$:
     - $32 - d = 21$
     - $d = 32 - 21 = 11$

4. **Conclusion:**
   - The fourth number $d$ is 11, which corresponds to Option 3.

5. **Explanation of Incorrect Options:**
   - **Option 1 (4):** If the fourth number were 4, the sum of all seven numbers would not match the given total of 21.
   - **Option 2 (3):** Similarly, if the fourth number were 3, the sum would not align with the total of 21.
   - **Option 4 (7):** If the fourth number were 7, the sum would again not match the required total of 21.

Therefore, based on the calculations and logical reasoning, Option 3 (11) is indeed the correct answer.