Out of seven given numbers, the average of the first four numbers is 4 and that of the last four numbers is also 4. If the average of all the seven numbers is 3, fourth number is

Options

A. 3
B. 4
C. 7
D. 11

Answer

To solve the problem step by step, let's break it down:

1. **Understanding the Given Information:**
- We have 7 numbers.
- The average of the first 4 numbers is 4.
- The average of the last 4 numbers is also 4.
- The average of all 7 numbers is 3.

2. **Calculating the Sum of the First 4 Numbers:**
- The average of the first 4 numbers is 4.
- Therefore, the sum of the first 4 numbers can be calculated as:
\[ \text{Sum of first 4 numbers} = \text{Average} \times \text{Number of elements} = 4 \times 4 = 16 \]

3. **Calculating the Sum of the Last 4 Numbers:**
- Similarly, the average of the last 4 numbers is also 4.
- Thus, the sum of the last 4 numbers is:
\[ \text{Sum of last 4 numbers} = 4 \times 4 = 16 \]

4. **Calculating the Sum of All 7 Numbers:**
- The average of all 7 numbers is 3.
- Therefore, the sum of all 7 numbers can be calculated as:
\[ \text{Sum of all 7 numbers} = \text{Average} \times \text{Number of elements} = 3 \times 7 = 21 \]

5. **Setting Up the Equation:**
- Let the 7 numbers be represented as \( a_1, a_2, a_3, a_4, a_5, a_6, a_7 \).
- From the above calculations, we have:
- \( a_1 + a_2 + a_3 + a_4 = 16 \) (Sum of first 4 numbers)
- \( a_4 + a_5 + a_6 + a_7 = 16 \) (Sum of last 4 numbers)
- \( a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = 21 \) (Sum of all 7 numbers)

6. **Finding the Fourth Number:**
- To find \( a_4 \), we can add the sums of the first four and last four numbers and then subtract the total sum of all seven numbers:
\[ (a_1 + a_2 + a_3 + a_4) + (a_4 + a_5 + a_6 + a_7) - a_4 = 16 + 16 - a_4 \]
- This simplifies to:
\[ 32 - a_4 = 21 \]
- Rearranging gives:
\[ a_4 = 32 - 21 = 11 \]

7. **Conclusion:**
- The fourth number \( a_4 \) is 11.