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

Options

A. a) 3
B. b) 4
C. c) 7
D. d) 11

Answer

To find the fourth number among the seven given numbers, we can follow these steps:

### Step 1: Calculate the sum of the first four numbers.
Given that the average of the first four numbers is 4, we can find the sum of these numbers:
\[ \text{Sum of first 4 numbers} = \text{Average} \times \text{Number of elements} = 4 \times 4 = 16 \]

### Step 2: Calculate the sum of the last four numbers.
Similarly, since the average of the last four numbers is also 4, we can calculate their sum:
\[ \text{Sum of last 4 numbers} = 4 \times 4 = 16 \]

### Step 3: Calculate the total sum of all seven numbers.
The average of all seven numbers is given as 3. Therefore, the total sum of these seven numbers can be calculated as:
\[ \text{Total sum of 7 numbers} = \text{Average} \times \text{Number of elements} = 3 \times 7 = 21 \]

### Step 4: Set up an equation to find the fourth number.
Let the seven numbers be represented as \( a_1, a_2, a_3, a_4, a_5, a_6, a_7 \). The fourth number is \( a_4 \). From our previous calculations, we know:
- The sum of the first four numbers \( a_1 + a_2 + a_3 + a_4 = 16 \)
- The sum of the last four numbers \( a_4 + a_5 + a_6 + a_7 = 16 \)
- The total sum of all seven numbers \( a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = 21 \)

### Step 5: Combine the equations.
From the first equation, we can express \( a_1 + a_2 + a_3 \) as:
\[ a_1 + a_2 + a_3 = 16 - a_4 \]
From the second equation, we can express \( a_5 + a_6 + a_7 \) as:
\[ a_5 + a_6 + a_7 = 16 - a_4 \]

Now substituting these into the total sum equation:
\[ (16 - a_4) + a_4 + (16 - a_4) = 21 \]

### Step 6: Simplify the equation.
This simplifies to:
\[ 16 - a_4 + a_4 + 16 - a_4 = 21 \]
\[ 32 - a_4 = 21 \]

### Step 7: Solve for \( a_4 \).
Rearranging gives:
\[ a_4 = 32 - 21 = 11 \]

### Conclusion
The fourth number \( a_4 \) is 11.