The average of 6 numbers is 30 . If the average of the first four is 25 and that of the last three is 35 , the fourth number is

Options

A. 25
B. 30
C. 35
D. 40

Answer

To find the fourth number given the average of six numbers and the averages of subsets of those numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given averages:**
- The average of 6 numbers is 30.
- The average of the first 4 numbers is 25.
- The average of the last 3 numbers is 35.

2. **Calculate the total sum of the 6 numbers:**
- The formula for average is:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Total numbers}} \]
- Therefore, the total sum of the 6 numbers can be calculated as:
\[ \text{Sum of 6 numbers} = \text{Average} \times \text{Total numbers} = 30 \times 6 = 180 \]

3. **Calculate the total sum of the first 4 numbers:**
- Using the average of the first 4 numbers:
\[ \text{Sum of first 4 numbers} = \text{Average} \times \text{Total numbers} = 25 \times 4 = 100 \]

4. **Calculate the total sum of the last 3 numbers:**
- Using the average of the last 3 numbers:
\[ \text{Sum of last 3 numbers} = \text{Average} \times \text{Total numbers} = 35 \times 3 = 105 \]

5. **Set up the equation for the fourth number:**
- Let the numbers be \( A_1, A_2, A_3, A_4, A_5, A_6 \).
- From the sums calculated:
- The sum of the first 4 numbers is \( A_1 + A_2 + A_3 + A_4 = 100 \).
- The sum of the last 3 numbers is \( A_4 + A_5 + A_6 = 105 \).
- We can express the total sum of the 6 numbers as:
\[ A_1 + A_2 + A_3 + A_4 + A_5 + A_6 = 180 \]

6. **Combine the equations:**
- From the first 4 numbers:
\[ A_1 + A_2 + A_3 = 100 - A_4 \]
- From the last 3 numbers:
\[ A_5 + A_6 = 105 - A_4 \]
- Substitute these into the total sum equation:
\[ (100 - A_4) + A_4 + (105 - A_4) = 180 \]

7. **Simplify the equation:**
- Combine like terms:
\[ 100 + 105 - A_4 = 180 \]
- This simplifies to:
\[ 205 - A_4 = 180 \]

8. **Solve for \( A_4 \):**
- Rearranging gives:
\[ A_4 = 205 - 180 = 25 \]

### Final Answer:
The fourth number \( A_4 \) is **25**.