The sum of five numbers is 290. The average of the first two numbers is 48.5 and the average of last two numbers is 53.5. What is the third number?

Options

A. 86
B. 96
C. 84
D. 72

Answer

To find the third number in the given problem, we can follow these steps:

1. **Understand the given information**:
- The sum of five numbers \( a + b + c + d + e = 290 \).
- The average of the first two numbers \( \frac{a + b}{2} = 48.5 \).
- The average of the last two numbers \( \frac{d + e}{2} = 53.5 \).

2. **Calculate the sum of the first two numbers**:
- From the average of the first two numbers, we can find their sum:
\[ a + b = 2 \times 48.5 = 97 \]

3. **Calculate the sum of the last two numbers**:
- From the average of the last two numbers, we can find their sum:
\[ d + e = 2 \times 53.5 = 107 \]

4. **Substitute the sums into the total sum equation**:
- Now we can substitute the values of \( a + b \) and \( d + e \) into the total sum equation:
\[ a + b + c + d + e = 290 \]
becomes
\[ 97 + c + 107 = 290 \]

5. **Simplify the equation**:
- Combine the known sums:
\[ 204 + c = 290 \]

6. **Solve for the third number \( c \)**:
- Rearranging gives:
\[ c = 290 - 204 \]
\[ c = 86 \]

7. **Conclusion**:
- Therefore, the third number is \( 86 \).