The average of five numbers is 612. If the average of the first two numbers is 418 and the average of the last two numbers is 521. find the third number.

Options

A. 1172
B. 1182
C. 1185
D. 1180

Answer

To solve the problem step by step, we can follow these calculations:

1. **Calculate the total sum of the five numbers.**
- Given that the average of five numbers is 612, we can use the formula for average:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Total numbers}} \]
- Therefore, the sum of the five numbers (let's denote them as \(x_1, x_2, x_3, x_4, x_5\)) can be calculated as:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 612 \times 5 = 3060 \]

2. **Calculate the sum of the first two numbers.**
- The average of the first two numbers \(x_1\) and \(x_2\) is given as 418. Thus, we can calculate their sum:
\[ x_1 + x_2 = 418 \times 2 = 836 \]

3. **Calculate the sum of the last two numbers.**
- The average of the last two numbers \(x_4\) and \(x_5\) is given as 521. Thus, we can calculate their sum:
\[ x_4 + x_5 = 521 \times 2 = 1042 \]

4. **Set up the equation to find the third number.**
- Now we know the total sum of all five numbers, the sum of the first two, and the sum of the last two. We can express the total sum as:
\[ (x_1 + x_2) + x_3 + (x_4 + x_5) = 3060 \]
- Substituting the known sums into the equation:
\[ 836 + x_3 + 1042 = 3060 \]

5. **Solve for the third number \(x_3\).**
- Combine the known sums:
\[ 1878 + x_3 = 3060 \]
- Now, isolate \(x_3\):
\[ x_3 = 3060 - 1878 = 1182 \]

Thus, the third number \(x_3\) is **1182**.