There are three positive integers. If the average of any two of them is added to the third, the resulting sums are 181, 172 and 160. The average of the given integer is:

Options

A. 78.5
B. 85.5
C. 86
D. 84

Answer

To solve the problem step by step, we will denote the three positive integers as \( A \), \( B \), and \( C \). According to the problem, we have the following equations based on the provided sums:

1. The average of \( A \) and \( B \) added to \( C \) equals 181:
\[ \frac{A + B}{2} + C = 181 \]
Multiplying through by 2 gives:
\[ A + B + 2C = 362 \quad \text{(Equation 1)} \]

2. The average of \( B \) and \( C \) added to \( A \) equals 172:
\[ \frac{B + C}{2} + A = 172 \]
Multiplying through by 2 gives:
\[ B + C + 2A = 344 \quad \text{(Equation 2)} \]

3. The average of \( C \) and \( A \) added to \( B \) equals 160:
\[ \frac{C + A}{2} + B = 160 \]
Multiplying through by 2 gives:
\[ C + A + 2B = 320 \quad \text{(Equation 3)} \]

Now we have three equations:
1. \( A + B + 2C = 362 \)
2. \( B + C + 2A = 344 \)
3. \( C + A + 2B = 320 \)

Next, we will add all three equations together:
\[ (A + B + 2C) + (B + C + 2A) + (C + A + 2B) = 362 + 344 + 320 \]
This simplifies to:
\[ 4A + 4B + 4C = 1026 \]
Dividing the entire equation by 4 gives:
\[ A + B + C = 256.5 \]

To find the average of the three integers, we divide the sum by 3:
\[ \text{Average} = \frac{A + B + C}{3} = \frac{256.5}{3} = 85.5 \]

Thus, the average of the given integers is \( \boxed{85.5} \).