Three positive number are given.If the average of any two of them is added to the third number, the sums obtained are 172, 216 and 180. What is the average ofthe given three numbers?

Options

A. 93
B. ` 95(1)/(3)`
C. `94 (2)/(3)`
D. `96`

Answer

To solve the problem, we need to find the average of three positive numbers \( A \), \( B \), and \( C \) given the conditions about their sums. Let's break it down step by step.

### Step 1: Set Up the Equations
We know that:
1. The average of \( A \) and \( B \) added to \( C \) gives 172:
\[ \frac{A + B}{2} + C = 172 \]
Multiplying through by 2 gives:
\[ A + B + 2C = 344 \quad \text{(Equation 1)} \]

2. The average of \( B \) and \( C \) added to \( A \) gives 216:
\[ \frac{B + C}{2} + A = 216 \]
Multiplying through by 2 gives:
\[ B + C + 2A = 432 \quad \text{(Equation 2)} \]

3. The average of \( C \) and \( A \) added to \( B \) gives 180:
\[ \frac{C + A}{2} + B = 180 \]
Multiplying through by 2 gives:
\[ C + A + 2B = 360 \quad \text{(Equation 3)} \]

### Step 2: Add the Equations
Now, we will add all three equations together:
\[ (A + B + 2C) + (B + C + 2A) + (C + A + 2B) = 344 + 432 + 360 \]

This simplifies to:
\[ 4A + 4B + 4C = 1136 \]

### Step 3: Simplify the Equation
Now we can divide the entire equation by 4:
\[ A + B + C = \frac{1136}{4} = 284 \]

### Step 4: Find the Average
To find the average of the three numbers \( A \), \( B \), and \( C \):
\[ \text{Average} = \frac{A + B + C}{3} = \frac{284}{3} \]

Calculating this gives:
\[ \text{Average} = 94.6667 \quad \text{or} \quad 94 \frac{2}{3} \]

Thus, the average of the given three numbers is \( 94 \frac{2}{3} \).

### Final Answer
The average of the given three numbers is \( 94 \frac{2}{3} \).

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