Let a, b and c be numbers with `altbltc` such that the average of a and b is 2, average of b and c is 4, and the average of a and c is 3. What is the average of a, b and c?

Answer

To solve the problem step by step, we will use the information given about the averages of pairs of numbers \(a\), \(b\), and \(c\).

### Step 1: Set up the equations based on the averages given.

1. The average of \(a\) and \(b\) is 2:
\[ \frac{a + b}{2} = 2 \]
Multiplying both sides by 2 gives:
\[ a + b = 4 \quad \text{(Equation 1)} \]

2. The average of \(b\) and \(c\) is 4:
\[ \frac{b + c}{2} = 4 \]
Multiplying both sides by 2 gives:
\[ b + c = 8 \quad \text{(Equation 2)} \]

3. The average of \(a\) and \(c\) is 3:
\[ \frac{a + c}{2} = 3 \]
Multiplying both sides by 2 gives:
\[ a + c = 6 \quad \text{(Equation 3)} \]

### Step 2: Add the equations together.

Now, we will add all three equations:
\[ (a + b) + (b + c) + (a + c) = 4 + 8 + 6 \]
This simplifies to:
\[ 2a + 2b + 2c = 18 \]

### Step 3: Simplify the equation.

Dividing the entire equation by 2 gives:
\[ a + b + c = 9 \quad \text{(Equation 4)} \]

### Step 4: Calculate the average of \(a\), \(b\), and \(c\).

The average of \(a\), \(b\), and \(c\) is given by:
\[ \text{Average} = \frac{a + b + c}{3} \]
Substituting the value from Equation 4:
\[ \text{Average} = \frac{9}{3} = 3 \]

### Conclusion

Thus, the average of \(a\), \(b\), and \(c\) is \(3\).

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