In a set of three numbers, the average of the first two numbers is 7, the average of the last two numbers is 10, and the average of the first and the last numbers is 14. What is the average of the three numbers?

Options

A. 29/4
B. 31/3
C. 25/4
D. 37/3

Answer

To solve the problem step by step, we will denote the three numbers as \( a \), \( b \), and \( c \).

### Step 1: Set up the equations based on the given averages.
1. The average of the first two numbers \( a \) and \( b \) is 7:
\[ \frac{a + b}{2} = 7 \]
Multiplying both sides by 2 gives:
\[ a + b = 14 \quad \text{(Equation 1)} \]

2. The average of the last two numbers \( b \) and \( c \) is 10:
\[ \frac{b + c}{2} = 10 \]
Multiplying both sides by 2 gives:
\[ b + c = 20 \quad \text{(Equation 2)} \]

3. The average of the first and last numbers \( a \) and \( c \) is 14:
\[ \frac{a + c}{2} = 14 \]
Multiplying both sides by 2 gives:
\[ a + c = 28 \quad \text{(Equation 3)} \]

### Step 2: Solve the system of equations.
Now we have the following three equations:
1. \( a + b = 14 \)
2. \( b + c = 20 \)
3. \( a + c = 28 \)

To find \( a + b + c \), we can add all three equations together:
\[ (a + b) + (b + c) + (a + c) = 14 + 20 + 28 \]
This simplifies to:
\[ 2a + 2b + 2c = 62 \]
Dividing both sides by 2 gives:
\[ a + b + c = 31 \quad \text{(Equation 4)} \]

### Step 3: Calculate the average of the three numbers.
The average of the three numbers \( a \), \( b \), and \( c \) is given by:
\[ \text{Average} = \frac{a + b + c}{3} \]
Substituting the value from Equation 4:
\[ \text{Average} = \frac{31}{3} \approx 10.33 \]

### Conclusion
Thus, the average of the three numbers is \( \frac{31}{3} \) or approximately \( 10.33 \).