Three numbers are such that the average of first two numbers is 2, the average of the last two numbers is 3 and the average of the first and the last numbers is 4, then the average of three numbers is equal to:

Options

A. 2
B. 3.5
C. 3
D. 2.5

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 averages given.

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

3. The average of the first and last numbers \( a \) and \( c \) is 4:
\[ \frac{a + c}{2} = 4 \]
Multiplying both sides by 2 gives:
\[ a + c = 8 \quad \text{(Equation 3)} \]

### Step 2: Solve the system of equations.

Now we have a system of three equations:
1. \( a + b = 4 \) (Equation 1)
2. \( b + c = 6 \) (Equation 2)
3. \( a + c = 8 \) (Equation 3)

We can solve these equations step by step.

### Step 3: Express \( c \) in terms of \( a \) using Equation 3.

From Equation 3:
\[ c = 8 - a \quad \text{(Equation 4)} \]

### Step 4: Substitute \( c \) into Equation 2.

Substituting Equation 4 into Equation 2:
\[ b + (8 - a) = 6 \]
Simplifying this gives:
\[ b + 8 - a = 6 \]
\[ b - a = -2 \]
Thus, we can express \( b \) in terms of \( a \):
\[ b = a - 2 \quad \text{(Equation 5)} \]

### Step 5: Substitute \( b \) into Equation 1.

Now, substitute Equation 5 into Equation 1:
\[ a + (a - 2) = 4 \]
Simplifying this gives:
\[ 2a - 2 = 4 \]
\[ 2a = 6 \]
\[ a = 3 \]

### Step 6: Find \( b \) and \( c \).

Using \( a = 3 \) in Equation 5:
\[ b = 3 - 2 = 1 \]

Using \( a = 3 \) in Equation 4:
\[ c = 8 - 3 = 5 \]

### Step 7: Calculate the average of the three numbers.

Now we have:
- \( a = 3 \)
- \( b = 1 \)
- \( c = 5 \)

The sum of the three numbers is:
\[ a + b + c = 3 + 1 + 5 = 9 \]

The average of the three numbers is:
\[ \text{Average} = \frac{a + b + c}{3} = \frac{9}{3} = 3 \]

### Final Answer:
The average of the three numbers is \( 3 \).

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