Question

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

• A. 6
• B. 5
• C. 7
• D. 4

Answer 1

The Correct Option is A

Given two equations with variables A, B, and C:

• \( A + \frac{B+C}{2} = 5 \)
• \( B + \frac{A+C}{2} = 7 \)

Step 1: Eliminate Fractions

Multiply both equations by 2 to remove fractions:

• Equation 1: \( 2A + B + C = 10 \)
• Equation 2: \( 2B + A + C = 14 \)

Step 2: Subtract Equations

Subtract Equation 1 from Equation 2:

\[ (2B + A + C) - (2A + B + C) = 14 - 10 \] \[ B - A = 4 \]

This yields:

\[ B = A + 4 \quad \text{(Equation 3)} \]

Step 3: Substitute into Equation 1

Substitute Equation 3 into Equation 1:

\[ 2A + (A + 4) + C = 10 \] \[ 3A + C = 6 \quad \text{(Equation 4)} \]

Step 4: Test Integer Values

Test with integer values:

• Assume \( A = 1 \)
• From Equation 4: \( 3(1) + C = 6 \Rightarrow C = 3 \)
• From Equation 3: \( B = 1 + 4 = 5 \)

The values are:

• \( A = 1 \)
• \( B = 5 \)
• \( C = 3 \)

Step 5: Final Answer

Sum of A and B: \[ A + B = 1 + 5 = \boxed{6} \]