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

We are given two equations involving variables A, B, and C:

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

Step 1: Eliminate Fractions

We multiply both equations by 2 to remove the fractions:

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

Step 2: Subtract Equations

Subtract Equation 1 from Equation 2:

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

So, we have:

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

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

Let us test with integer values:

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

So we have:

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

 

Step 5: Final Answer

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

Answer 2

We are given two equations:

1. \(A + \frac{B + C}{2} = 5\) ⇒ Multiply both sides by 2:
   \(2A + B + C = 10\)  ......(1)
2. \(B + \frac{A + C}{2} = 7\) ⇒ Multiply both sides by 2:
   \(A + 2B + C = 14\)  ......(2)

Subtracting (1) from (2):

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

🔸 Testing integer values for A, B, and C:

• If \(A = 1\), then \(B = 5\)

Now substitute A and B into equation (1):

\(2A + B + C = 10\)
\(2(1) + 5 + C = 10\)
\(7 + C = 10 \Rightarrow C = 3\)
All values are positive integers.

Let’s test \(A = 2\) ⇒ \(B = 6\), then:

Equation (1):
\(2(2) + 6 + C = 10 \Rightarrow 10 + C = 10 \Rightarrow C = 0\)
 But C must be a positive integer, so this is invalid.

Similarly, for \(A > 2\), C becomes negative, which is also invalid.

Final Conclusion:

The only valid solution with positive integers is:

• \(A = 1\)
• \(B = 5\)
• \(C = 3\)

 

Thus, A + B = \(1 + 5 = 6\)