CAT 2020

Slot 1

QA

Question & Solution

Given:
$A + \frac{B + C}{2} = 5$
$\Rightarrow 2A + B + C = 10 \quad \text{(i)}$

$ \frac{A + C}{2} + B = 7 $
$\Rightarrow A + 2B + C = 14 \quad \text{(ii)}$

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Step 1: Subtract equation (ii) from equation (i):

$$ (2A + B + C) - (A + 2B + C) = 10 - 14 $$

$$ B - A = 4 \quad \Rightarrow \quad B = A + 4 $$

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Step 2: Check the values of $A$, $B$, and $C$:

Given that $A$, $B$, and $C$ are positive integers:

• If $A = 1$, then $B = 5$, and from equation (i):

$$ 2A + B + C = 10 \quad \Rightarrow \quad 2(1) + 5 + C = 10 \quad \Rightarrow \quad C = 3 $$

Thus, $A = 1$, $B = 5$, and $C = 3$ is a valid solution.

• If $A = 2$, then $B = 6$, and from equation (i):

$$ 2A + B + C = 10 \quad \Rightarrow \quad 2(2) + 6 + C = 10 \quad \Rightarrow \quad C = 0 $$

But this is invalid since $C$ must be positive.

• If $A > 2$, then $C$ would be negative, which is also invalid.

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Conclusion:

Thus, the valid solution is $A = 1$, $B = 5$, and $C = 3$, and therefore:

$$ A + B = 6 $$

Final Answer: $6$