Question

Average ages of Ram and Manohar is 15, average ages of Manohar and Shyam is 12 and average ages of Ram and Shyam is 13, then the age of Manohar is?

• A. 13
• B. 16
• C. 14
• D. 12

Hint:

Use systems of linear equations for solving average age problems.
Sum of pairwise averages helps to find the sum of all individuals.

Answer 1

The Correct Option is C

Let the ages of Ram, Manohar, and Shyam be $R$, $M$, and $S$ respectively.
Step 1: Translate average ages into equations
Average age of Ram and Manohar = 15
$\Rightarrow \frac{R + M}{2} = 15 \implies R + M = 30$
Average age of Manohar and Shyam = 12
$\Rightarrow \frac{M + S}{2} = 12 \implies M + S = 24$
Average age of Ram and Shyam = 13
$\Rightarrow \frac{R + S}{2} = 13 \implies R + S = 26$
Step 2: Add all three equations
$(R + M) + (M + S) + (R + S) = 30 + 24 + 26 = 80$
Simplify:
$2(R + M + S) = 80 \implies R + M + S = 40$
Step 3: Use substitution to find $M$
From $R + M = 30$, we get $R = 30 - M$.
From $R + S = 26$, substituting for $R$,
$(30 - M) + S = 26 \implies S = 26 - 30 + M = M - 4$
Using $M + S = 24$, substitute $S$:
$M + (M - 4) = 24 \implies 2M - 4 = 24$
$2M = 28 \implies M = 14$
Step 4: Conclusion
Manohar's age is 14.