The average of the ages of Sumit, Krishna and Rishabh is 43 and the average of the ages of Sumit, Rishabh and Rohit is 49. If Rohit is 54 years old, what is Krishna's age?

### Concept & Strategy When given the averages of two groups that share members, setting up algebraic equations for their sums allows you to eliminate the shared members via subtraction, revealing the relationship between the unshared members. $$\text{Difference in Sums} = \text{Difference between Unshared Elements}$$ ### Step-by-Step Solution * **Given:** * Average of Sumit (S), Krishna (K), and Rishabh (Ri) = $43$. * Average of Sumit (S), Rishabh (Ri), and Rohit (Ro) = $49$. * Rohit (Ro) = $54$ years. * **Calculation:** 1. Find the sum of the ages of the first group: $$S + K + Ri = 3 \times 43 = 129$$ (Equation 1) 2. Find the sum of the ages of the second group: $$S + Ri + Ro = 3 \times 49 = 147$$ (Equation 2) 3. Subtract Equation 1 from Equation 2 to eliminate the overlapping variables ($S$ and $Ri$): $$(S + Ri + Ro) - (S + K + Ri) = 147 - 129$$ $$Ro - K = 18$$ 4. This tells us Rohit is $18$ years older than Krishna. 5. Substitute Rohit's given age ($54$) into the equation: $$54 - K = 18$$ $$K = 54 - 18 = 36$$ ### Exam Strategy & Shortcut **Average Difference Multiplier:** Instead of calculating full sums, observe the shift in the average. When Krishna is swapped out for Rohit, the average of the $3$ people increases from $43$ to $49$. Increase in average = $6$ years. Since there are $3$ people, the total sum must have increased by $3 \times 6 = 18$ years. This means the new person (Rohit) must be exactly $18$ years older than the old person (Krishna). Rohit is $54$. Therefore, Krishna is $54 - 18 = 36$. This skips multiplying $43 \times 3$ and $49 \times 3$ completely! ### Common Pitfall Getting overwhelmed by the multiple names and trying to find the individual ages of Sumit and Rishabh. You do not need to know their individual ages because they cancel each other out in the equations. ### Final Answer **Therefore, the correct answer is 36 years.**