Aman's expense is 30% more than Vimal's and Vimal's expense is 10% less than Raman's. If the sum of their expenses is ₹ 6447, then what would be Aman's expense?

### Concept & Strategy This is a successive comparison problem. When a chain of comparisons is given (A to V, V to R), the best strategy is to set the base variable (the one at the end of the chain, Raman) to 100 or a single variable $x$, and express the others in terms of it. ### Step-by-step Solution Let Raman's expense be $R$. Vimal's expense is 10% less than Raman's: $$ V = R - 0.10R = 0.90R $$ Aman's expense is 30% more than Vimal's: $$ A = V + 0.30V = 1.30V $$ Substitute $V = 0.90R$ into Aman's equation: $$ A = 1.30 \times (0.90R) = 1.17R $$ We are given the sum of their expenses: $$ A + V + R = 6447 $$ Substitute the variables in terms of $R$: $$ 1.17R + 0.90R + 1.00R = 6447 $$ $$ 3.07R = 6447 $$ Solve for $R$: $$ R = \frac{6447}{3.07} = \frac{644700}{307} $$ $ 307 \times 20 = 6140 $. The remainder is $ 6447 - 6140 = 307 $. $ 307 \times 21 = 6447 $. So, $ R = 2100 $. The question asks for Aman's expense: $$ A = 1.17R $$ $$ A = 1.17 \times 2100 = 117 \times 21 $$ Calculate $ 117 \times 21 $: $ 117 \times 20 = 2340 $. $ 2340 + 117 = 2457 $. ### Exam Strategy & Shortcut Use the **Base 100 Ratio Method**. Let Raman = 100 units. Vimal = 90 units. Aman = 130% of 90 = $ 1.3 \times 90 = 117 $ units. Total units = $ 100 + 90 + 117 = 307 $ units. We know 307 units = ₹ 6447. 1 unit = $ \frac{6447}{307} = 21 $. Aman = 117 units = $ 117 \times 21 = 2457 $. This avoids decimals entirely until the final division step. ### Common Pitfall A frequent trap is calculating Aman's percentage increase directly against Raman's base. Students might mistakenly think Aman is $ -10\% + 30\% = +20\% $ compared to Raman. This is false because the 30% increase is based on *Vimal's* reduced value, not Raman's original value. ### Final Answer **Therefore, the correct answer is ₹ 2457.**