A trader purchases a watch and a wall clock for ₹ 390. He sells them making a profit of 10% on the watch and 15% on the wall clock. He earns a profit of ₹ 51.50. The difference between the original prices of the wall clock and the watch is equal to

### Concept & Simultaneous Equations When given the combined cost and combined profit from two different percentage rates, we can set up a system of linear equations to solve for individual variables. ### Step-by-Step Solution * Let the original price of the watch be $W$ and the wall clock be $C$. * Total cost: $W + C = 390$ (Equation 1) * Total profit from 10% on watch and 15% on clock: $0.10W + 0.15C = 51.50$ * Multiply the profit equation by 100 to clear decimals: $10W + 15C = 5150$ * Divide by 5 for simpler numbers: $2W + 3C = 1030$ (Equation 2) * Multiply Equation 1 by 2: $2W + 2C = 780$ (Equation 3) * Subtract Equation 3 from Equation 2: $(2W + 3C) - (2W + 2C) = 1030 - 780 \Rightarrow C = 250$. * Substitute $C$ back into Equation 1: $W = 390 - 250 = 140$. * Difference between original prices: $C - W = 250 - 140 = 110$. ### Exam Strategy & Shortcut Use the Percentage Shifting method. If both items were sold at a 10% profit, the total profit would be 10% of 390 = ₹ 39. The extra profit $51.50 - 39 = 12.50$ comes purely from the extra 5% profit on the wall clock. So, $5\%$ of $C = 12.50 \Rightarrow C = 250$. Then $W = 140$. $250 - 140 = 110$. ### Common Pitfall Misaligning the percentages with the items or making arithmetic errors when solving the simultaneous equations. ### Final Answer Therefore, the correct answer is **₹ 110**.