A shopkeeper sells two items, A and B. Item B's cost price is twice as that of item A. The shopkeeper sells item A at 10% profit and item B at 20% profit. Which one of the following is the value of net profit?
- (a)15%
- (b)13.33%
- (c)18%
- (d)16.66%
Correct — D, 16.66%. Let item A cost C, so item B costs 2C and the total outlay is 3C. Profit on A is 10 per cent of C, that is 0.1C. Profit on B is 20 per cent of 2C, that is 0.4C. Total profit is 0.5C on a total cost of 3C, and 0.5 ÷ 3 = 1/6 = 16.666…, printed as 16·66. The result is a weighted average of 10 and 20 per cent with weights 1 and 2, which pulls it two-thirds of the way from 10 towards 20.
- (a)15% — 15 is the plain average of 10 and 20, which would be right only if the two items cost the same. B costs twice as much, so its 20 per cent carries twice the weight.
- (b)13.33% — 13.33 is the weighted average with the weights the wrong way round, applying the 10 per cent to the costlier item.
- (c)18% — 18 exceeds the weighted mean. No weighting of 10 and 20 in the ratio 1 : 2 can pass 16.67 without changing the cost ratio.
An overall profit percentage is total profit divided by total cost, never the simple average of the individual rates. It is a weighted mean whose weights are the cost prices, so the item with the larger outlay pulls the combined figure towards its own rate.
No rupee value appears in the stem, which is the signal to set the cheaper cost at 1 unit — or at 100 for easy arithmetic — and let everything else follow. Taking A at ₹100 and B at ₹200 gives profits of ₹10 and ₹40 on a total cost of ₹300, that is ₹50 on ₹300, or 16.67 per cent. Option (a) is placed for candidates who average the rates, and the shape of that trap repeats across profit, speed and concentration questions.
- Overall profit per cent = total profit ÷ total cost price × 100.
- With costs in the ratio 1 : 2, the combined rate is (10 × 1 + 20 × 2) ÷ 3 = 16.67 per cent.
- A simple average of the two rates is valid only when the cost prices are equal.
- Taking the cheaper item at ₹100 makes the whole calculation ₹50 profit on ₹300 of cost.
- The combined rate always lies between the two individual rates, closer to the one with the bigger cost.
The plain average of 10 and 20 is 15, and it is wrong because the two items do not cost the same.
- Averaging the two profit rates and answering 15 per cent.
- Weighting by the number of items instead of by their cost.
- Taking the overall profit on selling price rather than on cost price.
A weighted-average item in profit-and-loss clothing, with the unweighted average sitting in the option list as the bait.
If the average of the first four of five numbers in decreasing order is 25 and the average of the last four numbers is 20, then what is the difference between the first and the last number?
- (a) 5
- (b) 10
- (c) 15
- (d) 20
Answer(d) 20
Averages of overlapping groups on an earlier CAPF paper. Both items punish the reflex of averaging the averages, and both are solved by converting each average back into a total before combining.
- practice — not a real PYQ
Two articles cost ₹200 and ₹300. The first is sold at a 20 per cent profit and the second at a 10 per cent profit. The overall profit per cent is
- (a)13 per cent
- (b)14 per cent
- (c)15 per cent
- (d)16 per cent
Answer(b) 14 per cent — profit is ₹40 + ₹30 = ₹70 on a cost of ₹500.
- practice — not a real PYQ
A trader earns 25 per cent on one-fifth of his stock and 10 per cent on the rest. His overall profit per cent is
- (a)12 per cent
- (b)13 per cent
- (c)15 per cent
- (d)17.5 per cent
Answer(b) 13 per cent — (25 × 1 + 10 × 4) ÷ 5 = 13.