2.8 Selling Part of the Goods at a Loss, Part at a Profit (Averaging)

Here a trader buys a bulk quantity at a uniform cost per unit, then sells a fraction of it at a profit% and the remaining fraction at a loss% (or vice versa). Overall profit/loss must be computed on total CP vs total SP, weighted correctly by quantity — never by taking a plain average of the two percentages unless the two portions are exactly equal in cost value.

Worked Example 18: A trader buys 100 kg of rice at Rs 40/kg. He sells 60 kg at a profit of 20% and the remaining 40 kg at a loss of 10%. Find his overall profit or loss percentage.

Solution: Total CP = 100 × 40 = Rs 4000. SP of 60 kg = 60 × (40 × 1.20) = 60 × 48 = Rs 2880. SP of 40 kg = 40 × (40 × 0.90) = 40 × 36 = Rs 1440. Total SP = 2880 + 1440 = Rs 4320. Profit = 4320 − 4000 = 320. Profit% = (320/4000) × 100 = 8%.

Worked Example 19: A dealer bought 200 pens at Rs 12 each. He sold 150 of them at a profit of 20% and the rest at a loss of 5%. Find his overall profit percentage.

Solution: Total CP = 200 × 12 = Rs 2400. SP of 150 = 150 × 12 × 1.20 = 150 × 14.4 = Rs 2160. SP of 50 = 50 × 12 × 0.95 = 50 × 11.4 = Rs 570. Total SP = 2160 + 570 = Rs 2730. Profit = 2730 − 2400 = 330. Profit% = (330/2400) × 100 = 13.75%.

Worked Example 20 (mixed-rate purchase): A retailer buys 30 kg of rice at Rs 20/kg and 20 kg at Rs 25/kg, mixes them, and sells the entire mixture at Rs 26/kg. Find his overall profit percentage.

Solution: Total CP = (30×20) + (20×25) = 600 + 500 = Rs 1100. Total quantity = 50 kg. Total SP = 50 × 26 = Rs 1300. Profit = 1300 − 1100 = 200. Profit% = (200/1100) × 100 = 200/11 % ≈ 18.18%.

Worked Example 20A (reverse — find the ratio from the overall %): A trader sells a part of his goods at a profit of 25% and the remaining part at a loss of 10%. If his overall gain works out to 5%, find the ratio of the quantity (by cost value) sold at profit to that sold at loss.

Solution: Let the ratio be m : n. Using the weighted-average shortcut (Trick 12, Section 3): (25m − 10n)/(m+n) = 5 ⇒ 25m − 10n = 5m + 5n ⇒ 20m = 15n ⇒ m/n = 15/20 = 3 : 4. Check: With m=3, n=4: (3×25 − 4×10)/(3+4) = (75−40)/7 = 35/7 = 5 ✓.

Worked Example 20B (three-way split — edge case with three portions): A shopkeeper bought 50 kg of sugar at Rs 30/kg. He sold 20 kg at a profit of 10%, another 20 kg at a profit of 20%, and the remaining 10 kg at a loss of 5%. Find his overall profit percentage.

Solution: Total CP = 50 × 30 = Rs 1500. SP of first 20 kg = 20 × 30 × 1.10 = Rs 660. SP of next 20 kg = 20 × 30 × 1.20 = Rs 720. SP of last 10 kg = 10 × 30 × 0.95 = Rs 285. Total SP = 660 + 720 + 285 = Rs 1665. Profit = 1665 − 1500 = 165. Profit% = (165/1500) × 100 = 11%. The averaging method extends without change to three (or more) portions — just keep adding each portion’s SP to the running total before comparing with total CP.

Worked Example 20C (wastage/spoilage — a portion contributes zero SP, edge case): A trader buys 40 kg of tea at Rs 100/kg. During storage, 4 kg is spoiled and cannot be sold at all. He sells the remaining 36 kg at Rs 120/kg. Find his overall profit or loss percentage on the whole transaction.

Solution: Total CP = 40 × 100 = Rs 4000 (the CP of the spoiled 4 kg was still paid for, so it is not excluded from Total CP). Total SP = 36 × 120 = Rs 4320 (the spoiled 4 kg contributes Rs 0 to SP). Profit = 4320 − 4000 = 320. Profit% = (320/4000) × 100 = 8%. This “wastage” variant is a favourite Tier-II/NTPC trap: many aspirants mistakenly reduce the Total CP to account for only the 36 kg actually sold — but the trader paid for all 40 kg, so the full amount must remain in the denominator (Total CP), while only the sold portion contributes to Total SP.