Lesson 39 of 100 | Quantitative Aptitude / Commercial Arithmetic / वाणिज्यिक अंकगणित
Combined profit, loss and discount situations
Learning outcome
Combine actual costs and receipts across transactions, allowing for discounts and overheads without misusing special profit-and-loss shortcuts.
Totals come before percentages
Add effective costs and actual receipts separately. Include stated overheads once; assume all listed items sell, without unstated charges.
Profit equals total SP minus total CP; a negative result means loss. Divide the gain or shortfall by total CP and multiply by 100. All costs and counts here are positive.
Why not average percentages? Their cost bases may differ. Combining amounts correctly weights each transaction by its cost.
Equal costs: Equal profit and loss percentages create equal monetary changes, so they cancel. With unequal rates, calculate the difference in amounts.
Equal selling prices: Matching positive profit and loss percentages do not cancel: the loss-making item has a larger cost than the profit-making item.
The shortcut loss percentage = (r × r)/100 applies only to two items with equal SP, r% profit and r% loss, where 0 < r < 100. Both rates must use effective costs. Unequal prices or rates require separate calculations.
Use discounted selling prices when combining receipts; costs remain unchanged.
Worked examples
Example 1 — Equal costs. Two items cost ₹1,250 each. One gains 20%, the other loses 20%. Selling prices are 1250 × 1.20 = ₹1,500 and 1250 × 0.80 = ₹1,000. Total cost and total receipts both equal ₹2,500: neither profit nor loss.
Example 2 — Equal selling prices. Two items sell for ₹1,440 each, one at 20% profit and one at 20% loss. Costs are 1440 ÷ 1.20 = ₹1,200 and 1440 ÷ 0.80 = ₹1,800. Total cost = ₹3,000; receipts = ₹2,880. Loss = ₹120, or (120 ÷ 3000) × 100 = 4%.
Example 3 — Discounts and overheads. Ten lamps cost ₹360 each, plus ₹400 batch delivery. Total CP = 10 × 360 + 400 = ₹4,000. Each is marked ₹500; six sell at 10% discount and four at 30%. Receipts = 6 × 450 + 4 × 350 = ₹4,100. Profit = ₹100; profit percentage = (100 ÷ 4000) × 100 = 2.5%.
Common mistakes
Equal SP does not mean equal CP. Unequal rates can even produce profit; verify a shortcut’s conditions.
Practice questions
- Two items cost ₹875 each. One gains 16%, the other loses 8%. Find the overall profit percentage.
- Two items sell for ₹1,980 each, one at 10% profit and one at 10% loss. Find the overall loss percentage.
- Two items sell for ₹1,600 each, one at 60% profit and one at 20% loss. Find the overall result and percentage.
- Twenty lunchboxes cost ₹180 each, plus ₹400 batch freight. All are marked ₹250 each and sold after successive 10% discounts twice. Find the overall profit percentage.
Worked answers
- Gain = 875 × 0.16 = ₹140; loss = 875 × 0.08 = ₹70. Net profit = ₹70 on total cost ₹1,750, giving (70 ÷ 1750) × 100 = 4%.
- Costs = 1980 ÷ 1.10 = ₹1,800 and 1980 ÷ 0.90 = ₹2,200. Loss = 4000 - 3960 = ₹40, giving (40 ÷ 4000) × 100 = 1%.
- Costs = 1600 ÷ 1.60 = ₹1,000 and 1600 ÷ 0.80 = ₹2,000. Profit = 3200 - 3000 = ₹200, giving (200 ÷ 3000) × 100 = 6 2/3%, exactly. Unequal rates allow profit.
- Total CP = 20 × 180 + 400 = ₹4,000. Unit SP = 250 × 0.90 × 0.90 = ₹202.50. Total receipts = ₹4,050; profit = ₹50, giving (50 ÷ 4000) × 100 = 1.25%.