Lesson 3 of 3 | Commercial Arithmetic Applications
Profit, Loss, Markup, and Successive Discounts
Learning outcomes
This lesson connects percentages with the commercial-arithmetic questions that frequently appear in SSC CGL. You will learn to identify the correct base for profit or loss, separate marked price from cost price, handle successive discounts, and solve dishonest-dealer questions with a consistent method.
1. The four prices
- Cost price (CP): what the seller pays.
- Selling price (SP): what the customer pays.
- Marked price (MP): the displayed price before discount.
- Profit or loss: the difference between SP and CP.
Unless a question explicitly says otherwise:
[ \text{Profit percentage}=\frac{SP-CP}{CP}\times100 ]
[ \text{Loss percentage}=\frac{CP-SP}{CP}\times100 ]
Profit and loss percentages use cost price as the base. Discount percentage uses marked price as the base.
2. Multipliers make calculations faster
Replace each percentage operation with a multiplier:
| Change | Multiplier |
|---|---|
| 20% profit | 1.20 |
| 15% loss | 0.85 |
| 25% discount | 0.75 |
| 12.5% profit | 1.125 or 9/8 |
Worked example
An item costs ₹1,600 and is sold at 17.5% profit.
[ SP=1600\times1.175=₹1,880 ]
Because 17.5% = 7/40, you can also calculate profit as 1600 × 7/40 = 280.
3. Finding the original cost
A product is sold for ₹2,760 at a 15% profit. Here SP is 115% of CP:
[ CP=2760\times\frac{100}{115}=₹2,400 ]
Do not subtract 15% of ₹2,760. The profit percentage was based on CP, not SP.
4. Markup and discount
A shopkeeper marks an item 40% above CP and gives a 10% discount.
Use a base CP of 100:
- MP = 140
- SP = 140 × 0.90 = 126
- profit = 26%
Net profit is 26%, not 30%, because markup and discount use different bases.
Required markup for a target profit
A shop wants 20% profit after a 25% discount. Let CP = 100, so target SP = 120. Since SP is 75% of MP:
[ MP=120\times\frac{100}{75}=160 ]
Required markup is 60%.
5. Successive discounts
Two discounts of 20% and 10% produce a single equivalent discount:
[ 20+10-\frac{20\times10}{100}=28% ]
With MP = 100, the customer pays 80 × 0.90 = 72.
Three discounts are safest with multipliers. Discounts of 10%, 20%, and 5% leave:
[ 0.90\times0.80\times0.95=0.684 ]
Equivalent discount = 31.6%.
6. Same selling price, one profit and one loss
Two articles sell for the same price. One is sold at (x%) profit and the other at (x%) loss. There is always an overall loss of:
[ \frac{x^2}{100}% ]
For 20%, overall loss = 4%.
Proof with SP = ₹120 each:
- first CP = 120/1.20 = 100
- second CP = 120/0.80 = 150
- total CP = 250, total SP = 240
- loss = 10/250 × 100 = 4%
This shortcut applies only when selling prices are equal and the profit/loss rates are equal.
7. False weights and dishonest dealers
A dealer claims to sell at CP but gives 900 g instead of 1 kg.
Assume CP of 1 kg = ₹100. The dealer charges ₹100 but gives goods costing ₹90.
[ \text{Profit percentage}=\frac{10}{90}\times100=11\frac{1}{9}% ]
The base is the actual cost of 900 g.
If the dealer also marks up or discounts, combine the price multiplier and the quantity shortage carefully rather than using an isolated shortcut.
8. Profit after damaged goods
A trader buys 100 items at ₹80 each. Ten are damaged and cannot be sold. To earn 12.5% overall profit, total required revenue is:
[ 100\times80\times1.125=₹9,000 ]
Revenue must come from 90 items, so SP per sellable item = ₹100.
Overall calculations must include the cost of damaged or unsold units.
9. Common exam traps
- Using SP instead of CP as the profit base.
- Subtracting a discount directly from the markup percentage.
- Adding successive discounts.
- Applying the equal-SP shortcut when the selling prices differ.
- Ignoring damaged inventory in total cost.
- Using the stated kilogram as the cost base when the dealer delivers less.
10. Timed solution checklist
- Label every number CP, SP, or MP.
- Choose CP = 100 when only percentages are given.
- Convert changes into multipliers.
- Apply operations in the order stated.
- Calculate the final SP and compare it with CP.
- Check that profit gives SP > CP and loss gives SP < CP.
Check your understanding
- CP ₹750, profit 16% → SP = ₹870.
- SP ₹1,020 at 15% loss → CP = 1020 × 100/85 = ₹1,200.
- Markup 50%, discount 20% → SP on CP 100 is 120, so 20% profit.
- Discounts 25% and 12% → equivalent discount = 25 + 12 − 3 = 34%.
- A dealer gives 800 g for the price of 1 kg at claimed CP → profit = 200/800 × 100 = 25%.
Revision recap
Profit and loss use CP; discount uses MP. Multipliers prevent base errors. Successive discounts are multiplied, not added. For inventory loss and false weights, compare total revenue with the actual total cost.