Profit and loss

1. Basic concepts and the correct percentage base

Cost price, abbreviated CP, is the amount spent to acquire an article. Selling price, or SP, is the amount received on its sale. Marked price, or MP, is the price displayed before a discount. A transaction produces profit when SP exceeds CP, loss when SP is below CP, and neither profit nor loss when the two are equal. In arithmetic problems, profit and gain are usually interchangeable terms.

The relevant cost may include more than the purchase price. If a question specifies transport, repairs, packaging or other expenses incurred to make an article saleable, add these to the purchase price unless the wording indicates otherwise. For example, a bicycle bought for ₹2,400 and repaired for ₹600 has an effective CP of ₹3,000. Selling it for ₹3,450 yields ₹450 profit, or 15%.

The denominator is decisive. Profit percentage = Profit/CP × 100, while loss percentage = Loss/CP × 100. A ₹20 profit on an article costing ₹80 represents 25% profit, although the profit is only 20% of its ₹100 selling price. The latter measure is a profit margin on sales, not the ordinary profit percentage used in CSAT questions.

Reading discipline therefore comes before calculation. Identify whether the question mentions profit on cost, profit as a proportion of sales, or discount on marked price. These percentages describe different relationships even when their numerical values look similar.

• If CP is ₹100 and profit is 20%, SP is ₹120.
• If SP is ₹120 and profit is 20%, CP is ₹120/1.20 = ₹100, not ₹96.
• When only percentages matter and no absolute amount is fixed, assuming CP = ₹100 often simplifies the calculation.

2. Core formulas, ratios and reverse calculations

For a profit of p%, SP = CP × (100 + p)/100. Reversing the equation gives CP = SP × 100/(100 + p). For a loss of l%, SP = CP × (100 − l)/100 and CP = SP × 100/(100 − l). Reverse calculations require division by the relevant factor; subtracting the percentage from SP incorrectly changes the base.

Ratios frequently make these formulas easier to use. A 25% profit gives CP:SP = 4:5. A 20% loss gives CP:SP = 5:4. Thus, an article sold for ₹960 at a 20% loss cost ₹1,200. Useful fraction equivalents include 12.5% = 1/8, 16⅔% = 1/6, 20% = 1/5 and 25% = 1/4.

When alternative selling prices produce different profit or loss rates on the same article, their difference can reveal CP directly. If selling at ₹540 gives a 10% loss, while selling at ₹660 gives a 10% profit, the ₹120 difference equals 20% of CP. Hence CP is ₹600. A profit and a loss lie on opposite sides of cost, so their rates are added when finding this gap.

Quantity equivalence is another common pattern. If the cost price of 12 identical articles equals the selling price of 10, then 12CP = 10SP. Consequently, SP/CP = 6/5 and profit is 20%. Always distinguish the number of articles from the price per article.

• Same CP, two profit rates: difference in SP equals CP multiplied by the difference between the rates.
• Same CP, one profit and one loss: difference in SP equals CP multiplied by the sum of the rates.
• For identical articles, convert quantity statements into equations using unit CP and unit SP.

3. Markup, discounts and successive transactions

Markup raises cost price to marked price; discount lowers marked price to selling price. If markup is m% and discount is d%, then SP/CP = (1 + m/100)(1 − d/100). An article marked 40% above cost and discounted by 10% sells at 1.40 × 0.90 = 1.26 times cost, giving 26% profit rather than 30%.

Successive discounts act on progressively reduced prices. Two discounts of a% and b% have an equivalent discount of a + b − ab/100 percent. Discounts of 20% and 10% therefore equal a single 28% discount. On an MP of ₹1,000, the price becomes ₹800 and then ₹720. Reversing their order does not alter the final price when both apply to the whole remaining amount.

To determine the markup needed for a target profit, work backwards. If a trader wants 20% profit after offering a 25% discount, MP/CP = 1.20/0.75 = 1.60. The required markup is 60%. Distinguishing the target SP from MP prevents the common mistake of merely adding profit and discount rates.

In a chain of sales, each buyer's purchase price becomes the next cost base. If A sells to B at 20% profit and B sells to C at 25% profit, C pays 1.20 × 1.25 = 1.50 times A's original cost. This is a 50% increase across the chain, not a 50% profit earned by either trader individually.

• Use multiplication factors: 15% increase becomes 1.15; 15% decrease becomes 0.85.
• Do not treat tax collected for remittance as trader profit. Follow the question's stated treatment of taxes and costs.
• A percentage discount and a fixed-rupee coupon may produce different results when their order changes.

4. Equal prices, combined transactions and false weights

Equal percentage gains and losses do not necessarily cancel. Suppose two articles are each sold for ₹960, one at 20% profit and the other at 20% loss. Their costs are ₹800 and ₹1,200 respectively. Total CP is ₹2,000 against total SP of ₹1,920, producing a 4% loss. The shortcut x²/100% loss applies only when selling prices are equal and the gain and loss percentages are equal.

If the two articles instead have equal cost prices, equal percentage profit and loss cancel. This contrast is a frequent test of comprehension. For any collection of transactions, the safest general method is to total all costs and all selling prices, then calculate the net difference as a percentage of total cost.

Overall profit percentage is a cost-weighted average of individual profit percentages, with losses treated as negative. Spending ₹1,000 on goods earning 20% and ₹3,000 on goods losing 10% gives ₹200 profit and ₹300 loss. The net loss is ₹100 on ₹4,000, or 2.5%, not the simple average of the two rates.

False-weight problems compare receipts with the cost of the quantity actually supplied. Suppose a trader charges the purchase cost of 1 kilogram but supplies only 800 grams. Taking the cost of 1 kilogram as ₹100, revenue is ₹100 and actual cost is ₹80. Profit is therefore 25%. The calculation describes deceptive conduct, not a legitimate trading practice.

• When charging the cost-price rate for 1,000 grams but supplying w grams, gain percentage = (1,000 − w)/w × 100.
• If a trader also changes the quoted price, incorporate both the price factor and the actual quantity supplied.
• For damaged stock or unsold goods, check whether the question assigns any recovery value to the remaining stock.

5. CSAT problem-solving strategy and checks

Begin by labelling CP, MP and SP, then mark the base attached to every percentage. Choose a convenient assumed value for purely proportional questions; use the actual figures when the problem fixes a rupee difference. Write one equation before inserting numbers. This reduces unnecessary arithmetic and makes hidden assumptions visible.

Use fractions and cancellation wherever possible. A 20% profit followed by a 16⅔% reduction multiplies the original value by 6/5 and then 5/6, restoring it exactly. By contrast, a 20% increase followed by a 20% decrease leaves 96% of the original value. Equal percentage increases and decreases are not inverses.

Finally, check direction and magnitude. A positive discount must reduce MP; a profitable sale must exceed effective CP. In equal-selling-price gain–loss questions, expect a net loss. Use approximation to eliminate implausible options, but retain exact ratios when answer choices are close. Timed practice should build accuracy in recognising the structure rather than dependence on isolated shortcuts.

• Frequent traps: wrong denominator, omitted expenses, added successive percentages and unweighted averaging.
• For recovery after an l% loss, the required increase on the reduced value is 100l/(100 − l)%.
• A 25% fall therefore requires a 33⅓% increase to return to the starting value.

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