Chapter 13 — Profit-Loss

Cost Price (CP) is the amount a seller pays to buy or make an item. Selling Price (SP) is the amount at which the seller sells the item to a buyer. These two values are the starting point for profit and loss. In many real-life situations a person first pays money to get a product and later receives money when they sell it; that first amount is the CP and the second is the SP.

When we read a story sum, the very first step is to identify which number is the CP and which is the SP. Write them down as CP = … and SP = … so you do not mix them. If a shopkeeper buys an item from a wholesaler, the price he pays is the CP. When he puts a price tag and sells it to the customer, that final price is the SP.

If SP is more than CP, the seller earns a profit. If SP is less than CP, the seller faces a loss. If SP equals CP it is called no profit no loss or break-even. Always compare SP and CP by subtraction to find the amount: profit = SP − CP, loss = CP − SP. These are simple operations but mistakes happen if CP and SP are confused: practise reading words carefully. For example, "sold at Rs. 90" gives SP = 90; "bought for Rs. 60" gives CP = 60.

Also notice that units (rupees, paise) must match. If CP is given per item and many items are sold, multiply CP and SP by number of items before subtracting. For example, if CP = 40 per pen and 10 pens are sold at SP = 50 each, total CP = 400 and total SP = 500 so total profit = 100. Label each step to avoid errors.

Finally, get used to writing a short concluding sentence after calculation, e.g., "Profit = Rs.20". This habit helps in exams and when checking work. Build accuracy by practising small examples and always naming CP and SP clearly before performing any operation.

After identifying CP and SP, the next step is to calculate how much money is gained or lost. The amount of profit or loss is always found by subtracting the smaller value from the larger one. If SP > CP then Profit = SP − CP. If CP > SP then Loss = CP − SP. These give positive numbers which are the actual rupees gained or lost.

Write the operation clearly. For example, if CP = 120 and SP = 150, then Profit = 150 − 120 = 30 rupees. If CP = 200 and SP = 180, then Loss = 200 − 180 = 20 rupees. Practise doing subtraction carefully; errors in subtraction give wrong profit or loss. Always indicate whether the result is a profit or a loss by comparing SP and CP first.

When many identical items are bought and sold, calculate profit or loss per item, then multiply by the number of items. For instance, if a shopkeeper gains 5 rupees on one pencil and sells 12 pencils, the total gain is 5 × 12 = 60 rupees. This method prevents repeated subtraction and keeps work neat.

Word problems may hide CP or SP inside phrases. Look for words like "bought for", "cost", or "paid" to find CP, and words like "sold for", "received" or "gave" to find SP. Sometimes the problem gives discount or profit percent — in those cases compute SP from the given percent before subtracting.

Also learn to state the answer in a sentence: "Profit = Rs.30" or "Loss = Rs.20". This helps the teacher or examiner to see you understand the result. Check your work by adding CP and profit to see if you get SP, or subtracting loss from CP to see if you get SP. This reverse-check is a good habit to find mistakes quickly.

Profits and losses become more useful when expressed as percentages because percentages show size compared to the cost price. Profit percentage tells how big the profit is relative to the cost. Loss percentage does the same for loss. The general rules are: Profit% = (Profit / CP) × 100 and Loss% = (Loss / CP) × 100. Always divide by CP — this is important.

Work step by step: first find the amount of profit or loss (SP − CP or CP − SP), then divide this amount by the CP and multiply by 100 to change the fraction into a percentage. For example, if CP = 200 and profit = 40, Profit% = (40 / 200) × 100 = 20%. This means the seller earned 20 paise for every rupee spent (or 20% of the cost).

When calculating percentages, you can simplify the fraction before multiplying by 100 to make mental arithmetic easier. For example (30/150)×100 → simplify 30/150 = 1/5 then multiply by 100 = 20%. If division does not come out even, give the percent to one or two decimal places as required by the problem. For class 6, most numbers are chosen so the percent is whole or a simple decimal.

Percent is very useful for comparing two deals. Suppose one shopkeeper earns 15 rupees on CP 75 and another earns 20 rupees on CP 150. Convert to percent: 15/75 = 20% and 20/150 = 13.33%. So the first shopkeeper’s profit percent is higher even though the amount is smaller. Practise converting profit and loss amounts into percentages and always label the answer correctly with % sign. Also show the intermediate steps so the teacher can follow your method.

Some problems give a percentage and ask you to find the missing price. Use the percent formulas in reverse. If an item is sold at P% profit, then SP = CP + Profit. Since Profit = (P/100)×CP, SP = CP + (P/100)×CP = CP×(1 + P/100). Therefore CP = SP ÷ (1 + P/100). Similarly, if an item is sold at L% loss, SP = CP×(1 − L/100) and CP = SP ÷ (1 − L/100).

Follow these steps when solving such problems: write the percent as a decimal or fraction (for P% use P/100), form the multiplier (for profit 1 + P/100, for loss 1 − L/100), then divide SP by this multiplier to find CP or multiply CP by the multiplier to find SP. Use a calculator only if necessary; most class 6 problems are clean divisions.

Example: An article sold for Rs.240 at 20% profit. We know SP = CP×1.20 so CP = 240 ÷ 1.20 = 200. For loss example: sold for Rs.90 at 10% loss gives CP = 90 ÷ 0.90 = 100. These methods avoid guessing. Always show the intermediate step that converts percent to multiplier so marks are clear in exams.

When percent is a whole number like 25% or 50%, mental calculations are easy: 25% = 1/4 so SP = CP×1.25 or CP = SP÷1.25. If you get a repeating decimal, give answer to required decimal places or nearest rupee where instructed. Check by calculating the profit or loss from your CP to confirm it yields the given percentage. This reverse-check keeps answers correct and improves confidence in handling percent problems.

In shops, the seller often first writes a higher price on an item called the Marked Price (MP). The shop may then offer a Discount — a reduction from the MP — to attract customers. The price the customer actually pays is the Selling Price (SP). Discount is usually given as a percent of the Marked Price. If discount percentage is d%, then Discount amount = (d/100)×MP and SP = MP − Discount = MP×(1 − d/100).

It is important to understand that discount is different from loss. Discount is a reduction from the seller’s chosen marked price; loss is when SP is less than CP. A seller may give a discount and still make profit if the MP was higher than CP. For example, if CP = 150, MP = 240 and discount = 10% then SP = 240 − 24 = 216. Since SP (216) > CP (150) the seller still makes profit.

Successive discounts are common in sales: for example, "20% and then 10% off". Apply them one after the other. First reduce MP by first discount to get a new price, then apply the second discount on that new price. Mathematically SP = MP×(1 − d1/100)×(1 − d2/100). This product is not equal to MP×(1 − (d1+d2)/100) unless you are asked to approximate: always multiply successive multipliers.

Practice converting percent to decimal multipliers: a 25% discount uses 0.75 multiplier, 10% discount uses 0.90 multiplier. When CP is given too, compare SP after discount with CP to decide profit or loss. Always show steps: compute discount amount, then SP, then compare with CP. This clear method prevents mistakes in examinations and daily shopping problems.

Discount changes the selling price but profit or loss depends on the cost price. To find whether the seller earns profit or suffers loss after giving discount, follow these steps: (1) calculate the selling price after discount from the marked price; (2) compare this SP with the CP; (3) if SP > CP there is profit, if SP < CP there is loss. Always write the arithmetic clearly so the examiner can follow each step.

Example: CP = 300, MP = 400, Discount = 25%. Discount amount = 0.25×400 = 100, so SP = 300. Since SP equals CP there is no profit no loss. Another situation: CP = 250, MP = 400, Discount = 10% → SP = 360; profit = 110 rupees and profit% = (110/250)×100 = 44%.

Many students confuse discount and loss because both reduce price; remember discount is taken from MP while loss is measured with respect to CP. A seller can set a high MP so that after giving discount the SP still leaves room for profit. Conversely, a small MP and a large discount can cause a loss. For successive discounts, compute SP after all discounts then compare with CP.

Also practise reversing the question: sometimes you are given desired profit% and asked to find the minimum marked price if a certain discount will be offered. Start from CP, find SP needed for the profit, then set MP so that after discount SP is obtained. Breaking the problem into these steps makes such questions manageable and clear to the reader.

Goods may be bought and sold more than once. To find the overall gain or loss after two or more transactions, treat each transaction separately and then combine the results. For each sale compute the amount of profit or loss using SP − CP for that transaction. Then add gains and subtract losses algebraically to get the net result.

Important: the CP of the second seller equals the SP of the first seller, because the next seller bought the item at the price the earlier seller sold it. That means when you convert a percentage of profit or loss in the second step you must use the correct base: the CP for that seller, not the original CP. Write each CP and SP clearly for every step to avoid confusion.

Example: A buys a bike for 6000 and sells to B at 10% profit. A’s SP = 6000×1.10 = 6600, so A’s profit = 600. B sells the same bike at 5% loss: B’s SP = 6600×0.95 = 6270, so B’s loss = 330. Overall from original buyer’s viewpoint, if we compare the original CP 6000 and final SP 6270, there is a net gain of 270. Alternatively, add A’s gain 600 and B’s loss (−330) to get 270 net.

When there are several steps, repeat the same method and add all signed gains/losses. Also note that percent changes applied successively are multiplicative: overall multiplier = (1 ± p1/100)×(1 ± p2/100)×... where signs indicate profit or loss. Practise several two-step problems to gain confidence and always show intermediate prices to get full marks in exams.

Ratios help compare profits, discounts and prices without using decimals. A profit percent itself is a ratio (Profit : CP) multiplied by 100. To compare which of two trades is better, form the ratio of profit to cost for both and compare the two fractions. Converting to percent makes the comparison easy to understand.

Example: Trade A gives profit 20 on CP 80; profit ratio = 20/80 = 1/4 = 25%. Trade B gives profit 30 on CP 150; profit ratio = 30/150 = 1/5 = 20%. So Trade A is better by percent even though the absolute amount 30 is larger in B. Using ratios prevents being misled by absolute amounts when costs differ.

Ratios are also useful for comparing final price to marked price. SP/MP shows the fraction of marked price paid after discount. For instance SP/MP = 0.85 means the customer pays 85% of the marked price. When several discounts are applied, multiply the fractions: SP/MP = (1 − d1/100)×(1 − d2/100). Simplify ratios where possible before converting to percent to keep calculations simple for mental math.

If problems involve many items, use ratios to scale results: if profit per item is known, total profit = profit per item × number of items. Practise turning stories into ratios and percentages step-by-step. This strengthens number sense and prepares students for algebraic methods in later years.

Word problems combine several ideas: identification of CP, SP or MP, calculation of discount, conversion of percent to decimal, and then computation of profit or loss. Start by reading the problem slowly and underlining key words: "bought for", "sold for", "discount", "profit", "loss", "percent". Translate the text into simple statements like CP = …, SP = …, MP = … before starting calculations.

Use these steps systematically: (1) Write down the given numbers and what they represent. (2) Decide which formula applies: profit = SP − CP, profit% = (profit/CP)×100, SP = MP×(1 − d/100), etc. (3) Convert percent values to fractions or decimals for multiplication. (4) Compute carefully showing each step. (5) Check the final result by reverse calculation: if you found CP from SP and profit%, multiply CP by the multiplier to see if SP returns.

Many problems include multiple parts, for example "find SP and profit%" or "find CP given SP and profit%". Solve parts in order and reuse intermediate results. When several items are involved, find per-item values first and then multiply. For successive transactions, keep track of the changing CP for each seller — the next seller’s CP equals the previous SP.

Practice translating common phrases: "sold at x% profit" means SP = CP×(1 + x/100), while "sold at x% discount" means SP = MP×(1 − x/100). Drawing a small table with headings CP, MP, Discount%, SP helps organise information. Show answers with sentences that include units and percent symbols to make grading easy and show you understand the solution fully.

Rounding and estimation are useful skills in everyday shopping and for checking exam answers quickly. Estimation helps you see whether an exact answer is reasonable. For example, if CP = 249 and SP = 299, exact profit = 50. Estimating CP ≈ 250 and SP ≈ 300 gives profit ≈ 50, which confirms the exact calculation. Practice rounding to the nearest 10 or 100 to make quick mental checks.

Use estimation to check percentage answers too. If profit = 40 on CP = 200, estimated profit% = (40/200)×100 = 20% exactly; rounding is not needed. If numbers are awkward, round CP and profit before dividing to check that the answer is near the expected value. For instance, CP = 378 and profit = 56; round to 380 and 60 to see profit% ≈ (60/380)×100 ≈ 15.8% so your exact result should be close.

When discounts are involved, estimate the discount by using easy percentages. For example, 20% of 499 ≈ 20% of 500 = 100, so SP ≈ 400. This quick check tells you if you made a calculation error. However, in exams give exact answers unless the question asks for an estimate. Always show both: a quick estimate to check and then the exact working for marks.

Teach the habit of final checking: after solving precisely, do a rough mental check. Estimation reduces silly mistakes and builds number sense. It also helps in time-limited tests where a quick idea of correctness lets you move on with confidence. Practise estimation frequently with different sizes of numbers to become faster and more accurate.

Students often make a few repeating mistakes in profit and loss problems. The main ones are: using SP as denominator when finding percent instead of CP, confusing discount with loss, forgetting to change percent to a fraction or decimal, and mixing up CP and MP. Recognising these errors helps you avoid them and score full marks.

To prevent mistakes, follow a fixed routine for each problem: Step 1 — read carefully and write what CP, SP and MP are; Step 2 — choose the correct formula and write it; Step 3 — convert percent into P/100 before multiplying; Step 4 — compute and then check by reverse calculation. Using the same step-by-step method makes fewer errors and helps in exams where space for working is limited.

Common specific errors include calculating profit% as (Profit/SP)×100 — this is wrong. Always divide by CP. Another frequent error is treating a given discount as loss; check by comparing SP with CP to decide. For successive discounts many students add percentages instead of multiplying factors; practise examples so that you use SP = MP×(1 − d1/100)×(1 − d2/100).

Finally, use small checks: after finding CP from SP and percent, multiply back to get SP and see if it matches the given value. If it does not match, rework your steps. Keep neat work and label numbers as CP, SP or MP. Good notation and careful checking catch most mistakes before you submit your answer.

This combined topic is for quick revision of all formulas and for practicing mixed exercises. Revision: remember the basic relations — Profit = SP − CP, Loss = CP − SP; Profit% = (Profit/CP)×100, Loss% = (Loss/CP)×100. For percent changes use multipliers: SP = CP×(1 + P/100) for profit and SP = CP×(1 − L/100) for loss. For discount use SP = MP×(1 − d/100) and for successive discounts multiply factors.

Practice strategy: start by classifying the problem — is it asking for amount (profit/loss), percent, CP, SP or MP? Label given values. Use fractions or decimal multipliers for percent calculations. Always show intermediate steps. For multi-step problems write every intermediate price: start CP1, SP1, then CP2 = SP1, SP2, and so on. This prevents mixing bases and makes it easy to compute overall gain or loss by summing signed amounts.

Example exercises help combine ideas: (1) find profit% when many items are sold; (2) find CP from SP and profit%; (3) calculate SP after one or more discounts and check profit against CP; (4) solve two-step transactions for net gain or loss. Time yourself on a set of 6–8 mixed problems to build speed while keeping accuracy. After solving, check each answer by doing the reverse calculation (e.g., multiply CP by multiplier to get SP).

Keep a short list of useful mental multipliers for common percentages: 10%→0.90 or 1.10, 25%→0.75 or 1.25, 50%→0.50 or 1.50. Practise with these to speed up written work. Finally, write a one-line final answer for each question with units and % symbol so the examiner can see your result clearly.