Profit and Loss Tricks for SSC, RRB and Bank Exams

Profit and loss questions look wordy, and that is most of the difficulty. The maths underneath is one idea: everything is measured against the cost price, unless the question says otherwise. Once a student really believes that, the chapter turns from "sixteen formulas" into about four ideas plus a habit.

The trouble is that textbooks throw all sixteen formulas at you on page one, and coaching notes add a few more. You memorise, forget under pressure, and then stare at a question about "marked price is 40% above cost, a discount of 25% is given" trying to recall which formula to use. This post skips the memorisation and builds the chapter from ratios. Every number in the examples has been calculated and checked, so if you follow the method you will get the same answers.

If you have not yet built a solid base in percentages, read the percentage tricks post first. Profit and loss is percentage with labels attached, and the fraction table from that post does most of the work here.

The only definitions you need

Cost price (CP) is what the seller paid. Selling price (SP) is what the seller got. Profit = SP - CP when SP is more; loss = CP - SP when CP is more. Profit percent and loss percent are always calculated on cost price, unless the question says "on selling price".

Example: an article bought at 800 and sold at 920. Profit = 120. Profit percent = 120/800 = 15%. Nothing more.

Marked price (MP) is the price printed on the label. Discount is a reduction from MP, and discount percent is measured on MP, not on CP. SP = MP - discount. These two rules, "profit on CP, discount on MP", explain about half the errors students make.

That is the whole vocabulary. The other terms (overhead, transportation, repair) are only additions to CP.

The ratio method: the tool that replaces most formulas

Stop writing formulas. Write CP : SP as a ratio, and read every profit or loss percentage off it.

Turn the profit or loss into a fraction of CP:

• Profit 20% = 1/5, so CP : SP = 5 : 6.
• Profit 25% = 1/4, so CP : SP = 4 : 5.
• Profit 12.5% = 1/8, so CP : SP = 8 : 9.
• Loss 10% = 1/10, so CP : SP = 10 : 9.
• Loss 20% = 1/5, so CP : SP = 5 : 4.
• Loss 16.67% = 1/6, so CP : SP = 6 : 5.

The rule: for profit 1/n, CP : SP = n : (n+1). For loss 1/n, CP : SP = n : (n-1). If you know the fraction table from percentages, you are done.

Here is how quick it becomes. "An article is sold at 12.5% profit for 1,350. Find the cost price." 12.5% = 1/8, so CP : SP = 8 : 9. SP is 1,350 which is 9 parts, so one part is 150, and CP is 8 parts = 1,200. You did no percentage multiplication.

Another: "By selling an article for 960 a shopkeeper makes a 20% profit. What is the CP?" 20% profit means CP : SP = 5 : 6. SP = 960 = 6 parts, so one part is 160 and CP = 800. And if he had made a 20% loss on the same 960, CP : SP = 5 : 4, so one part is 240 and CP = 1,200. You will see both of these numbers again in a minute.

Two-step problems: the "if he sold it for more" type

"A man sells an article at a loss of 12%. If he had sold it for 90 more, he would have gained 3%. Find the cost price."

Do not set up two equations. The two selling prices differ by 15% of CP (from -12% to +3%), and that difference is 90. So 15% of CP = 90, and CP = 600. Check: 12% loss gives SP = 600 x 0.88 = 528; 3% gain gives SP = 600 x 1.03 = 618; 618 - 528 = 90. Correct.

This is the "gap method". Whenever two scenarios differ only in profit percentage, the gap in rupees equals the gap in percentages times CP. The same pattern shows up in interest questions and it is one of the neatest tricks in the chapter.

Same selling price, one gain and one loss

This is the most tested rule of the whole chapter, and there is nothing to memorise if you can do the ratio method.

"Two articles are sold at 960 each. On one there is a gain of 20% and on the other a loss of 20%. What is the overall result?"

The 20% gain article has CP 800 (from 5 : 6). The 20% loss article has CP 1,200 (from 5 : 4). Total CP = 2,000. Total SP = 1,920. Loss = 80, and 80/2,000 = 4%.

The shortcut: whenever two articles are sold at the same price, one at x% gain and the other at x% loss, the overall result is always a loss of x^2/100 percent. Here that is 400/100 = 4%. It is a loss, always, because the article with the loss had a larger cost. Note that the rule needs the same SP and the same percentage. If the percentages differ, use the ratio method.

Let me show that. Two articles sold at 1,000 each, one at 25% gain, the other at 20% loss. CP of the first = 1,000/1.25 = 800. CP of the second = 1,000/0.8 = 1,250. Total CP = 2,050, total SP = 2,000, so a loss of 50, which is 50/2,050 = about 2.44%. The formula x^2/100 would not apply here; the ratio method does.

The opposite rule: if the two articles have the same cost price, the overall percentage is just the average of the two percentages. A man buys two articles at the same price, sells one at 20% profit and the other at 10% loss. Overall profit = (20 - 10)/2 = 5%. That is valid only when CP is equal. Read the question for which of the two prices (cost or selling) is equal.

Marked price and discount

This is where the wording gets thick. The template is: an article is marked at some percentage above CP, a discount is offered, find profit or loss.

"A shopkeeper marks an article 40% above cost and offers a 25% discount. Find the profit percent." Take CP = 100. MP = 140. SP = 140 x 0.75 = 105. Profit = 5%. Notice that it is the same as a successive change: +40% then -25% = 40 - 25 - 10 = +5%. Yes, the successive percentage rule from the percentage chapter applies directly. Markup then discount = successive change.

"A shopkeeper marks a product 25% above cost and gives a 20% discount. What's the result?" 1.25 x 0.8 = 1.0. He earns nothing; SP equals CP. This pair (25% up, 20% down) is also the price-and-consumption pair, and a very popular question in the exam. Memorise: mark 25% above, discount 20%, no profit no loss.

Some more combinations worth knowing:

• Mark 20% above, discount 10%: 1.2 x 0.9 = 1.08, so 8% profit. If CP is 500, MP is 600, SP is 540, profit 40, which is 8% of 500. Correct.
• Mark 40% above, discount 20%: 1.4 x 0.8 = 1.12, so 12% profit.
• Mark 50% above, discount 20%: 1.5 x 0.8 = 1.2, so 20% profit.

Now the reverse. "A shopkeeper wants a 12% profit after giving a discount. He marks the goods 40% above cost. What discount does he give?" 1.12 / 1.4 = 0.8. So SP is 80% of MP, and the discount is 20%. Check: CP 100, MP 140, SP 112, 140 x 0.8 = 112.

And the marking question: "A trader allows a 10% discount and still wants a 20% profit. How much above cost should he mark?" MP = 1.2 / 0.9 = 1.3333 of CP. So he should mark 33.33% above cost. Check with CP 900: MP = 1,200, SP = 1,200 x 0.9 = 1,080, and 1,080 = 900 x 1.2. Correct. The fraction 33.33% is 1/3, which you can see if you write the ratio 12 : 9 = 4 : 3.

Direct discount from MP and SP

"An item marked at 500 is sold for 425. What is the discount percent?" Discount = 75. 75/500 = 15%. Use MP as the base. Do not divide by CP, which is not even given.

Successive discounts

If an item is offered at two successive discounts of a% and b%, the equivalent single discount is a + b - ab/100. So 20% and 10% together give 20 + 10 - 2 = 28%. Check: on a marked price of 1,200, the first discount brings it to 960, and the second to 864. The total discount is 336, and 336/1,200 = 28%. Correct.

Two 10% discounts are not 20% but 19%. A discount of "buy 3 get 1 free" means you pay for 3 out of 4, which is a 25% discount. "Buy 2 get 1 free" means paying for 2 out of 3, which is 33.33% off. These come up in shopping-style questions, and the correct discount is always the free items divided by the total items you get.

Profit percent on selling price

The words "profit percent on selling price" appear perhaps once in a paper, and they are a trap. The rule: profit % on SP = (profit / SP) x 100.

If profit is 25% on cost, then CP : SP = 4 : 5, profit is 1 part of 5 on SP, which is 20%. If profit is 20% on SP, then profit is 1 part of SP = 5 parts, so CP = 4 parts and profit on cost = 25%.

Quick conversion: profit 1/n on cost is 1/(n+1) on SP. That is the same ratio trick again. Never try to plug these into the ordinary formula, which assumes cost as base.

Chain transactions

"A sells an article to B at 20% profit, and B sells it to C at 25% profit. If C pays 1,500, what did A pay?" Work backwards in ratio: C's price = 1.25 x B's price, and B's price = 1.2 x A's cost. So 1,500 = 1.5 x A's cost, and A's cost = 1,000. That is an example of two successive percentages, and you should not compute the individual amounts unless the question asks for them. If you do, B's price = 1,500/1.25 = 1,200 and A's cost = 1,200/1.2 = 1,000, which agrees.

False weights and dishonest dealers

A dishonest shopkeeper sells goods claiming to weigh a kilogram but actually giving less. The gain is on the goods he keeps.

"A trader sells at cost price but uses a weight of 900 g instead of 1 kg. What is his gain percent?" He gives 900 g and charges for 1,000 g. The goods he saves are 100 g, and the base is what he actually gave, which is 900 g. Gain = 100/900 = 11.11%.

The rule: gain % = (error / true weight given) x 100, where the base is the weight he actually delivered, not the weight he claims. If he uses 800 g for 1 kg, gain = 200/800 = 25%. The base is 800, not 1,000. Students who divide by 1,000 get 20%, which is a classic trap, and an option for 20% is almost always present.

If the trader also makes a stated profit on the actual weight, combine the two with the successive-change rule. If he uses 900 g for 1 kg and also marks his goods 20% above cost, the profit is (1,000/900) x 1.2 - 1, and you evaluate the multiplier. Do not add 11.11 and 20.

"CP of x articles equals SP of y articles"

"The cost price of 12 articles equals the selling price of 10 articles. Find the profit percent." Assign a unit: let CP of one article be c. Then 12c = 10 s, where s is the SP of one. So s/c = 12/10 = 1.2, a profit of 20%. In general, if CP of x = SP of y, then profit % = (x - y)/y x 100 when x > y.

Another version: "the cost of 15 articles equals the selling price of 12". Profit = (15 - 12)/12 = 25%. And "the cost of 10 articles equals the selling price of 8" gives (10 - 8)/8 = 25% again. Check: 10c = 8s gives s/c = 1.25.

The base here is the number of articles sold, y. If you use x, you get the wrong percentage. This is the same idea as the dishonest weight rule: divide by the smaller count, the one on the selling side.

Mixed selling

Some questions mix part sale at one price and part at another. The way to solve them is to write total CP and total target SP, compute what has already been recovered and find what is left.

"A trader buys 100 kg of rice at 40 per kg. He sells 60 kg at a profit of 10%. At what price per kg must he sell the remaining 40 kg to make an overall profit of 20%?"

• Total CP = 100 x 40 = 4,000.
• Target SP for overall 20% profit = 4,800.
• The first 60 kg sold at 44 per kg (10% above 40) = 2,640.
• Remaining SP needed = 4,800 - 2,640 = 2,160 for 40 kg, which is 54 per kg.

Check: 54 is 35% above 40, and 40 kg at 54 = 2,160. Total SP = 2,640 + 2,160 = 4,800. Correct. If the question asked for the profit percent on the second lot, it would be 35%. Notice it is high because the first lot earned less than average.

A related type is mixing two varieties. A shopkeeper mixes 3 kg of tea at 60 per kg with 2 kg at 80 per kg. The mix costs (3 x 60 + 2 x 80)/5 = 340/5 = 68 per kg. To make 10% profit, he must sell at 68 x 1.1 = 74.80 per kg. Use weighted average for cost, then apply the profit percentage to that.

Profit questions with fixed unit prices

A common shopping pattern: "A man buys pens at 2 for 5 and sells at ..." Convert to a per-unit price first and only then compare. Selling 5 pens for 10 and buying 3 pens for 5 is a comparison between 2 per pen and 1.67 per pen. Convert to a common quantity (the LCM, here 15 pens) and compare costs and revenues. It is much safer than juggling fractions.

Overheads, repairs and freight

The cost price in the profit formula includes everything the seller spent to get the article ready to sell. If a man buys a bicycle at 3,000, spends 500 on repair and sells at 4,200, his CP is 3,500, and profit = 700, which is 20% of 3,500. Students who forget the repair cost give the answer as 1,200/3,000 = 40%. That is wrong. The question will often mention the extra in a different sentence, so read to the end before starting.

A little on simple and compound interest overlap

Profit and loss and interest share ideas. A trader who buys goods and sells on credit is really earning interest. A percentage rise followed by another is the same shape as compound interest, and the a + b + ab/100 rule sits underneath both. If you master one, you get the other faster. A profit problem will not always tell you it is secretly successive percentages, so train yourself to spot the shape.

Worked walk-through of a typical question

Let me take a single question and solve it three ways, to show how much time the ratio method saves.

"A shopkeeper marks his goods 30% above cost. He allows a discount of 10% on the marked price. If he sold the item for 1,170, what was the cost price?"

Way one: write equations. MP = 1.3 CP. SP = 0.9 MP = 0.9 x 1.3 CP = 1.17 CP. So 1.17 CP = 1,170 and CP = 1,000. Fine, but you spent time writing.

Way two: markup then discount is successive change, +30 -10 -3 = +17%. CP : SP = 100 : 117. SP = 1,170 so CP = 1,000.

Way three: 1,170 is 117 x 10. You see that and CP is 100 x 10 = 1,000. 5 seconds.

The third way is not magic. You just know that 30% up and 10% down is 17% up, and 1,170 divided by 1.17 is 1,000. That knowledge comes from doing enough problems in the ratio style.

Handling the messy questions

Some questions hide the structure in the wording. "A fruit seller gets 120 kg for the price of 100 kg" means his cost per kg is 100/120, about 83.3% of the normal price. When a question gives a free quantity, convert it to a discount on price first. A dealer who gives 1 free for every 4 bought is giving a discount of 1/5, or 20%, on the total count. If he also marks up 25%, the two cancel (the 25% up, 20% down pair again).

When you see "loss", "profit" and "discount" all in the same sentence, draw three lines on your rough sheet: CP, MP, SP. Write the percentage relationship between them along each line. Once you have the three numbers as ratios (say, 100, 130, 117), the question answers itself. If a question feels tangled, it is usually because you are trying to keep all three in your head.

Habits and mistakes

The first mistake is choosing the wrong base. Profit is on CP; discount is on MP; false-weight gain is on the weight given. If your answer looks close to an option but not exactly, you have probably used the wrong base.

The second is treating markup and discount as if they cancel. They never do unless the pair is exactly matched (like 25% up, 20% down). A 20% markup followed by a 20% discount is a 4% loss, not zero.

The third is skipping the check. Every profit and loss question can be verified in 10 seconds by plugging in a friendly CP. For the marked-price question above, CP = 900 gave MP = 1,200. Write this down and test the option.

The fourth is over-reading. Profit and loss questions are wordy, but the words map onto three quantities. If you cannot find CP, MP and SP in the sentence, read the sentence again.

The fifth is calculating decimals when a fraction is available. 12.5% profit is 1/8; do not multiply by 1.125.

How to practise this chapter

A useful plan is one week, then revisit.

Days one and two: learn the ratio table for profit and loss (1/n gives n : n+1; loss 1/n gives n : n-1). Do 20 questions using only the ratio method. Do not write formulas.

Days three and four: markup and discount. Do 20 questions where you write CP = 100 and compute MP and SP. Then do 10 where you go backwards from a target profit.

Day five: the special cases: same SP with x% gain and loss, false weight, "CP of x = SP of y", partial sale. These are the ones that repeat.

Day six: a timed set of 25 mixed questions in about 20 minutes. Note which ones took over 50 seconds.

Day seven: revisit those slow ones, redo them with ratios, then take a sectional test. The free sectional practice at /practice is enough at this stage. Once you can do a sectional test on arithmetic comfortably, try it inside a full mock test, where you will see how the same skill holds up when you are tired in the third section.

Then return to the chapter again after two weeks, because profit and loss is one of those topics where speed decays if you leave it. It ties in with time and work, since both use the same "assume a convenient number" approach, and with data interpretation, where profit and cost columns are common.

What to skip

Skip questions with three or four moving parts at the start of the section if you are short on time. Profit and loss has a small number of long, layered questions in the tougher papers, and you can usually leave them for the end. Attempt the direct ones (find CP, find profit percent, discount chains) first.

Skip memorising a table of 25 formulas. If you understand the ratio method and the successive change rule, you can rebuild everything.

If CP comes out as a large non-round number, you probably mis-read the question. Real exam questions are designed to produce clean answers, so a messy intermediate value is a hint to re-read. And keep chains as one multiplier (1.2 x 0.9), evaluated once, instead of computing each stage separately.

Your next step

Write down the ratio table (profit 1/n gives n : n+1) and the three combinations that come up most: 25% markup with 20% discount for no profit, 20% markup with 10% discount for 8% profit, and same SP with x% gain and loss for a loss of x^2/100 percent. Then do 25 mixed questions with a timer and mark any that took longer than a minute.

FAQ

Is profit always calculated on cost price? Yes, unless the question explicitly says "profit percent on selling price". Discount is always on marked price.

Why is the same-SP, equal-percentage case always a loss? Because the article sold at a loss had a higher cost price, so the loss in rupees is larger than the gain in rupees. The net loss is x^2/100 percent of the combined cost.

How do I convert a profit on cost to a profit on selling price? If the profit is 1/n of cost, it is 1/(n+1) of the selling price. So 25% on cost is 20% on selling price.

What if the question gives no actual price? Assume CP = 100, or a number that keeps all the percentages whole (like 200 or 900). The answer in percent will not change.

Do I need to learn separate formulas for false weights? No. Use gain = error / weight actually given. The only care is to divide by the weight given, not the weight claimed.

Is this chapter common to SSC, RRB and bank exams? The concepts are the same. SSC papers tend to include more markup and discount chains, and bank papers more often bury it inside data interpretation or word problems. Check the latest notification for the exact pattern of your exam.