Profit and Loss

This is the starting point. CP and SP are known (or one of them can be found in one step), and you have to find the profit, the loss or their percentage.

Profit = SP − CP, Profit% = ProfitCP × 100
Loss = CP − SP, Loss% = LossCP × 100

The denominator is always CP. Read that line again, because the exam is built to test exactly this. In "SP = ₹3,300 and profit = ₹300, find profit%", CP is not given directly. First find CP = 3,300 − 300 = ₹3,000, and then profit% = 3003000 × 100 = 10%. A student who writes 3003300 gets 9.09% and loses the mark.

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Use the fraction chart. Profit% = ProfitCP is just a fraction. Cancel it down and read it as a percentage: 1236 = 13 = 3313%, and 45450 = 110 = 10%.

When there are many units. If a workshop makes 20 tables and earns ₹350 profit on each, total profit = profit per unit × number of units = 20 × 350 = ₹7,000. There is no need to work out the full CP and full SP unless the question asks for them.

Break-even. No profit, no loss means total SP = total CP. If a food stall at a fair costs ₹7,500 to set up and a plate sells for ₹150, the stall must sell 7500150 = 50 plates just to recover the cost.

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"Profit is 27% of the cost." This only checks whether you know the definition. Profit% means profit as a percentage of cost, so the answer is 27%. Nothing is to be added or subtracted.

Radhika buys a diary for ₹48 and sells it for ₹56, a profit of ₹8. Since 848 = 16, her profit% is 1623%.

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Profit per rupee. Sometimes the question does not ask for a percentage at all. It says "how much does he earn on every rupee he spends?" That is the same profit%, only read out of 1 instead of 100. A profit of 25% means 25 paise on every rupee of CP, and 40 paise on every rupee means a profit of 40%. If a stall owner buys goods for ₹600 and sells them for ₹690, the profit is ₹90 on ₹600, which is 90600 = ₹0.15, that is 15 paise on every rupee.

Read the base carefully. "15 paise on every rupee of sale" is a profit on SP. Take SP = 100: profit = 15, CP = 85, so the profit on CP is 1585 × 100 = 17.65%, not 15%.

Now one price and the percentage are given, and the other price is asked. It can go in two directions, and both run on the multiplying factor.

CP → SP (going forward). Take CP = 100. Profit p% ⇒ SP = 100 + p. Loss l% ⇒ SP = 100 − l. So

SP = CP × 100 + p100 or SP = CP × 100 − l100

CP ₹350 at 20% profit ⇒ SP = 350 × 120100 = ₹420. With the fraction it is quicker: 20% = 15, CP : SP = 5 : 6, so SP = 350 × 65 = ₹420.

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SP → CP (going back). This is where most mistakes happen. SP is ₹960 at 20% loss, and that 20% was on CP, not on 960. Take CP = 100, so SP = 80. Now 80 units = ₹960, so 1 unit = ₹12 and CP = ₹1,200.

CP = SP × 100100 + p or CP = SP × 100100 − l

Why can't we add 20% of 960 to 960? Because 20% of 960 is 192, and 960 + 192 = 1,152, which is not the CP. A percentage is always of its own base: going back from SP to CP, divide by the factor and never add or subtract the same percentage of SP.

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Fractions let you do it mentally. 12.5% profit = 18 ⇒ CP : SP = 8 : 9. SP ₹1,080 is 9 parts, so 1 part = 120 and CP = 8 × 120 = ₹960.

Radhika sells her old cycle for ₹1,170 at 10% loss. CP = 100 ⇒ SP = 90. So 90 units = ₹1,170, 1 unit = ₹13 and her CP was ₹1,300.

Take a question like this: "An article is sold at a loss of ₹180. The loss is 6%. Find the CP." Here the rupee amount is the profit or loss itself, not SP or CP, and the percentage tells you what part of CP that profit or loss is.

So take CP = 100 units. A 6% loss means loss = 6 units. Now 6 units = ₹180, so 1 unit = ₹30 and CP = 100 units = ₹3,000. This is the same percent-to-amount pair you used in the Percentage chapter: 6% ↔ 180, so 100% ↔ 3,000.

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CP = Profit (or Loss) × 100Profit% (or Loss%)

Once CP is known, SP is one more step: SP = CP + profit, or SP = CP − loss.

"The difference between CP and SP is ₹630." Do not be confused by the wording. The difference between CP and SP is the profit or the loss. So "difference = ₹630 and loss = 18%" means loss = ₹630 = 18% of CP, which gives CP = ₹3,500.

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"Find the value of profit." If CP = ₹640 and profit = 15%, the profit is 15% of 640 = ₹96. That is Type 1 read backwards.

Radhika makes a profit of ₹45 on a photo frame, and that is a 15% profit. 15% ↔ 45 ⇒ 1% ↔ 3 ⇒ 100% ↔ 300. So the frame cost her ₹300, and she sold it for ₹345.

In real business the buying price is not the whole cost. A trader pays for transport, storage, packing and advertising, and a man who buys an old car pays for its repair. Every expense made before the sale is part of CP.

Effective CP = Purchase price + Repair + Transport + Storage + all other expenses

Profit% is then calculated on this effective CP, never on the purchase price alone. If an old bike is bought for ₹24,000 and ₹3,000 is spent on repairs, CP = ₹27,000. Selling it for ₹29,700 gives a profit of ₹2,700, that is 270027000 = 10%, not 570024000.

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Working backwards. "Sold for ₹9,000 at 20% profit after spending ₹1,500 on polishing. Find the purchase price." First find the total cost from SP: 9,000 = 120% of total cost, so total cost = ₹7,500. Then take out the extra expense: purchase price = 7,500 − 1,500 = ₹6,000. Note that ₹7,500 is usually among the options for anyone who stops early.

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Wastage and damaged goods. When some articles spoil, the money spent on them does not come back. The trader still wants his profit on the whole amount he spent, but he can sell only the good pieces. So

Required total SP = Total cost × 100 + p100, and Price per piece = Required total SPNumber of good pieces

Radhika buys 50 cupcakes for ₹1,000 for a school stall and wants a 20% profit. Five cupcakes get crushed. She must collect ₹1,200 from the remaining 45, so each must be sold at 120045 = ₹26.67. The crushed cupcakes don't reduce her cost. They only reduce the number of pieces that have to carry it.

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Damaged pieces sold at a loss. If the damaged pieces are sold at some loss and the rest at a profit, apply each group's percentage to its own purchase price, add up all the SPs, and compare the sum with the total cost including overheads.

A part of the goods is stolen or lost. Theft, fire and flood work like wastage, but the question usually gives the lost part as a fraction of the stock and asks for the overall result. The money spent on the lost part is gone, and only the rest is sold. So work with factors on the total cost: if a fraction f is lost and the rest is sold at p% profit,

Total SP = Total CP × (1 − f) × 100 + p100

If a quarter of a stock of shawls is stolen and the rest is sold at 20% profit, total SP = CP × 34 × 1.2 = 0.9 × CP, so there is a 10% loss overall, even though every shawl that was sold made a profit. Going the other way, to make 8% profit overall after losing 110 of the stock, the rest must be sold at 1.080.9 = 1.2 times its cost, that is at 20% profit.

Many questions don't give any rupee amount at all. They give a relation between CP and SP and ask for the profit or loss percent, or for a ratio. The rule is the one from the foundation table:

If CP : SP = a : b, then profit (or loss) = (b − a) parts on a parts, so Profit% = b − aa × 100

CP : SP = 4 : 7 ⇒ profit = 3 parts on 4 ⇒ profit : CP = 3 : 4 and profit% = 75%.
CP : SP = 13 : 11 ⇒ loss = 2 parts on 13 ⇒ loss% = 213 = 15513%.

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Profit : CP given. Profit : CP = 3 : 10 means CP = 10 parts and profit = 3 parts, so SP = 13 parts. Any ratio can now be read off: SP : profit = 13 : 3, SP : CP = 13 : 10.

"x% of CP = y% of SP". Write it as x × CP = y × SP. Then CPSP = yx, that is CP : SP = y : x, the two numbers cross over. So 45% of CP = 30% of SP ⇒ CP : SP = 30 : 45 = 2 : 3 ⇒ 50% profit.

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The same crossing works for "7 times SP = 8 times CP": 7 × SP = 8 × CP ⇒ CP : SP = 7 : 8 ⇒ profit = 17 = 1427%.

When one rupee amount comes with the ratio. CP : SP = 3 : 8 and profit = ₹750. Profit = 5 parts = ₹750, so 1 part = ₹150 and CP = 3 parts = ₹450. It is the same unit method again.

Radhika notices that 15% of the price she paid for a book equals 12% of the price she got for it. So CP : SP = 12 : 15 = 4 : 5. She gained 1 part on 4, a profit of 25%.

Till now every percentage was on CP. Sometimes a question clearly says "profit is 40% on the selling price" or "of the selling price". Then the base changes: SP becomes 100.

Profit = 40% of SP. Take SP = 100, so profit = 40 and CP = 100 − 40 = 60.
Profit% on CP = 4060 × 100 = 6623%.

Loss = 20% of SP. Take SP = 100, so loss = 20 and CP = 100 + 20 = 120.
Loss% on CP = 20120 × 100 = 1623%.

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Look at the pattern. A profit on SP becomes a bigger percentage on CP, and a loss on SP becomes a smaller percentage on CP, because CP is smaller than SP when there is a profit and bigger when there is a loss.

p% profit on SP = p100 − p × 100% profit on CP
l% loss on SP = l100 + l × 100% loss on CP

Fractions make it even quicker. 10% on SP = 110: SP 10 parts, profit 1 part, CP 9 parts ⇒ 19 = 1119% on CP. 3712% on SP = 38: SP 8, profit 3, CP 5 ⇒ profit : CP = 3 : 5, that is 60%.

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Radhika runs a juice stall and says, "I keep one-sixth of every sale as my profit." On a ₹60 glass she keeps ₹10, so the cost was ₹50. On cost her profit is 1050 = 20%, even though she calls it 1623%.

"A fan is sold for ₹2,000 at a loss of 20%. At what price should it be sold to gain 10%?" The article is the same, so the CP is the same in both situations. Only the multiplying factor changes.

Take CP = 100 units. At 20% loss, SP = 80 units. At 10% gain, SP = 110 units. The given price tells us 80 units = ₹2,000, so 1 unit = ₹25, and the required SP = 110 × 25 = ₹2,750.

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In one line: New SP = Old SP × 100 ± new%100 ± old%
New SP = 2000 × 11080 = ₹2,750.

You don't even need the CP in rupees. Keep the reasoning in mind, though: both SPs are percentages of the same CP, and the next type is built on exactly this.

Radhika sells her old phone for ₹4,600 at a 15% profit. Her friend says she could have asked for a 25% profit. CP = 100 units, SP = 115 units = ₹4,600 ⇒ 1 unit = ₹40. At 25% profit, SP = 125 × 40 = ₹5,000.

The same article is sold (or could have been sold) at two different prices, and the question gives the difference between those two prices in rupees. CP is the same in both cases, so the difference of the two SPs comes only from the difference of the two percentages.

Take CP = 100 units.

• 20% profit and 35% profit: SPs are 120 and 135. Difference = 15 units.
• 15% loss and 10% profit: SPs are 85 and 110. Difference = 25 units.

So the rule is: if both are profits (or both losses), subtract the percentages; if one is a profit and the other a loss, add them. That many units equal the given rupee difference.

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CP = Difference in SP × 100Difference in percentages

Radhika sells a cushion at 20% profit. She realises that for ₹36 more she would have made 32%. So 12 units = ₹36, 1 unit = ₹3 and CP = ₹300.

"Profit at one price = loss at another price." Look at the number line in the figure. CP sits between the two SPs. If the profit at ₹1,250 equals the loss at ₹950, CP is exactly in the middle:

CP = SP₁ + SP₂2 = 1250 + 9502 = ₹1,100

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"Profit at one price = n times the loss at another." CP is still between the two SPs, but now the gap is divided in the ratio profit : loss = n : 1. Suppose the profit at ₹1,500 is 4 times the loss at ₹1,000. The gap of 500 is divided 4 : 1, so the loss part = 5005 = 100 and CP = 1,000 + 100 = ₹1,100. As a formula:

CP = SP₁ + n × SP₂n + 1, where SP₁ gives the profit and SP₂ the loss.

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Once CP is found, answer what is actually asked. It may be a new SP for some profit, or the profit% at a third price.

This type runs entirely on the multiplying factor, just like successive percentage change.

Successive selling (A → B → C). A sells to B at 25% profit, B sells to C at 20% loss, and C sells to D at 10% profit. Each sale multiplies the price by its own factor, and the SP of one person is the CP of the next. So

Final price = A's CP × 125100 × 80100 × 110100 = A's CP × 1110

If the final price is known, divide by the same factors to get back to A's CP. Do not add and subtract the percentages (25 − 20 + 10 = 15%). The true overall rise is 10%, because each percentage is on a different base.

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CP and SP both change by the same percentage. Profit% = SP − CPCP. If SP and CP are both multiplied by the same factor, the ratio SPCP does not change, so the profit% stays exactly the same. A 25% profit stays 25% whether both prices rise by 10% or fall by 30%. Options that add or subtract the change are there only to catch you.

[[fig:1]]

CP and SP change by different amounts. Take CP = 100 and write the old SP. Apply each change to its own price, then compare the new SP with the new CP.
Profit 25%: CP = 100, SP = 125. SP rises 20% ⇒ 150. If the new profit is 20%, new CP = 1501.2 = 125, a rise of 25% in CP.

When the change is in rupees (₹40 added to CP and ₹20 to SP, or "bought 20% cheaper and sold ₹20 cheaper"), write both situations in units of the same CP = 100 and find how many units the rupee change is worth. You can also take CP = x and form an equation.
"Sold at 20% profit. Had he bought it 20% cheaper and sold it for ₹20 less, he would have gained 40%." CP = 100, SP = 120. New CP = 80, new SP = 80 × 1.4 = 112. The ₹20 is the fall in SP: 120 − 112 = 8 units = ₹20, so 1 unit = ₹2.50 and CP = ₹250.

[[fig:2]]

Radhika buys a toy and sells it to her cousin at 10% profit, and the cousin sells it to a friend at 10% profit. The friend pays 1.1 × 1.1 = 1.21 times Radhika's cost, which is 21% over her cost, not 20%.

Sometimes the profit is given neither in rupees nor in percent but in articles: "By selling 15 articles he gains the SP of 3 articles", or "the CP of 20 articles equals the SP of 15 articles". Take the CP of one article = c and the SP of one article = s, and everything becomes a ratio.

CP of x articles = SP of y articles. x × c = y × s ⇒ c : s = y : x. CP of 18 = SP of 15 ⇒ c : s = 15 : 18 = 5 : 6 ⇒ profit = 15 = 20%. So profit (or loss) % = x − yy × 100, where y is the number of articles sold. When x > y it is a profit, when x < y it is a loss.

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Why divide by y? The y articles are sold for the money that x articles cost. The gain, worth (x − y) articles at cost, is compared with the CP of those y articles.

Profit (or loss) equal to the price of some articles. Out of N articles sold:

• Profit = SP of n articles ⇒ N × s − N × c = n × s ⇒ CP of N = SP of (N − n) ⇒ profit% = nN − n × 100.
• Profit = CP of n articles ⇒ profit% = nN × 100 (directly, since the profit is measured in CP).
• Loss = SP of n articles ⇒ CP of N = SP of (N + n) ⇒ loss% = nN + n × 100.
• Loss = CP of n articles ⇒ loss% = nN × 100.

There is no need to memorise four formulas. Check one thing: is the profit measured in CP or in SP? If it is in CP, divide by N. If it is in SP, first change it into CP of articles.

"CP of 27 articles is ₹540 and SP of 30 articles is ₹540." The same ₹540 buys 27 and sells 30, so CP of 27 = SP of 30 ⇒ loss = 30 − 2730 = 10%.

Radhika buys 10 rakhis and sells all of them. She notices that her profit equals the selling price of 2 rakhis. So the money from 8 rakhis covers the cost of all 10, and the money from the other 2 is pure profit. Profit% = 28 = 25%.

[[fig:1]]

"Buys pencils at 8 for ₹60 and sells them at 6 for ₹54." The buying rate and the selling rate are given for different numbers of articles, so ₹60 cannot be compared with ₹54 directly. We must compare the same number of articles.

Method 1: price of one article. CP of one pencil = 608 = ₹7.50. SP of one pencil = 546 = ₹9. Profit = ₹1.50 on ₹7.50 ⇒ 1.57.5 × 100 = 20%.

Method 2: LCM of the counts. LCM of 8 and 6 = 24. CP of 24 pencils = 3 × 60 = ₹180. SP of 24 pencils = 4 × 54 = ₹216. Profit = 36180 = 15 = 20%. This way avoids decimals, and it is the better choice when the price of one article is awkward (like 73 or 116).

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Buying at two rates and selling at one. Find the total CP of all the articles and the total SP of all the articles, and compare. Average the two rates only if the same number of articles is bought at each rate.

"How many should he sell for ₹A to make p% profit?" First find the CP of one article from the first deal, then the SP of one article needed for p% profit. Number of articles = ASP of one article.

Radhika sold 12 handmade cards for ₹1,500 and made a 25% profit. SP of one card = ₹125 = 125% of CP, so CP of one card = ₹100. For a 20% profit she must sell each card at ₹120, so for ₹1,800 she should sell 1800120 = 15 cards.

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Dozens. If the rate is per dozen, divide by 12 to get the rate per piece before comparing.

When several articles or lots are involved, one rule is always safe: overall profit% = Total SP − Total CPTotal CP × 100. Never add or average the separate percentages blindly.

Same CP base: the weighted average. A shopkeeper buys 200 bulbs at ₹50 each, sells 150 at 12% profit and 50 at 8% loss. The 150 bulbs carry 75% of the cost and the 50 bulbs carry 25%. Overall profit% = 0.75 × 12 + 0.25 × (−8) = 9 − 2 = 7%. In general,

Overall % = CP₁ × p₁ + CP₂ × p₂ + …CP₁ + CP₂ + …

The weights are the costs of the lots, not the number of articles. The two are the same only when every article costs the same.

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Alligation: finding the split. "Some sold at 10% gain and the rest at 25% gain; overall gain 16%." The overall figure is a weighted average, so use alligation: cost sold at 10% : cost sold at 25% = (25 − 16) : (16 − 10) = 9 : 6 = 3 : 2.

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Mixtures of two varieties. Mixing is just buying two lots and selling them together. CP of the mixture per kg = total costtotal quantity. Tea at ₹200 and ₹300 per kg mixed 3 : 2 ⇒ CP = 3 × 200 + 2 × 3005 = ₹240 per kg. Then compare it with the SP per kg.

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Two investments with a net result. If ₹6,000 is split between a 15% profit and a 5% loss and the net is a ₹300 profit, write 0.15x − 0.05(6000 − x) = 300, which gives x = ₹3,000. Or apply alligation to the net percentage 3006000 = 5%: money at +15% : money at −5% = (5 − (−5)) : (15 − 5) = 10 : 10 = 1 : 1, so the profit part is ₹3,000.

Radhika buys 4 identical lamps. She sells three at 20% profit and one at 20% loss. Each lamp carries one-fourth of the cost, so overall = 3 × 20 − 204 = 10% profit.

Break-even split. For no overall profit or loss, the profit on one part must equal the loss on the other: 12% of A = 8% of B ⇒ A : B = 8 : 12 = 2 : 3.

Three varieties in one mixture. Nothing new is needed when a third variety joins. Add the cost of every lot, add any transport or other expense, and divide by the total quantity. Suppose 10 kg of dal at ₹90, 15 kg at ₹100 and 25 kg at ₹80 are mixed. Total cost = 900 + 1,500 + 2,000 = ₹4,400 for 50 kg, so the mixture costs ₹88 per kg. Selling it at ₹99 per kg gives 1188 = 12.5% profit. When the quantities are in a ratio such as 2 : 3 : 5, use the ratio numbers as kilograms; the CP per kg comes out the same.

Here the selling prices are equal (or in a given ratio), not the cost prices. That changes everything, because the percentages are on CP and the CPs are now different.

Same SP, x% profit on one and x% loss on the other. Many students expect "no profit, no loss". Look closer: at 20% profit, CP = SP1.2; at 20% loss, CP = SP0.8. The article sold at a loss had the bigger CP, so the loss in rupees is bigger than the profit in rupees. There is always a loss, and it is x²100%.

Radhika sells two bags at ₹600 each, one at 20% profit and the other at 20% loss. The CPs are 6001.2 = 500 and 6000.8 = 750. Total CP = 1250 and total SP = 1200, so the loss is 50 on 1250, that is 4% = 20²100%.

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Same SP, different percentages. Make the SPs equal using the CP : SP ratios.
25% profit ⇒ CP : SP = 4 : 5. 10% loss ⇒ CP : SP = 10 : 9.
To make SP equal, take the LCM of 5 and 9 = 45 ⇒ first 36 : 45, second 50 : 45.
Total CP = 86, total SP = 90 ⇒ profit = 486 ≈ 4.65%.
You can also take the actual SP and divide: CP₁ = SP1.25, CP₂ = SP0.90, then compare the totals.

[[fig:1]]

SPs in a ratio. Suppose two SPs are in the ratio 2 : 3, the first at 25% profit and the second at 25% loss. Each CP = SP ÷ its factor: 21.25 and 30.75. Multiply the ratio by a number that clears the decimals (here 5): SPs 10 and 15 ⇒ CPs 8 and 20. Total CP = 28, total SP = 25 ⇒ loss = 328 ≈ 10.71%.

Loss in rupees at the same SP. Once you have the overall loss%, total CP = Total SP × 100100 − loss%, and the loss in rupees = total CP − total SP.

A dishonest dealer says "I sell at cost price", but gives 900 g when the customer pays for 1 kg. Where is his profit? Take the price of 1 g = ₹1.

• He gives 900 g, so his cost = ₹900.
• The customer pays for 1000 g, so he receives ₹1000.
• Profit = ₹100 on a cost of ₹900 ⇒ 100900 × 100 = 1119%.

[[fig:0]]

Profit% = True weight − False weightFalse weight × 100 (when he claims to sell at cost price). The denominator is the false weight, because that is all he actually paid for.

False weight plus a claimed profit or loss. Now two effects come together. He charges (100 ± p)% of the true price of 1000 g, but hands over only the false weight.

SP : CP = (100 ± p) × 1000 : 100 × False weight

He claims a 4% loss and uses 840 g: SP = 0.96 × 1000 = 960, CP = 840 ⇒ profit = 120840 = 17 = 1427%. The man who "sells at a loss" is in fact making a profit.

[[fig:1]]

Uses a weight 20% less than the actual means the false weight = 800 g for every 1000 g.

Free articles while buying. "Buys 9 caps for the marked price of 8." Here the scheme works on the buying side: CP of 9 caps = price of 8 caps. If he then sells each cap at a 5% discount on the marked price M, SP of 9 = 9 × 0.95 M = 8.55 M and CP of 9 = 8 M ⇒ profit = 0.558 = 6.875%. Discount is covered fully in the next chapter; here we only need SP = (100 − d)% of the marked price.

[[fig:2]]

Radhika's neighbour sells sugar "at cost price", but his 1 kg weight actually weighs 960 g. For every ₹960 worth of sugar he hands over, he collects ₹1,000. His hidden profit is 40960 = 124 = 416%.

Cheating on both sides: while buying and while selling. Some dealers cheat their supplier too. "He cheats 10% while buying" means he takes 10% more weight than he pays for: for the price of 1,000 g he gets 1,100 g. "He cheats 10% while selling" means he gives 10% less: for the price of 1,000 g he hands over only 900 g. Take ₹1 as the price of 1 g. His ₹1,000 brings in 1,100 g, and every 900 g he gives out brings back ₹1,000.

[[fig:3]]

So the goods multiply by 1100900. If he also claims a profit or a loss on the price, multiply that factor in as well:

SP : CP = (100 ± p) × (100 + x) : 100 × (100 − y)

where x% is the cheating while buying and y% the cheating while selling. With 10% on both sides and an honest-looking price, the factor is 11090 = 1.2222, a hidden profit of 22.22%.

Between the trader and the customer there is often a third person: a salesman, an agent or a broker. He is paid a commission, a percentage of the amount he handles. The whole question is about which amount that percentage is taken on, and who pays it.

Commission paid by the seller. If an agent sells goods for the owner and keeps c% of the selling price, the owner receives only (100 − c)% of the SP. The owner's real SP is what reaches his hand. Profit is then counted on this net amount. An article sold for ₹1,000 with 15% commission brings the owner ₹850. If it cost him ₹680, his profit is ₹170, which is 25% of the cost.

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Going backwards. "After paying 15% commission he still gains 25%" means 0.85 × SP = 1.25 × CP. So SP = CP × 1.250.85, and every such question becomes one line of factors, just like the overheads in an earlier type.

Brokerage on a purchase or a sale. A broker who arranges a deal charges a percentage of the deal price. If the buyer pays b% brokerage, his CP becomes price × (1 + b100). If the seller pays it, he receives price × (1 − b100). Often the broker charges both sides, so he earns from the buyer and from the seller on the same price.

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Salary plus commission, and slabs. A salesman may get a fixed salary and a commission only on the sales above a target, or different rates on different portions of his sales. Charge each rate only on its own portion, exactly as in slab tax.

Tax on sales. When a bill includes GST, the tax goes to the government, not to the seller. His SP is the price before tax: bill × 100100 + g. Profit is always counted on this price. Radhika buys a lamp for ₹800 and sells it for ₹1,180 including 18% GST. Her real SP is ₹1,000, so her profit is ₹200, which is 25%, not 47.5%.