Profit, loss and discount

A shopkeeper buys a bicycle for ₹4,000 and sells it for ₹4,600. Did they do well? You cannot tell from the raw numbers alone — ₹600 sounds small next to a ₹40,000 car deal and large next to a ₹100 toy. What you need is a percentage, a number that compares the gain to what was originally spent, so that deals of any size can be judged on the same scale. This is the whole idea behind profit, loss and discount: three everyday percentage comparisons that describe a single sale from different angles. Profit and loss compare what a shopkeeper earns against what the goods cost them. Discount compares what a customer actually pays against the price originally marked on the tag. And when GST is added on top, the final bill depends on getting the order of these steps exactly right — marked price, then discount, then tax. This article builds each formula from first principles and works through a full multi-step transaction end to end.

The core insight

Profit, loss and discount are all percentage comparisons — profit and loss are measured against the cost price, discount is measured against the marked price, and a real transaction always applies discount before GST, never the other way round.

The core idea

Profit and loss compare selling price to cost price. The Cost Price (CP) is what the shopkeeper paid to acquire or make the goods; the Selling Price (SP) is what they sold it for. When SP exceeds CP there is a profit; when CP exceeds SP there is a loss.

Profit=SP−CP(SP>CP),Loss=CP−SP(CP>SP)\text{Profit} = SP - CP \quad (SP > CP), \qquad \text{Loss} = CP - SP \quad (CP > SP)

where: SPSP is the selling price and CPCP is the cost price, both in rupees. To compare deals of different sizes fairly, both profit and loss are converted to percentages — and always relative to the cost price, since that is the shopkeeper's original investment:

Profit %=ProfitCP×100,Loss %=LossCP×100\text{Profit \%} = \frac{\text{Profit}}{CP} \times 100, \qquad \text{Loss \%} = \frac{\text{Loss}}{CP} \times 100

where: the numerator is the profit or loss in rupees, and the denominator is always CPCP, never SPSP.

Discount compares selling price to marked price. The Marked Price (MP), also called the list price, is the price printed on the tag before any reduction. A discount is the amount knocked off that tag price to arrive at the actual selling price:

Discount=MP−SP,Discount %=DiscountMP×100\text{Discount} = MP - SP, \qquad \text{Discount \%} = \frac{\text{Discount}}{MP} \times 100

where: MPMP is the marked price and the discount percentage's denominator is always MPMP, never CPCP — this is the single most common mix-up in the whole topic.

GST sits on top, at the very end. Goods and Services Tax is a percentage added to the price the customer is actually about to pay — that is, after any discount has already been subtracted:

Amount paid=SP+GST%100×SP=SP(1+GST%100)\text{Amount paid} = SP + \frac{GST\%}{100} \times SP = SP\left(1 + \frac{GST\%}{100}\right)

where: SPSP here means the discounted selling price, and GST%GST\% is the tax rate applied to it. The three ideas describe two different comparisons happening in one sale. Discount and GST both move along the chain Marked Price → Selling Price → Amount paid, in that fixed order. Profit and loss are a separate comparison entirely — SP against CP, the shopkeeper's margin over what the goods originally cost — and never involve the marked price at all.

Figure
A flow diagram showing marked price reduced by a discount to give selling price then increased by GST to give the amount paid
One transaction, three prices, fixed order Marked Price MP = 2500 minus 20% discount Selling Price SP = 2000 plus 5% GST Amount paid 2100 Discount always acts first, on the marked price; GST always acts second, on the discounted price
Derivation AThe final amount paid after discount then GST

The goal is to build a single formula that takes a marked price, a discount percentage and a GST percentage and delivers the final amount a customer pays — while making it algebraically obvious why discount must be applied before GST and never the reverse, since the two steps use different bases and swapping their order changes the answer.

Step 1. Start from the marked price MPMP and subtract the discount. A discount of d%d\% removes d100×MP\frac{d}{100} \times MP from the marked price, leaving the selling price SP=MP−d100MP=MP(1−d100)SP = MP - \frac{d}{100}MP = MP\left(1 - \frac{d}{100}\right). Notice the discount's base is MPMP, exactly as defined earlier — it can never be applied to any other price.

Step 2. Add GST to this selling price, not to the marked price. A GST rate of g%g\% adds g100×SP\frac{g}{100} \times SP on top of SPSP, giving the amount paid A=SP+g100SP=SP(1+g100)A = SP + \frac{g}{100}SP = SP\left(1 + \frac{g}{100}\right). This step uses SPSP, the already-discounted price, as its base — GST is a tax on what the customer is actually about to pay, not on the original tag price.

Step 3. Combine the two steps into one chain by substituting the Step 1 result into Step 2: A=MP(1−d100)(1+g100)A = MP\left(1 - \frac{d}{100}\right)\left(1 + \frac{g}{100}\right). This single expression carries the discount and GST as two separate multiplying factors applied in a fixed sequence, which is why the two percentages can never simply be added or subtracted from each other.

Step 4. Check the formula against the known numbers: MP=2500MP = 2500, d=20d = 20, g=5g = 5 gives A=2500×(1−0.20)×(1+0.05)=2500×0.80×1.05=2100A = 2500 \times (1 - 0.20) \times (1 + 0.05) = 2500 \times 0.80 \times 1.05 = 2100, matching the transaction worked out directly below.

Worked example

A jacket has a marked price of ₹2,500. The shopkeeper offers a 20% discount, and then GST of 5% is added on the discounted price. Find the final amount the customer pays, and also find what percentage of the marked price this final amount represents.

Step 1. Find the discount amount on the marked price. Discount=20100×2500=500\text{Discount} = \frac{20}{100} \times 2500 = 500.

Step 2. Subtract to get the selling price. SP=MP−Discount=2500−500=2000SP = MP - \text{Discount} = 2500 - 500 = 2000.

Step 3. Find the GST amount on this selling price, not on the marked price. GST=5100×2000=100\text{GST} = \frac{5}{100} \times 2000 = 100.

Step 4. Add GST to the selling price to get the final amount paid. Final amount=2000+100=2100\text{Final amount} = 2000 + 100 = 2100.

Step 5. Express the final amount as a percentage of the marked price. 21002500×100=84\frac{2100}{2500} \times 100 = 84. So even though the discount is 20% and GST is 5%, the customer ends up paying 84% of the marked price, not 100−20+5=85%100 - 20 + 5 = 85\% — because the GST is calculated on the already-reduced price, not on the marked price. So the customer finally pays ₹2,100, which is 84% of the original marked price. This confirms the earlier warning: discount and GST percentages cannot simply be added or subtracted, since they use different bases.

Competitive stretch

A shirt marked at ₹1,800 is sold after a discount of d%d\%, and GST of 12% is then added, so that the customer pays exactly ₹1,747.20. Find dd. Using the derivation's formula, 1747.20=1800(1−d100)(1.12)1747.20 = 1800\left(1 - \frac{d}{100}\right)(1.12), so (1−d100)=1747.201800×1.12=1747.202016=0.8667\left(1 - \frac{d}{100}\right) = \frac{1747.20}{1800 \times 1.12} = \frac{1747.20}{2016} = 0.8667, giving d100=0.1333\frac{d}{100} = 0.1333, so d≈13.3%d \approx 13.3\%.

Common traps

  • Dividing profit or loss by the selling price instead of the cost price — the base for these percentages is always CP.
  • Dividing discount by the cost price instead of the marked price — the base for discount is always MP.
  • Adding GST to the marked price instead of the discounted selling price, or adding the discount and GST percentages together as if they combine linearly.
  • Reversing the order of operations — applying GST first and then discount gives a different, incorrect final amount, since each percentage is calculated on a different base price.

Ask the tutor: Not sure which price — CP, MP, or SP — a given percentage should be divided by in a mixed word problem? Ask your AI tutor to walk through the transaction chain with you step by step.

Why is discount percentage based on marked price and not cost price?

Discount is a promise the shopkeeper makes to the customer about the tag price they are seeing in the shop — "20% off this list price" — so it is naturally measured against that same list price, the marked price. The cost price is a private figure between the shopkeeper and their supplier that the customer never sees on the tag, so it plays no role in defining what "20% off" means. Profit and loss, by contrast, are the shopkeeper's own internal measure of how the sale went compared to what they paid, which is exactly why that comparison uses CP instead.

If discount is 20% and GST is 5%, why isn't the final price simply 85% of the marked price?

Because the two percentages are taken of different amounts, not added together. The 20% discount is subtracted from the marked price first, shrinking it to 80% of MP. The 5% GST is then added to that already-smaller amount, not to the original marked price, so it only adds 5% of 80% of MP, which is 4% of MP, not 5% of MP. Adding those correctly gives 80%+4%=84%80\% + 4\% = 84\% of the marked price, matching the worked example, not the naive 100−20+5=85%100 - 20 + 5 = 85\%.

Can profit or loss percentage and discount percentage ever be equal for the same item?

They can come out numerically equal by coincidence, but they are measuring completely different things and are calculated on different base prices, so there is no rule connecting them. Profit or loss percentage compares the selling price to what the shopkeeper paid (CP); discount percentage compares the selling price to the price displayed on the tag (MP). A shopkeeper can even offer a large discount on the marked price and still make a healthy profit over the cost price, if the marked price was set well above the cost price to begin with.

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