Aptitude · Profit, Loss and Discount · Tax and GST

Discount first, then tax on what is left

Add a tax to a discount and you have two percentages pulling in opposite directions on different bases. The order is fixed, the two never cancel, and the “bill or no bill” question built on this is a reliable exam favourite.

Set a discount and a tax and watch them fail to cancel →

The order is discount first, then tax on the discounted figure — because tax is charged on what you actually pay, not on the label.

01 The idea

Two percentages, opposite directions, different bases

A laptop is priced at ₹50,000. The trader takes 15% off, then adds 18% GST when he makes out the bill. The customer pays ₹50,150 — slightly more than the original price, even though a discount was given.

The order is what makes it work. The discount comes off the marked price, giving ₹42,500. The tax is then charged on that ₹42,500 rather than on the ₹50,000 — because tax is levied on the amount actually transacted. Adding 18% gives ₹50,150.

Notice the two percentages do not cancel and cannot be netted off. A 15% discount and an 18% tax do not make a 3% increase. The multipliers are 0.85 and 1.18, and their product is 1.003 — a rise of just 0.3%, not 3%.

The second form of this question is more interesting. A shopkeeper offers either a bill with 18% tax, or 10% off with no bill. A customer takes the discount and pays ₹3,600. How much did he save? You have to recover the actual price from the discounted payment first — ₹3,600 is 90% of it, so the price is ₹4,000 — and only then compute the taxed alternative.

Final price = MP × (100 − discount)/100 × (100 + tax)/100. Two multipliers, in that order, on different bases.

Taxable valueThe price after the discount has been applied. This is the base the tax is computed on, not the marked price.

GST or sales taxA percentage added to the taxable value. It raises the price, so its multiplier is (100 + t)/100.

SavingIn “bill or no bill” questions, the difference between what the taxed route would have cost and what was actually paid. Both must be computed from the same actual price.

02 Worked example

A ₹50,000 laptop, 15% off, 18% GST

The source question, worked in order. The price of a laptop is ₹50,000. A trader sold it at a 15% discount and then charged 18% GST on the bill. Find the final price paid by the customer.

Discount off the marked priceFifteen per cent off ₹50,000 leaves 85% of it.50,000 × 85/100 = ₹42,500

This is now the taxable valueTax is charged on what the customer actually pays for the goods, which is the discounted figure.taxable value = ₹42,500, not ₹50,000

Add the GSTEighteen per cent on ₹42,500 takes it to 118% of that figure.42,500 × 118/100 = ₹50,150

Compare with where you startedThe customer pays slightly more than the original marked price, despite the discount.50,150 − 50,000 = ₹150 more, a rise of 0.3%

The two multipliers togetherOne line, and it explains the small net rise.0.85 × 1.18 = 1.003 ⇒ 100.3% of the marked price

The 0.3% is the point. A 15% discount followed by an 18% tax feels like it should be a 3% increase, and it is a tenth of that, because the 18% is charged on a base the discount already cut by 15%. Whenever two percentages act in sequence, multiply the multipliers — adding or subtracting the percentages is only ever right when they share a base, which here they do not.

03 The method

The order, and the reverse question

The forward direction is two multiplications. The reverse direction — recovering a price from what someone paid — is where the marks are.

Final = MP × (100 − d)/100 × (100 + t)/100. Backwards, to recover the actual price from a discounted payment: MP = paid × 100/(100 − d).

The “bill or no bill” question. A customer pays P after a discount of d%. First recover the actual price: P × 100/(100 − d). Then the taxed alternative is that price times (100 + t)/100, and the saving is the difference. Do not compute the saving as a percentage of what was paid — the two routes must be compared from the same base price.

Discount	Tax	Net multiplier on MP
15%	18%	`0.85 × 1.18 = 1.003`
12.5%	18%	`0.875 × 1.18 = 1.0325`
20%	18%	`0.80 × 1.18 = 0.944`
10%	18%	`0.90 × 1.18 = 1.062`
18%	18%	`0.82 × 1.18 = 0.9676`
Tax then discount	`wrong order`	different answer
Netting 18 − 15 = 3%	`wrong`	actual rise is 0.3%

05 Cheat sheet

Tax and discount on one page

The forward pair, the reverse move, and the two orderings that get confused.

Case	Route	On 50,000 with 15% and 18%
Final price	`MP(100−d)/100 × (100+t)/100`	50,000 → 50,150
Taxable value	`MP(100−d)/100`	₹42,500
Net multiplier	`(100−d)(100+t)/10000`	1.003
Recover price from payment	`paid × 100/(100−d)`	3,600 → 4,000
Saving, bill vs no bill	`price(100+t)/100 − paid`	4,720 − 3,600 = 1,120
Tax before discount	`wrong order`	not how bills work
Netting the percentages	`wrong`	3% vs the true 0.3%

Discount first, alwaysTax is levied on the transacted amount, so the discount comes off before the tax goes on. This is both the accounting reality and the exam convention.

The percentages never net offA 15% discount and an 18% tax give a 0.3% rise, not 3%. Multiply the multipliers; the percentages are taken on different bases.

Recover the price before comparingIn bill-or-no-bill questions, both routes must be measured from the same actual price. Get that price from the payment first.

06 Where & why

Where this shows up

A modern favourite, because GST made the arithmetic familiar to everyone and the base confusion is easy to set.

Bank PO · SSC CGL

Discount then GST, final price wanted

The straightforward direction. Two multiplications, and the trap is netting the percentages into one.

TCS NQT · Infosys

“Bill or no bill” savings

The reverse form: recover the actual price from the discounted payment, then compare with the taxed route. This is where the marks are.

Everyday purchases

Why a discounted bill still exceeds the label

Exactly the laptop case. A modest discount against 18% GST leaves you paying slightly more than the tag, which surprises people at the till.

Comparison questions

Which offer is cheaper

Two schemes with different discount and tax combinations. Multiply out each net multiplier and compare; no prices are needed.

The habit is the same one this whole module rests on: before combining two percentages, ask what each is a percentage of. Here they are percentages of two different amounts, so they multiply and never add.

07 Interview questions

What gets asked

Ten, and the savings question is the one most likely to appear in a mains paper.

Which comes first, the discount or the tax?

The discount. Tax is charged on the amount actually transacted, so the discount comes off the marked price and the tax goes on the reduced figure. That is both how a real bill works and the convention every exam uses.

A ₹50,000 laptop gets 15% off and then 18% GST. What does the customer pay?

₹50,150. The discount takes it to ₹42,500, and 18% on that gives 42,500 × 1.18 = ₹50,150. Slightly more than the original price, despite the discount.

Why doesn't a 15% discount cancel most of an 18% tax?

Because they act on different bases. The multipliers are 0.85 and 1.18, and their product is 1.003 — a rise of 0.3%, not the 3% you get by subtracting the percentages. The tax is levied on a base the discount already shrank.

A shop offers 18% tax with a bill or 12% off without one. A customer pays ₹3,520 for the discount option. How much did he save?

₹1,200. First recover the actual price: ₹3,520 is 88% of it, so the price is ₹4,000. The taxed route would have cost 4,000 × 1.18 = ₹4,720, so the saving is 4,720 − 3,520 = ₹1,200.

Why must you recover the actual price in that question?

Because the two routes have to be compared from the same base. The ₹3,520 is already discounted, so it cannot be taxed to give the alternative. Recovering the ₹4,000 is what makes the two options comparable.

Can the final price after a discount and a tax be below the marked price?

Yes, whenever the discount multiplier outweighs the tax multiplier. A 20% discount with 18% GST gives 0.8 × 1.18 = 0.944, so the customer pays 94.4% of the label. Which way it goes depends on the product of the multipliers, not on which percentage is numerically larger.

An 18% discount and an 18% tax — do they cancel?

No. The multipliers are 0.82 and 1.18, and their product is 0.9676, so the customer pays about 96.8% of the marked price. Equal percentages acting in opposite directions on different bases always leave you below where you started.

A ₹40,000 laptop with 12.5% off and 18% GST. Find the price paid.

₹41,300. The discount gives 40,000 × 0.875 = ₹35,000, and 18% on that is 35,000 × 1.18 = ₹41,300. Recognising 12.5% off as multiplying by 7/8 makes the first step immediate.

Would applying the tax before the discount give the same answer?

Hard

A government levies 18% GST on all sales. (a) Find the discount percentage a shop must offer so that the customer pays exactly the marked price. (b) Show that this discount is not 18%, and explain in terms of bases why it must be less. (c) Generalise: for a tax of t%, give a formula for the discount d that leaves the customer paying exactly the marked price, and verify it for t = 18.

Follow-up

Part (c) is the general statement of why opposite percentages never cancel. Setting the product of the two multipliers to 1 and solving gives d = 100t/(100 + t), which is always less than t — and seeing that inequality fall out of the algebra is the point.

Show the hint

You need (100 − d)/100 × (100 + t)/100 = 1. Solve that for d.