Introduction to Discount

Profit, Loss and Discount · 20 min

Aptitude · Profit, Loss and Discount · Discount

Discount is always on the marked price, and never on cost

Discount questions come in two kinds: those that mention the cost price and those that do not. Telling them apart on sight is most of the skill, because when a question is purely about discount the cost price is not merely unknown — it is irrelevant.

Set a marked price and a discount, and read the selling price →
Discount is calculated only on the marked price. If a question asks only about discount, you do not need the cost price at all.

01 The idea

Three prices, and when you can ignore one

You buy a piece of land for ₹25,000, put a price tag of ₹1,00,000 on it, and eventually sell it to a friend for ₹90,000. Three prices, and each has a name: the ₹25,000 is your cost price, the ₹1,00,000 is the marked price, and the ₹90,000 is the selling price.

The ₹10,000 gap between the marked price and the selling price is the discount. Notice what it is measured against: the marked price. A discount of ₹10,000 on a label of ₹1,00,000 is 10%, and the ₹25,000 you originally paid plays no part in that figure whatsoever.

That is the golden rule, and it cuts both ways. It tells you where to apply a percentage, and it also tells you when a question needs less information than it appears to. If you are asked only for a selling price after a discount, the cost price is decoration — and examiners include it precisely to see whether you know that.

The reverse is worth knowing too. Once a question asks about profit, the cost price comes straight back in, because profit is measured on cost. So the reading skill is to spot which of the two the final line actually wants.

Discount = MP − SP, and Discount% = (MP − SP)/MP × 100. The marked price is the base, always.
Cost price (CP)What the seller originally paid to buy or make the item. Needed for profit questions and irrelevant to pure discount questions.
Marked price (MP)The tag or label price fixed by the seller. Every discount is a percentage of this, which is why it is sometimes called the list price.
Selling price (SP)The final amount the customer actually pays after the discount has been applied. Below the marked price by exactly the discount.

02 Worked example

A book marked at ₹200, with 10% off

The simplest possible discount, worked so the base is unmistakable. The marked price of a book is ₹200 and the seller allows a discount of 10%. Find the discount amount and the selling price.

1
Identify the baseThe discount is a percentage of the marked price. No other price is mentioned, and none is needed.MP = 200, discount = 10% of the MP
2
Compute the discountTen per cent of two hundred.discount = 10% of 200 = ₹20
3
Subtract to get the selling priceThe customer pays the marked price less the discount.SP = 200 − 20 = ₹180
4
Do it in one step insteadA 10% discount leaves 90% of the marked price, which skips the subtraction entirely.SP = 200 × 90/100 = ₹180
5
Check the discount percentage backwardsFrom the two prices, recover the percentage — dividing by the marked price.(200 − 180)/200 × 100 = 10% ✓

The one-step form in step four is the one to build a habit around. A discount of d% means multiplying by (100 − d)/100, and that single multiplier is what makes successive discounts and reverse questions tractable in the next two lessons. Computing the discount and subtracting works, but it doubles the arithmetic every time.

03 The method

The formulas, and the two that save time

The first two are definitions. The derived pair below them are what you should actually use, because they go straight to the answer.

Discount = MP − SP and Discount% = (MP − SP)/MP × 100. Both have the marked price underneath, which is the definition of a discount.
Go direct, in either direction. SP = MP × (100 − D)/100 gets you the selling price without computing the discount, and MP = SP × 100/(100 − D) recovers the marked price from a selling price. Most competitive questions are faster still with fraction equivalents — 12.5% off is × 7/8, 20% off is × 4/5.
DiscountMultiplierAs a fraction
10%× 0.90× 9/10
12.5%× 0.875× 7/8
20%× 0.80× 4/5
25%× 0.75× 3/4
33⅓%× 0.6667× 2/3
37.5%× 0.625× 5/8
40%× 0.60× 3/5

05 Cheat sheet

Discount on one page

Two definitions, two shortcuts, and the two things students get wrong about the base.

CaseFormulaOn MP 200, 10% off
Discount amountMP − SP200 − 180 = 20
Discount%(MP−SP)/MP × 10020/200 = 10%
SP from MPMP(100−D)/100200 × 0.9 = 180
MP from SPSP × 100/(100−D)180 / 0.9 = 200
Profit% (needs CP)(SP−CP)/CP × 100needs the cost price
Discount on the cost priceneverthe base is MP
Discount% on the selling pricenever20/180 = 11.1% is not it
The marked price is the baseBoth the discount amount and the discount percentage are measured against the label price. Dividing by the selling price gives 11.1% here, which is not a discount percentage.
Pure discount questions need no cost priceIf the final line asks for a selling price or a discount, the cost price is decoration. It is included to test whether you know which base each percentage uses.
Learn the fraction equivalents12.5% off is × 7/8 and 37.5% off is × 5/8. Competitive papers choose these percentages precisely because the fractions cancel.

06 Where & why

Where this shows up

The base case, set as a warm-up question and as the foundation of every harder discount model in the two lessons that follow.

Every placement paper
One direct discount question

Marked price and discount given, selling price wanted. Worth ten seconds with the multiplier form.

Bank PO · SSC
Reverse: marked price from selling price

“After a 20% discount an article sells for ₹400 — find the marked price.” Divide by 0.8, do not add 20%.

Questions with a decoy cost price
Testing whether you know the base

A cost price appears in the question and is not needed for the answer. Using it is the error being probed.

Everything that follows
Successive discounts, tax, reverse models

All of the next two lessons are this multiplier applied more than once. Getting the single case automatic is what makes them easy.

The one line to carry forward: a discount of d% is a multiplication by (100 − d)/100. Every remaining lesson in this module is that idea used more than once.

07 Interview questions

What gets asked

Nine, and the first three cover almost everything examiners do with a single discount.

What is a discount calculated on?
The marked price, always. Discount = MP − SP, and the percentage is that difference divided by the marked price. The cost price never appears in a discount calculation.
An article marked at ₹200 is sold at a 10% discount. Find the selling price.
₹180. Either compute the discount as 10% of 200 = ₹20 and subtract, or multiply directly by 90/100. The second route is faster and is the one to build the habit on.
After a 20% discount an article sells for ₹400. Find the marked price.
₹500. The selling price is 80% of the marked price, so divide: 400 × 100/80 = 500. Adding 20% to ₹400 gives ₹480, which is wrong because the 20% was a percentage of the marked price, not of the selling price.
Do you need the cost price to answer a discount question?
Not if the question only asks about discount, selling price or marked price. The cost price is needed only when profit or loss is involved. Examiners often supply it anyway, to see whether you know which base each percentage uses.
What is the difference between a discount and a loss?
A discount is measured from the marked price and says nothing about whether the seller profited. A loss is measured from the cost price. A seller can give a large discount and still profit handsomely, provided the marked price was set high enough.
An article is marked ₹500 and sold for ₹425. Find the discount percentage.
15%. The discount is 500 − 425 = ₹75, and 75/500 × 100 = 15%. Dividing by ₹425 instead gives 17.6%, which is not a discount percentage because the base is wrong.
What fraction is a 12.5% discount?
Multiplying by 7/8. Similarly 37.5% off is × 5/8, 20% off is × 4/5 and 33⅓% off is × 2/3. Competitive papers choose these percentages because the fractions cancel against the marked price, so recognising them saves real time.
A shop marks an item up and then discounts it. Can it still make a profit?
Yes, and that is the entire point of having a marked price. If cost is ₹400, the marked price is ₹600 and a 20% discount brings it to ₹480, the seller still makes ₹80. The marked price exists so a discount can be advertised profitably.
What is the fastest general form for a discount question?
Treat the discount as a single multiplier: (100 − d)/100 forwards and its reciprocal backwards. That one idea handles single discounts, and multiplying several of them together handles everything in the next lesson.

08 Practice problems

Six single discounts

Use the multiplier rather than computing the discount and subtracting. Two of these include a cost price you do not need.

Straight off the label

Easy
The marked price of a shirt is ₹1,200 and the shop offers a 25% discount. Find the selling price.
Follow-up
A 25% discount is a multiplication by 3/4, which makes this mental arithmetic rather than long multiplication. Build that habit now.
Show the hint
Three quarters of ₹1,200.

The decoy

Easy
A trader buys a watch for ₹1,500 and marks it at ₹2,400. He sells it at a 15% discount. Find the selling price, and state which of the given figures you did not need.
Follow-up
One of the three numbers is irrelevant to what was asked. Identifying it is the actual point, because in the exam it will not be flagged.
Show the hint
Discount is on the marked price — ask yourself whether the ₹1,500 enters at all.

Reverse it

Medium
After a discount of 20%, an article is sold for ₹960. Find its marked price. Then state what answer you would get by adding 20% to ₹960, and why it is wrong.
Follow-up
Both routes give a plausible marked price and only division uses the correct base. Working out both is how you stop making the mistake under time pressure.
Show the hint
₹960 is 80% of the marked price.

Find the percentage

Medium
An article is marked at ₹750 and sold for ₹600. Find the discount percentage, and also compute the difference divided by the selling price, saying what that second figure is not.
Follow-up
The two candidate answers are 20% and 25%, and only one is a discount percentage. Knowing which base each corresponds to is the whole content of this lesson.
Show the hint
The discount is ₹150 — divide it by the marked price for the discount percentage.

Discount and profit together

Medium
A dealer buys a chair for ₹800, marks it at ₹1,250 and sells it at a 20% discount. Find (a) the selling price, (b) the discount amount, and (c) his profit percentage.
Follow-up
Parts (a) and (b) use the marked price as their base and part (c) uses the cost price. Three answers, two different bases, and switching between them deliberately is the skill.
Show the hint
Work out the selling price first; only then does the ₹800 become relevant.

When is a discount not a loss?

Hard
A shopkeeper marks his goods at 60% above cost. (a) Find the largest whole-number discount he can offer while still breaking even. (b) Find his profit percentage if he offers a 30% discount. (c) A customer argues that a 60% discount must exactly cancel a 60% markup, leaving the shopkeeper at cost. Compute what a 60% discount actually leaves him and explain, in terms of bases, why the customer's reasoning fails.
Follow-up
Part (c) is the bases question in its sharpest form: a 60% markup and a 60% discount are percentages of different amounts, so they cannot cancel. The result is a substantial loss, which surprises most people the first time they compute it.
Show the hint
Work in multipliers: the markup is × 1.6 and a discount of d% is × (1 − d/100). Ask what product equals 1.