Marked Price, Discount and the CP–MP–SP Chain — notes, question types & practice · Profit, Loss & Discount

01

The chain: CP to MP to SP

A shopkeeper does not sell at cost. He writes a higher price on the label, then offers a discount on it.

\text{Discount} = MP - SP, \qquad \text{Discount\% on MP}

Buy at Rs 800, tag at Rs 1,000, sell at Rs 900. The discount is Rs 100, which is $10\%$ of MP. The profit is Rs 100, which is $12.5\%$ of CP. Same hundred rupees, two different percents.

Rule: Discount per cent is on MP; profit per cent is on CP. Never mix the two bases.

02

The master relation

MP = CP \times \frac{100 + g}{100 - d}

Here $g$ is the final gain per cent and $d$ the discount per cent. Any two of markup, discount and gain fix the third.

Marked $40\%$ above CP with a $10\%$ discount: $\dfrac{14}{10} \times \dfrac{9}{10} = 1.26$ , a profit of $26\%$ .

"What markup allows a $10\%$ discount and still gains $17\%$ ?" $\dfrac{117}{90} = 1.3$ , so mark $30\%$ above CP.

03

One chain, any missing link

Marked Rs 1,200, sold at $15\%$ off: $SP = 1200 \times \dfrac{17}{20} = 1020$ .
Marked $25\%$ above CP, discount $10\%$ : $\dfrac{5}{4} \times \dfrac{9}{10} = \dfrac{9}{8}$ , a gain of $12.5\%$ .
SP Rs 1,275 after a $15\%$ discount and a $25\%$ profit: $MP = 1275 \times \dfrac{20}{17} = 1500$ and $CP = 1275 \times \dfrac{4}{5} = 1020$ .

Tip: Exam numbers come out whole. If yours do not, re-read the question before doubting the method.

04

Multi-level trade chains

Manufacturer to wholesaler to retailer to customer: each stage multiplies its own chip on its own purchase price.

A TV costs Rs 4,000 to make. The maker sells at $25\%$ profit, the wholesaler at $20\%$ , the retailer at $10\%$ :

4000 \times \frac{5}{4} \times \frac{6}{5} \times \frac{11}{10} = 6600

The customer pays $65\%$ above the making cost. To work backwards, divide by the chips in reverse order.

Watch: Never add the markups. $25 + 20 + 10 = 55\%$ is the trap; multiply chips to get $65\%$ .

05

Free-item offers

"Buy 5, get 1 free": pay for 5, receive 6. Discount $= \dfrac{1}{6} = 16\dfrac{2}{3}\%$ .
"Buy 4, get 1 free": discount $= \dfrac{1}{5} = 20\%$ .
"Buy 19, get 1 free": discount $= \dfrac{1}{20} = 5\%$ .

Key: Discount $=$ free items $\div$ items received. The free item counts in what you receive, not what you pay for.

06

Flat against successive

A flat $40\%$ leaves $\dfrac{60}{100}$ of the price. Successive $30\%$ and $10\%$ leave $\dfrac{7}{10} \times \dfrac{9}{10} = \dfrac{63}{100}$ .

The flat offer is always better for the buyer when the numbers add up the same.

07

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Discount per cent from MP and SP

How to spot it:

MP and SP (or the discount in rupees) are given; the discount per cent or amount is asked.

\text{Discount\%} = \frac{MP - SP}{MP} \times 100

Method

Discount = MP - SP.
Divide by MP, never by SP or CP.
Multiply by 100 for the per cent.

Why it works:

A discount is a cut offered on the tagged price, so MP is the base.

Try this

An article marked Rs 900 is sold for Rs 765. Find the discount per cent.

Show solution

Discount $= 900 - 765 = 135$ .
$\dfrac{135}{900} \times 100 = 15$ .

Answer

15%

Type 2very common2 practice Q

Marked price for a target profit after discount

How to spot it:

'At what price should he mark the article to allow a d% discount and still gain g%?'

MP = CP \times \frac{100 + g}{100 - d}

Method

Write the chain CP x markup chip x discount chip = SP.
Set SP to the wanted gain chip of CP.
Divide by the discount chip to get MP.

Why it works:

The MP must absorb both the discount and the desired gain.

Try this

An article costing Rs 800 is to be sold at a 20% discount and still gain 20%. Find the marked price.

Show solution

$MP = 800 \times \dfrac{120}{80} = 1200$ .
Check: $1200 \times \dfrac{4}{5} = 960 = 800 \times \dfrac{6}{5}$ .

Answer

Rs 1,200

Type 3very common2 practice Q

Markup per cent from discount and profit

How to spot it:

'Allows a d% discount and still gains g%. The marked price is what per cent above cost?'

\text{markup\%} = \left(\frac{100+g}{100-d} - 1\right) \times 100

Method

Compute the ratio of 100+g to 100-d.
Subtract 1 and multiply by 100.
Trace CP = 100 through MP to SP as a check.

Why it works:

The ratio of MP to CP depends only on the two percents.

Try this

A trader allows a 25% discount on the marked price and still gains 20%. The marked price is what per cent above the cost price?

Show solution

$\dfrac{120}{75} = \dfrac{8}{5}$ .
Markup $= \dfrac{3}{5} \times 100 = 60\%$ .
Check: CP 100, MP 160, SP $= \dfrac{3}{4} \times 160 = 120$ .

Answer

60%

Type 4common2 practice Q

Multi-level trade chain

How to spot it:

Manufacturer to wholesaler to retailer to customer, each at his own profit per cent.

\text{final} = \text{cost} \times \prod \frac{100 + x_i}{100}

Method

Write each stage's chip.
Multiply all chips with the base cost.
Working from the final price, divide by the chips in reverse order.

Why it works:

Every stage treats its buying price as its own cost.

Try this

A manufacturer sells a TV to a wholesaler at 25% profit, the wholesaler to a retailer at 20% profit, and the retailer to a customer at 10% profit. If it cost Rs 4,000 to make, what does the customer pay?

Show solution

Chips: $\dfrac{5}{4}, \dfrac{6}{5}, \dfrac{11}{10}$ .
$4000 \times \dfrac{5}{4} = 5000$ , then $6000$ , then $6600$ .
Total rise $= 65\%$ .

Answer

Rs 6,600 (65% above cost)

Type 5common2 practice Q

Free-item and flat offers

How to spot it:

'Buy 5 get 1 free', 'pay for 12 take 14', or comparing a scheme with a flat discount.

\text{discount\%} = \frac{\text{free items}}{\text{items received}} \times 100

Method

Count what the customer pays for and what he receives.
Discount per cent = free over received, times 100.
Compare schemes by the kept fraction, not the label.

Why it works:

The seller spreads the cost of the free items over the whole batch.

Try this

A shop offers 'buy 4 get 1 free' on pens. What discount per cent is this?

Show solution

Pay for 4, receive 5.
Discount $= \dfrac{1}{5} \times 100 = 20\%$ .

Answer

20%

08

Formula sheet

Discount per cent

\text{D\%} = \frac{MP - SP}{MP} \times 100

On the marked price.

Marked price for a target gain

MP = CP \times \frac{100 + g}{100 - d}

g = gain%, d = discount%.

SP through the chain

SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)

m = markup%.

Markup per cent

\text{markup\%} = \left(\frac{100+g}{100-d} - 1\right) \times 100

Multi-level markups

\text{final} = \text{cost} \times \prod \frac{100 + x_k}{100}

One chip per trader.

Free-item discount

\text{discount\%} = \frac{\text{free}}{\text{received}} \times 100

09

Shortcuts that save time

⚡ Build the MP-over-CP ratio

Gain $g\%$ and discount $d\%$ together fix MP as a multiple of CP: $\dfrac{100+g}{100-d}$ .

Example

After allowing a 10% discount a trader still gains 20%. The marked price is what per cent above cost?

Show solution

$\dfrac{MP}{CP} = \dfrac{120}{90} = \dfrac{4}{3}$ .
Markup $= 33\dfrac{1}{3}\%$ .

Answer

33 1/3% above CP

⚡ Markup chip then discount chip

Two fractions give the whole story from CP to SP in one line.

Example

An article costing Rs 1,000 is marked 25% above cost and sold at a 15% discount. Find the profit per cent.

Show solution

$SP = 1000 \times \dfrac{5}{4} \times \dfrac{17}{20} = 1062.50$ .
Profit $= \dfrac{62.5}{1000} = 6.25\%$ .

Answer

6.25%

⚡ Trade chains multiply

One chip per middleman. Work backwards from the final price by dividing in reverse order.

Example

A manufacturer sells to a wholesaler at 20% profit, the wholesaler to a retailer at 25% profit, and the retailer to a customer at 10% profit. The customer pays Rs 3,300. Find the manufacturing cost.

Show solution

Chips: $\dfrac{6}{5} \times \dfrac{5}{4} \times \dfrac{11}{10}$ .
$3300 \div \left(\dfrac{6}{5} \times \dfrac{5}{4} \times \dfrac{11}{10}\right) = 2000$ .

Answer

Rs 2,000

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Taking the discount per cent on CP.

The discount always acts on the marked price.

Mistake 02

Markup minus discount: $40\%$ up, $10\%$ off called $30\%$ up.

Multiply: $\dfrac{14}{10} \times \dfrac{9}{10} = 1.26$ , so $26\%$ up.

Mistake 03

Adding chain markups: $20 + 25 + 10 = 55\%$ .

Multiply chips: $1.2 \times 1.25 \times 1.1 = 1.65$ , a $65\%$ rise.

Mistake 04

Taking every trader's profit on the manufacturer's cost.

Each stage's per cent is on its own purchase price.

Mistake 05

Reading 'buy 4 get 1 free' as a $25\%$ discount.

Discount is $\dfrac{1}{5} = 20\%$ , free over received.

11

Quick revision

Read this the night before the exam.

Discount per cent on MP; profit per cent on CP.
$MP = CP \times \dfrac{100+g}{100-d}$ .
Markup $\% = \left(\dfrac{100+g}{100-d} - 1\right) \times 100$ .
Chains: one chip per stage, multiply; backwards, divide in reverse.
Free items: discount $=$ free $\div$ received.
Flat beats successive discounts of the same total.

12

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.