Profit, Loss & Discount

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Marked Price, Discount and the CP–MP–SP Chain

The full chain: trader marks up CP to MP, then allows a discount on MP to reach SP.

$\text{Discount} = MP - SP, \qquad \text{Discount\% is always on MP}$ $MP = CP \times \frac{100 + g}{100 - d}, \qquad SP = MP\left(\frac{100 - d}{100}\right)$

Multi-level trade (manufacturer → wholesaler → retailer) chains the same way: multiply each markup factor $\left(1 + \frac{x}{100}\right)$ .

If a discount of d% still leaves a gain of g%, the markup on CP must be $\frac{(100+g)}{(100-d)} - 1$ times CP.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 14 practice Q

Detailed notes

The full chain: CP → MP → SP

A shopkeeper does not sell at cost. He writes a higher price on the label — the marked price (MP), also called the list price or tag price — and then offers a discount on it. $\text{Discount} = MP - SP, \qquad \text{Discount\% is always calculated on MP}$ Buying at ₹800, tagging ₹1,000 and selling at ₹900: discount ₹100 = 10% of MP; profit ₹100 = 12.5% of CP. The discount is never a percentage of CP.

The master relation

$MP = CP \times \frac{100 + g}{100 - d}$ where g is the final gain% and d the discount%. Any one of markup, discount, gain fixes the other two. Marked 40% above CP with a 10% discount: $1.4 \times 0.9 = 1.26$ → profit 26%.

One-step and two-step versions

"What should the markup be to allow 10% discount and still gain 17%?" → $\frac{117}{90} = 1.3$ → mark 30% above CP.
Marked ₹1,200, sold at 15% off → SP $= ₹1{,}020$ , discount $= ₹180 = 15\%$ of MP.
Find the gain: marked 25% above CP, discount 10% → $1.25 \times 0.9 = 1.125$ → gain 12.5%.

Working backwards

MP ₹1,500 sold for ₹1,275 → discount 15%. If that left a gain of g%, build CP by dividing the SP by the gain chip: SP ₹1,275 at 15% discount and 25% profit → CP $= 1275 \div 1.25 = ₹1{,}020$ . Exam numbers are always chosen to come out whole — if yours do not, re-read the question.

Multi-level trade chains

Manufacturer → wholesaler → retailer → customer: each stage multiplies its own chip. Cost ₹4,000, then +25%, +20%, +10%: $4000 \times 1.25 \times 1.2 \times 1.1 = ₹6{,}600$ . Each trader's profit is on HIS own purchase price, not on the original cost. Working backwards, divide by the chips in reverse order.

Discount schemes

Flat d% is one step × $\frac{100-d}{100}$ .
"Buy 5 get 1 free" → pay for 5, receive 6 → discount $= \frac{1}{6} = 16\frac{2}{3}\%$ .
"Buy 19 get 1 free" → $\frac{1}{20} = 5\%$ — the free item counts against what you RECEIVE, not what you pay for.
A flat discount always beats successive discounts of the same total: 40% flat leaves 0.60, while 30% + 10% leaves $0.7 \times 0.9 = 0.63$ .

Common traps

Taking discount% on CP or SP instead of MP.
Subtracting the discount from the markup (40% up − 10% down is 26% up, not 30%: $1.4 \times 0.9 = 1.26$ ).
In a chain, applying every markup to the manufacturer's cost only.

Quick revision

Discount% is on MP; profit% is on CP.
$MP = CP \times \frac{100+g}{100-d}$ ; multiply chips through the chain.
Markup% $= \left(\frac{100+g}{100-d} - 1\right) \times 100$ .
Free-item offers: discount = free ÷ total received.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Discount per cent (and discount amount) from MP and SPvery common2 practice Q

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

Discount $= MP - SP$ .
Divide by MP (never by SP or CP) and multiply by 100.
For the rupee value, take that per cent of MP.

Why: a discount is a reduction offered on the tagged price.

Example: An article marked ₹900 is sold for ₹765. Find the discount per cent.

$\frac{135}{900} \times 100 = 15\%$ . Discount amount $= ₹135$ .

Type 2: Marked price for a target profit after discountvery common2 practice Q

How to spot it:

'At what price should he mark to allow a d% discount and still gain g%?' — CP given, MP asked.

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

Write the chain: CP × markup chip × discount chip = SP.
Set $SP = CP \times \frac{100+g}{100}$ and divide by the discount chip.
Or apply the formula directly.

Why: MP must absorb both the discount and the desired gain.

Example: An article costing ₹800 is to be sold at a 20% discount and still gain 20%. The marked price should be:

$MP = 800 \times \frac{120}{80} = ₹1{,}200$ . Check: $1200 \times 0.8 = 960 = 800 \times 1.2$ .

Type 3: Markup per cent from discount and profitvery common2 practice Q

How to spot it:

'Allows 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

Compute $\frac{100+g}{100-d}$ — this is MP as a multiple of CP.
Subtract 1 and multiply by 100.
Take CP = 100 and trace the rupees if the formula feels abstract.

Why: the ratio of MP to CP depends only on the two percentages.

Example: 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?

$\frac{120}{75} = 1.6$ → marked 60% above CP. (CP 100 → MP 160 → SP 120.)

Type 4: Multi-level trade chain (profit of the last trader)common2 practice Q

How to spot it:

Manufacturer sells to wholesaler, wholesaler to retailer, retailer to customer, each at a profit per cent; the final price or total rise is asked.

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

Write each stage's chip ( $1.25 = +25\%$ , $1.1 = +10\%$ …).
Multiply all chips with the base cost (or divide backwards from the final price).
Each trader's profit% is on his own purchase price.

Why: every stage treats its buying price as its cost.

Example: 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 ₹4,000 to make, the customer pays:

$4000 \times 1.25 \times 1.2 \times 1.1 = ₹6{,}600$ . Total rise $= 65\%$ .

Type 5: Discount schemes: free items and flat offerscommon2 practice Q

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

Count what the customer PAYS for and what he RECEIVES.
Discount% $=$ free ÷ received × 100.
Compare schemes by the kept fraction, not by the label.

Why: the seller spreads the cost of the free items over the whole batch.

Example: 'Buy 4 get 1 free' — what discount is the shop really giving?

Pay for 4, get 5 → discount $= \frac{1}{5} \times 100 = 20\%$ .

Formulas

Discount per cent

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

Marked price for target gain

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

SP through the chain

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

Multi-level markups

\text{Final} = \text{Base} \times \prod \left(1 + \frac{x_k}{100}\right)

Shortcut tricks

⚡ Build the MP/CP ratio

Gain g% and discount d% together fix MP as a multiple of CP.

Example: After allowing a discount of 10% on the marked price a trader still gains 20%. The marked price is how much per cent above the cost price?

$\frac{MP}{CP} = \frac{120}{90} = \frac{4}{3}$ ⇒ 33⅓% above CP.

⚡ Markup chip then discount chip

One chip for the markup, one for the discount.

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

$SP = 1000 \times 1.25 \times 0.85 = 1062.50$ ⇒ profit $6.25\%$ , i.e. ₹62.50.

⚡ Trade-chain multiplication

Each middleman's markup is one chip.

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. If the customer pays ₹3,300, the manufacturing cost is:

$2000 \times 1.2 \times 1.25 \times 1.1 = 3300$ ⇒ cost = ₹2,000.

Where students lose marks

Taking the discount on CP instead of MP.
Forgetting that profit % is still on CP while discount % is on MP — the two bases differ.
In trade chains, adding the markups (20 + 25 + 10 = 55%) instead of multiplying chips (65%).
Reading 'marked 25% above cost' as MP = 125 when CP = 100 — right — but then applying discount % to CP.

Practice sets — 14 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

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