🧮 Aptitude & Reasoning · Chapter 6

Profit, loss and discount chains

"50% off + extra 20% off" is not 70% off, and every profit-and-loss question in a placement test is built on someone believing it is.

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The basics

Profit and loss is percentages with three prices on one ladder: what the shop paid, what it wrote on the tag, and what you actually handed over.

The problem it solves: working out whether a deal, a markup or a "festival discount" actually makes or loses money — and by how much.

The mental model: picture a ladder. Cost price (CP) is the bottom rung. The shop climbs to the marked price (MP) with a markup, then steps down to the selling price (SP) with a discount. Each step is a multiplier, exactly as in the percentages chapter, so the whole chain is one multiplication: SP = CP × (1 + markup) × (1 − discount).

The one rule that decides most questions: profit % is measured on CP; discount % is measured on MP. Two different bases, in one question, on purpose.

Explained

A ₹2,000 kurta, Diwali sale: "Flat 50% off + extra 20% off".

Most people read 70% off and expect to pay ₹600. The bill says ₹800.

The extra 20% is taken off ₹1,000, not off ₹2,000. 0.5 × 0.8 = 0.4, so you pay 40% — a 60% discount. The shop never lied. It just let you add two percentages that have different bases.

Sharma ji's festival discount

Sharma ji runs a saree shop in Lajpat Nagar. He buys a saree for ₹1,000.

Before Navratri he writes ₹1,400 on the tag — a 40% markup. During the sale he puts up a big red board: 25% OFF.

A customer pays ₹1,050, feels she saved ₹350, and leaves happy.

Sharma ji still makes ₹50 — 5% profit on his ₹1,000.

Nothing is hidden. The markup was 40% of CP, the discount was 25% of MP. 40 − 25 looks like 15, but the discount bit into a bigger number: 1.4 × 0.75 = 1.05. Every markup-and-discount question is this saree.

The price ladder

            MP  ₹1,400   ← tag price
             ▲      │
 markup 40%  │      │  discount 25%
 (of CP)     │      │  (of MP = 350)
  ×1.40      │      ▼  ×0.75
             │     SP  ₹1,050   ← what she paid
             │      │
            CP  ₹1,000 ─┘  profit = 50 → 5% of CP

 discount chain on a ₹2,000 kurta:
 2,000 ──×0.50──▶ 1,000 ──×0.80──▶ 800
         (−1,000)          (−200)        total off: 60%, not 70%

Every arrow is a multiplier. Read the chain left to right and multiply; never add the labels.

The formulas, and the shortcut

Standard method — multipliers:

Profit %   = (SP − CP) / CP × 100
SP         = CP × (1 + p/100)          (loss: 1 − l/100)
SP         = MP × (1 − d/100)
MP / CP    = (1 + p/100) / (1 − d/100)
Discounts a%, b%  → pay (1 − a/100)(1 − b/100)

Shortcut — two steps in one line: net % = a + b + ab/100, signs included.

• Discounts 20 and 15: −20 − 15 + 3 = −32% (single 32% off)
• Markup 30, discount 15: 30 − 15 − 4.5 = +10.5% profit

Special results worth knowing:

• Two items sold at the same SP, one at x% profit and one at x% loss → always a loss of x²/100 %
• False weight: gives w grams for 1,000 → extra gain = (1000 − w)/w
• Buy x get y free → discount = y/(x + y)

When the shortcut fails: three or more steps (just multiply), and any step that is a flat ₹ amount — "₹500 off" is not a percentage of anything until you know the price.

Solving steps

Step 3 is where marks are lost. Say the base out loud before you write a single number.

Three questions, ninety seconds

Q1 (30 s). A trader buys at ₹600 and wants 20% profit after giving 10% discount. What should he mark?

SP = 600 × 1.2 = 720. MP × 0.9 = 720 → MP = ₹800. (Not 600 × 1.3 = 780 — the 10% is off MP, not added to CP.)

Q2 (20 s). Successive discounts of 20%, 10% and 5%. Single equivalent?

0.8 × 0.9 × 0.95 = 0.684 → 31.6%. Three steps, so skip the shortcut and multiply.

Q3 (40 s). Two phones sold at ₹990 each, one at 10% profit, one at 10% loss. Net?

CPs: 990/1.1 = 900 and 990/0.9 = 1,100. Total CP 2,000, total SP 1,980 → loss ₹20 = 1%. Shortcut: x²/100 = 100/100 = 1% loss, done in five seconds.

The honest-looking loss

A dry-fruit seller in Chandni Chowk swears he sells almonds at 10% below cost price. His scale, though, gives 800 g whenever it shows 1 kg.

Traps examiners set

• Adding successive discounts — 50% + 20% is 60% off, not 70%
• Discount on CP — the 10% off is on MP; "CP × 1.3" for "20% profit after 10% off" is wrong (it's ×1.2 ÷ 0.9)
• Profit as a % of SP — profit % is on CP unless the question explicitly says "on selling price"
• Same SP, ±x% — feels like break-even; it is always an x²/100 % loss
• Buy 3 get 2 free — that is 2 free out of 5 = 40%, not 2/3
• "Gain equals SP of 11 m when 33 m are sold" — the gain is compared to CP, so it's 11/22 = 50%, not 11/33
• Flat-rupee discounts — they break every percentage shortcut; convert to rupees first

Words to know

In two minutes

• Three prices sit on one ladder: cost price → (markup) → marked price → (discount) → selling price.
• Profit and loss are percentages of CP; discount is a percentage of MP — most traps swap those bases.
• Every step is a multiplier, so SP = CP × (1 + markup) × (1 − discount).
• Successive discounts multiply: 50% then 20% is 60% off, not 70%; for two steps use a + b + ab/100.
• Two items at the same SP, one at x% profit and one at x% loss, always give an x²/100 % net loss.
• False weights and 'buy x get y free' are just hidden multipliers: (1000 − w)/w and y/(x + y).
• Set CP = 100 when no rupee value fixes it, solve in plain numbers, then scale back.

Test yourself

There are 10 questions, 3 puzzles and 5 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.