Profit, Loss and Discount

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Profit and loss measure the gap between Cost Price (CP) and Selling Price (SP) in PKR; discount measures the gap between Marked Price (MP) and SP. Both profit% and loss% are always computed on CP, while discount% is computed on MP.

Profit = SP − CP; Profit % = (Profit ÷ CP) × 100
Loss = CP − SP; Loss % = (Loss ÷ CP) × 100
SP = CP × (1 + x/100) for x% profit, or CP × (1 − x/100) for x% loss
Discount % = ((MP − SP) ÷ MP) × 100, so SP = MP × (1 − d/100)

HAT-UG traps in two sentences: successive discounts of 20% and 30% give a single equivalent of 44%, not 50%. A profit of x% on CP equals only (100x)/(100+x)% on SP.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Definitions

Cost Price (CP) is what the seller pays; Selling Price (SP) is what the buyer pays; Marked Price (MP) is the listed price before any discount. The signed difference SP − CP is profit when positive and loss when negative; SP − MP is the discount, always non-positive.

Single-Transaction Relationships

When the seller marks up CP by x% to set MP and then offers a discount d%, the final SP becomes CP × (1 + x/100) × (1 − d/100). A positive result means profit, a negative one loss.

Successive Discounts

Two discounts of d₁% and d₂% applied one after the other collapse into a single equivalent discount d = d₁ + d₂ − (d₁·d₂)/100. Three successive discounts d₁, d₂, d₃ combine as d₁ + d₂ + d₃ − pairwise products + triple product, all divided by the appropriate power of 100.

Concept	Formula	Base
Profit %	(SP − CP) / CP × 100	on CP
Loss %	(CP − SP) / CP × 100	on CP
Discount %	(MP − SP) / MP × 100	on MP
Equivalent discount	d₁ + d₂ − d₁d₂/100	MP
Dishonest-dealer gain	(True − False) / False × 100	true weight

HAT-UG Pattern

HAT-UG Quantitative Reasoning allocates roughly 4% of marks to this topic, typically one MCQ per past paper. Items range from a single-step profit/loss percentage to a two-successive-discount equivalent or a marked-up-then-discounted transaction.

Spot the base first: profit/loss % sits on CP, discount % sits on MP.
Convert every percentage into the multiplier (1 ± x/100) before chaining.
A false-weight problem becomes a ratio in disguise: gain % = (true weight − false weight) / false weight × 100.
A profit of x% on CP translates to (100x)/(100+x)% on SP — useful when the question reverses the base.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Worked Example

A shopkeeper marks an article at MP = PKR 2,500 and offers successive discounts of 20% and 10%. Find the single equivalent discount and the final SP.

Step 1 — Apply 20%: 2,500 × 0.80 = 2,000. Step 2 — Apply 10%: 2,000 × 0.90 = 1,800, so SP = PKR 1,800. Step 3 — Single equivalent: d = 20 + 10 − (20×10)/100 = 28%. Check: 2,500 × (1 − 0.28) = 2,500 × 0.72 = 1,800 ✓.

If CP was PKR 1,500, profit = 1,800 − 1,500 = 300, profit % = (300/1,500) × 100 = 20%.

Edge Cases and Mechanism

Profit% and loss% are mirror operations: a 20% profit on CP corresponds to a loss of 16⅔% if the same SP were treated as a markup on SP instead of CP. Dishonest-dealer problems share the same structure as partnership-time ratios because both rely on (true − claimed)/claimed. Break-even pricing requires SP = CP × (1 + overhead fraction); anything above zero overhead guarantees a loss once discount exceeds the markup margin.

Common Traps in MCQs

Trap	What students write	Correct move
Base confusion	Computing profit % on SP	Always divide by CP
Additive discounts	20% + 30% = 50%	Use 20 + 30 − 6 = 44%
SP ↔ CP reversal	CP = SP × (100+x)/100	CP = SP × 100/(100+x)
Unit mismatch	Mixing kg and g in false weight	Convert to a single unit
Markup vs profit	Treating MP − CP as profit	Subtract discount first

Practice Prompts

An item bought at PKR 800 is marked at PKR 1,200 and sold after a 15% discount. Compute profit % and the equivalent single discount if a second 5% discount is then offered.
A dealer claims 1 kg but delivers 900 g. Buying at CP per kg and selling at MP per kg, find gain %; then find the discount he must offer on MP to wipe out the gain.

Tip: Always write the multiplier form (1 ± x/100) before you substitute numbers — it kills three out of four HAT-UG traps in this topic.

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Profit, Loss and Discount