Profit and Loss

Definitions:

• CP (Cost Price): what the seller paid.
• SP (Selling Price): what the customer pays.
• MP (Marked Price / List Price): displayed price; before discount.
• Profit = SP − CP. Loss = CP − SP.
• Profit % is always on CP. Discount % is always on MP.

Key formulas:

• Profit % = (Profit / CP) × 100
• SP = CP × (1 + Profit% / 100) (or × (1 − Loss%/100) for loss)
• SP = MP × (1 − Discount%/100)

Trick 1: convert percentages to fractions.

For a 25% profit, SP/CP = 5/4. Just multiply CP by 5/4 directly — much faster than (1 + 0.25).

Trick 2: successive markup-discount.
If MP is x% above CP and discount is y%, net profit% = x + (−y) + (x · −y / 100).

Example: MP = 50% above CP, discount 20%. Net = 50 − 20 − 10 = +20% profit.

Trick 3: "would have made x% more" types.
If actual SP1 gives p% profit, and SP2 gives p% loss, the question is often "by how much should SP1 be increased to give p% profit" — directly compute the difference.

Trick 4: false weight problems.
If a shopkeeper sells at CP but uses a 900g weight for 1 kg: gain = 100/900 × 100 = 11.11%.
General: gain % = (False weight error / True weight after error) × 100.

Worked example. A shopkeeper marks up by 25% and gives 12% discount. Find profit %.

Net = 25 + (−12) + (25 · −12 / 100) = 25 − 12 − 3 = +10%.

Worked example (false weight). Shopkeeper claims to sell at CP but uses a 940g weight for 1kg. Gain %?

Gain = 60/940 × 100 = 6.38%.

PROFIT AND LOSS — QUICK MASTERY

(See cp-sp-mp Pack 19 for cost/selling/marked prices; this focuses on tricks.)

FUNDAMENTAL RULES:

Profit % = (SP − CP)/CP × 100.
Loss % = (CP − SP)/CP × 100.

Always on CP.

FAST TECHNIQUES:

1. SP from CP & Profit%:
SP = CP × (1 + P/100).

E.g., CP = ₹100, profit 25% → SP = 125.
CP = ₹80, profit 15% → SP = 80 × 1.15 = 92.

2. CP from SP & Profit%:
CP = SP / (1 + P/100).

E.g., SP = ₹120, profit 20% → CP = 120/1.2 = ₹100.

3. Markup-Discount in one step:

Markup M%, discount D%, profit %:
Profit% = M − D − MD/100.

E.g., M=30%, D=10%: profit = 30 − 10 − 3 = 17%.
E.g., M=25%, D=20%: profit = 25 − 20 − 5 = 0%.

KEY OBSERVATIONS:

Two articles same SP, equal % profit & loss:
Net LOSS = (X/10)² %.

• 10%, 10% → 1% loss.
• 20%, 20% → 4% loss.

Total SP common, profit on one, loss on other:

• Sum of fractions (1+P/100) and (1−L/100) = 2.
• If profit P% on one, loss must be (P − P²/(100+P))%... complicated. Use direct calculation.

Two purchases at different rates → calculate effective profit:

• E.g., 60% sold at 20% profit; 40% sold at 30% loss.
• Net = 0.6 × 1.2 − 0.4 × 0.7 = 0.72 − 0.28 = 0.44.
• Wait, this gives net SP of 0.44; for net profit/loss, compare to CP of 1.0 → loss of 0.56? No wait, need to be careful.
• Actually: 60% of stock at 1.2 × CP_unit + 40% at 0.7 × CP_unit. Total cost paid = 1.0 (assuming uniform CP_unit). Total revenue = 0.6 × 1.2 + 0.4 × 0.7 = 0.72 + 0.28 = 1.00. Break-even.

WORKED EXAMPLES:

Q1. A man bought oranges at 16/₹10 and sold at 14/₹10. Loss %?

• CP/orange = 10/16 = ₹0.625.
• SP/orange = 10/14 = ₹0.714.
• Wait, that's profit not loss. Let me re-read.
• 16 oranges for ₹10; CP/orange = 10/16 = 0.625.
• Sold 14 oranges for ₹10; SP/orange = 10/14 ≈ 0.714.
• Profit per orange = 0.714 − 0.625 = 0.089.
• Profit % = 0.089 / 0.625 × 100 ≈ 14.28%.

Hmm — actually selling 14 for ₹10 means LESS oranges for same money → MORE per orange → PROFIT.

Wait, the problem said "loss %". Let me re-examine.

• Buying 16 oranges for ₹10 means each costs ₹10/16 = ₹0.625.
• Selling 14 oranges for ₹10 means each sells for ₹10/14 = ₹0.714.

So profit, not loss. The problem statement may be wrong, OR it's the reverse: 14/₹10 buying (₹0.714 CP), 16/₹10 selling (₹0.625 SP). That gives loss of 12.5%.

This shows: read units carefully.

Q2. A trader marks up 20% above CP, then offers 10% discount. Profit %?

• Markup 20%, discount 10%.
• Net = 20 − 10 − 20×10/100 = 10 − 2 = 8% profit.

Q3. Two articles sold at ₹500 each: one at 25% profit, other at 25% loss. Net %?

• Profit one: CP₁ = 500/1.25 = 400. Profit = 100.
• Loss one: CP₂ = 500/0.75 = 666.67. Loss = 166.67.
• Net: loss of 66.67 on combined CP of 1066.67.
• Loss % = 6.25%.

(Direct shortcut: (25/10)² = 6.25 ✓.)

Q4. A merchant buys 30 kg rice at ₹40/kg and 50 kg at ₹30/kg. Mixes and sells at ₹35/kg. Profit %?

• Total CP = 30×40 + 50×30 = 1200 + 1500 = 2700.
• Total SP = 80×35 = 2800.
• Profit = 100, Profit % = 100/2700 × 100 ≈ 3.7%.

CHAIN PROFIT:

A → B → C → D, with successive markups.

If A → B at 20% profit, B → C at 25% profit:

• C's CP relative to A's CP = 1.2 × 1.25 = 1.5 (50% net).

EXAM HOOKS:

• Profit %, discount % both on different bases (CP vs MP).
• Same SP, equal % profit/loss → net loss = (X/10)².
• For mixture sales: weighted average.
• Markup + discount formula: M − D − MD/100.
• Read CP/SP units carefully.
• Dishonest weight: use trick.