Profit and Loss

Profit and Loss

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Profit and Loss measures the gain or loss when selling price (SP) differs from cost price (CP). Profit = SP − CP, Loss = CP − SP, and every percentage is computed on CP, never on SP. The two most frequent NAT-I traps are (1) using SP as the base for percentage and (2) assuming two items sold at the same price — one at x% profit and one at x% loss — break even. They actually yield a guaranteed loss of x²/10 %.

• Profit % = (Profit / CP) × 100
• Loss % = (Loss / CP) × 100
• SP = CP × (1 + Profit%/100); CP = SP × 100 / (100 + Profit%)
• Discount sits on Marked Price, profit/loss sits on CP — different bases.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core definitions and the CP-as-base rule

The four commercial-mathematics price labels are CP (what the seller pays), MP (the list/tag price), SP (what the buyer finally pays), and Discount (MP − SP). Profit and loss percentages are always measured against CP, while discount is always measured against MP. Conflating these two bases is the single most common source of wrong answers in NAT-I.

The standard percentage conversions

The two formula pairs you must convert without hesitation:

Want	Given	Formula
SP	CP, Profit%	CP × (100 + g%) / 100
CP	SP, Profit%	SP × 100 / (100 + g%)
SP	CP, Loss%	CP × (100 − L%) / 100
CP	SP, Loss%	SP × 100 / (100 − L%)

The “two items at same SP” identity

When the same SP is charged for two articles with equal CP — one at x% profit, the other at x% loss — total CP = 2·CP_each, total SP = 2·SP_each, and the net loss simplifies to x²/10 % of CP. So 20% + 20% gives a 4% loss, not zero. This identity is tested at least once every NAT-I cycle.

Successive discounts

Two successive discounts of D₁% and D₂% on the same MP give an effective single discount of D₁ + D₂ − (D₁ × D₂)/100. So 20% followed by 30% is 44%, not 50%.

• Apply 20% to MP, then 30% to the reduced price — never add them straight.
• For three discounts, fold two at a time: combine D₁ & D₂ first, then with D₃.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge cases and overhead costs

Real NAT-I items sometimes embed transport, repair, or commission into the cost side. The true CP becomes purchase CP + overhead before any profit/loss arithmetic. A trader who buys a fan for Rs. 8,000, spends Rs. 1,000 on repair, then sells for Rs. 10,000 makes Rs. 1,000 profit (12.5%), not 25% — many students skip the overhead and double the real percentage.

The “a vs b articles” relationship

When the cost price of a articles equals the selling price of b articles, the resulting profit or loss % is computed as:

Direction	Result
a > b (fewer articles sold per rupee)	Profit % = ((a − b) / b) × 100
a < b (more articles sold per rupee)	Loss % = ((b − a) / b) × 100

For example, if CP of 12 pens equals SP of 15 pens, then loss % = (15 − 12)/15 × 100 = 20%.

Worked micro-example

A shopkeeper marks a shirt at Rs. 2,000 and offers successive discounts of 10% and 5%, then sells it. Effective discount = 10 + 5 − (10×5)/100 = 14.5%, so SP = 2,000 × (1 − 0.145) = Rs. 1,710. If the shirt’s CP was Rs. 1,500, profit = 1,710 − 1,500 = Rs. 210, and profit % = 210/1,500 × 100 = 14%.

Common NAT-I traps

• Mixing MP-based discount % with CP-based profit % in the same line.
• Forgetting to convert percentage to a decimal in successive discount chains.
• Treating “no profit no loss” as the default for matching x% profit and x% loss pairs.

Practice prompts

1. An article is sold at Rs. 1,200 with a 20% profit. Find CP and the new SP if profit is reduced to 10%.
2. CP of 9 toys equals SP of 11 toys. Compute the gain or loss percentage and verify with the x²/10 identity on equivalent rates.

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