Profit-Loss, SI & CI

Snapshot

• Profit-Loss, Simple & Compound Interest is applied percentage — if Part 1 of the Ratio/Percentage guide is solid, this is one of the fastest-scoring areas in the General Test.
• Two engines: profit/loss (a percentage on the cost price) and interest (simple grows in a straight line, compound curves upward). Almost every error here is a wrong base — knowing which price each percentage sits on is the whole game.
• This guide covers profit/loss with discounts and dishonest-dealer tricks, simple and compound interest with the shortcut formulas, and the successive-discount rule — with worked examples for each.
• Exam reality: +5 / −1.

Part 1 — Profit & Loss

Everything is a percentage of the cost price (CP) unless stated otherwise.

Two traps the exam loves: profit% is on CP but discount% is on MP — different bases; and a 20% profit then 20% loss is a net loss (the successive-change rule): 20 − 20 − 4 = 4% loss.

Successive discounts combine like successive change: two discounts of a% and b% give a single equivalent discount of a + b − (ab ÷ 100).

Dishonest dealer / false weights. A trader who sells at cost price but gives only 900 g for a kg makes a profit of (error ÷ true − error) = 100 ÷ 900 = 11.11%. The gain% = (true value − given value) ÷ given value × 100.

Part 2 — Simple vs Compound Interest

Simple Interest (SI) = P × R × T ÷ 100 — the same interest every year, so the amount grows in a straight line. Compound Interest: Amount = P × (1 + R/100)ᵀ, and CI = Amount − P. Each year's interest itself earns interest, so the amount curves upward and pulls away from SI:

For ₹10,000 at 10%, year 1 SI and CI are equal (₹1,000), but by year 4 CI (₹14,641) is well clear of SI (₹14,000). The high-value shortcuts:

• For 2 years, CI − SI = P × (R/100)² — a one-line classic.
• The 2-year equivalent CI rate is 2R + R²/100 (10% → 21%); the 3-year is 3R + 3R²/100 + R³/100⁴.
• Half-yearly compounding: halve the rate, double the time (10% for 1 year → 5% for 2 periods).
• Doubling: money doubles under SI in 100/R years; under CI, roughly 72/R years (the "rule of 72").

Part 3 — Speed techniques

1. Anchor profit/loss to CP = 100 — then SP is just "100 + profit%", all mental.
2. Use the successive formula for profit-then-loss and for two discounts (a + b − ab/100).
3. For 2-year CI, use 2R + R²/100 instead of expanding the bracket.
4. CI − SI (2 years) = P(R/100)² — recognise it on sight.
5. Convert the rate to a fraction (12.5% = 1/8) to simplify interest sums.
6. Match rate and time units before substituting (half-year ⇒ half the annual rate).

Part 4 — Worked examples

1. Profit%. Bought ₹400, sold ₹500. (100 ÷ 400) × 100 = 25%.

2. SP from profit%. CP ₹250, profit 18%. SP = 250 × 1.18 = ₹295.

3. Discount. MP ₹800, 15% discount. SP = 800 × 0.85 = ₹680.

4. Successive discount. Two discounts 10% and 20%. Single equivalent = 10 + 20 − (10×20÷100) = 28%.

5. Profit then loss. 20% profit then 20% loss on selling. Net = 20 − 20 − 4 = 4% loss.

6. False weight. A grocer sells at CP but uses a 900 g "kilogram". Gain% = 100 ÷ 900 × 100 = 11.11%.

7. Compound interest. CI on ₹5,000 at 10% for 2 years? 2-year rate 21% → CI = 21% of 5,000 = ₹1,050. (SI = ₹1,000; the ₹50 gap = P(R/100)².)

8. Find principal. SI for 3 years at 8% is ₹1,200. Principal? P = (1200 × 100) ÷ (8 × 3) = ₹5,000.

Part 5 — Common traps

• Profit/loss % is on CP; discount % is on MP — never the same base.
• Profit then equal loss is a net loss, not break-even.
• SI is linear, CI curves — never use the SI formula for a compound question.
• Units must match — a half-year at 10% p.a. uses R = 5% for that period.
• Successive discount ≠ sum — 10% + 20% is a 28% single discount, not 30%.

Part 6 — How to use this page

Anchor every profit/loss question to CP = 100, memorise the 2-year CI shortcuts and the successive-discount formula, re-solve the eight examples closed-book, then attempt the practice set and the timed test.

One-line revision: profit/loss % sits on CP, discount % on MP, SI grows straight while CI curves, two discounts combine as a + b − ab/100, and for 2 years CI − SI = P(R/100)².