Arithmetic — Percentage, Profit & Loss, Interest — IBPS Clerk / SBI Clerk

Percentage, Profit & Loss and Interest are taught together because they are, structurally, the same relationship applied to three different contexts. Profit and loss compare a selling price to a cost price; discount compares a selling price to a marked price; simple and compound interest compare a final amount to a principal — in every case, the percentage is computed relative to the ORIGINAL, unchanged value, never the final one.

1. Profit and loss

Profit percentage is the profit (selling price minus cost price) divided by the COST price, times 100 — never divided by the selling price. A product bought for ₹1,250 and sold for ₹1,500 earns a profit of ₹250, giving $\frac{250}{1250}\times 100=20\%$ profit.

$\text{Profit \%}=\frac{\text{SP}-\text{CP}}{\text{CP}}\times 100,\text{Loss \%}=\frac{\text{CP}-\text{SP}}{\text{CP}}\times 100$

2. Marked price and discount

A discount is calculated on the MARKED price, not the cost price — the selling price after a discount equals the marked price multiplied by (1 − discount rate). A ₹1,200 marked-price item with a 15% discount sells for $1200\times 0.85=₹1,020$; the seller's actual profit or loss then depends separately on how this selling price compares to the cost price, a distinct calculation.

3. Recovering cost price from a known loss

Given a selling price and a loss percentage, the cost price is recovered by dividing the selling price by (1 − loss rate), the same reverse-percentage logic used throughout this cluster. A ₹1,350 selling price at a 10% loss means the cost price was $\frac{1350}{1-0.10}=₹1,500$.

4. Successive percentage changes are not simply additive

Two successive percentage changes — a decrease followed by an increase, or vice versa — combine by MULTIPLYING their factors, and the result is frequently NOT what a naive addition would suggest. A 20% decrease followed by a 25% increase combines as $0.80\times 1.25=1.00$ — exactly the ORIGINAL value, a net change of 0%, even though a candidate expecting simple addition might guess +5% or −5%.

5. Simple interest

Simple interest grows by a FIXED amount every period, calculated once on the original principal and never on any interest already earned.

$\text{SI}=\frac{P\times R\times T}{100}$

₹8,000 at 10% per annum for 3 years earns $\text{SI}=\frac{8000\times 10\times 3}{100}=₹2,400$, regardless of which year is being considered — the same ₹800 accrues every single year.

6. Compound interest

Compound interest grows on the principal PLUS all previously accumulated interest, compounding each period rather than staying fixed.

$\text{CI}=P\left[{\left(1+\frac{R}{100}\right)}^T-1\right]$

The same ₹8,000 at 10% per annum for 2 years earns $\text{CI}=8000\left[(1.10{)}^2-1\right]=8000\times 0.21=₹1,680$ — more than the equivalent 2-year simple interest of ₹1,600, because the second year's interest is calculated on ₹8,800, not ₹8,000.

7. The CI−SI shortcut for exactly 2 years

For exactly 2 years, the difference between compound and simple interest equals the principal multiplied by the square of the rate (as a decimal) — a direct shortcut that avoids computing both values separately.

$\text{CI}-\text{SI (2 years)}=P\times {\left(\frac{R}{100}\right)}^2$

For the same ₹8,000 at 10%: $8000\times (0.10{)}^2=8000\times 0.01=₹80$ — matching the difference computed the long way (₹1,680 − ₹1,600 = ₹80).

Worked Examples

Example 1. A trader buys an item for ₹1,250 and sells it for ₹1,500. Find the profit percentage.

Profit $=1500-1250=250$. Profit % $=\frac{250}{1250}\times 100=20\%$.

Answer: 20%.

Example 2. An item marked at ₹1,200 is sold after a 15% discount. Find the selling price.

$1200\times (1-0.15)=1200\times 0.85=₹1,020$.

Answer: ₹1,020.

Example 3. An item is sold for ₹1,350 at a loss of 10%. Find the cost price.

$\text{CP}=\frac{1350}{1-0.10}=\frac{1350}{0.90}=₹1,500$.

Answer: ₹1,500.

Example 4. A price is decreased by 20% and then increased by 25%. Find the net percentage change from the original price.

Net multiplier $=0.80\times 1.25=1.00$ — exactly the original value.

Answer: 0% net change (not +5% or −5% as a naive addition might suggest).

Example 5. Find the simple interest on ₹8,000 at 10% per annum for 3 years.

$\text{SI}=\frac{8000\times 10\times 3}{100}=₹2,400$.

Answer: ₹2,400.

Example 6. Find the compound interest on ₹8,000 at 10% per annum for 2 years, compounded annually.

$\text{CI}=8000\left[(1.10{)}^2-1\right]=8000\times 0.21=₹1,680$.

Answer: ₹1,680.

Example 7. Find the difference between compound and simple interest on ₹8,000 at 10% per annum for 2 years, using the direct shortcut.

$\text{CI}-\text{SI}=8000\times (0.10{)}^2=8000\times 0.01=₹80$.

Answer: ₹80 (matches: ₹1,680 CI − ₹1,600 SI = ₹80).

Summary

Profit/loss percentage always divides by the COST price; discount is always calculated on the MARKED price; both are distinct calculations even when they appear in the same question.

Recovering a cost price from a known selling price and loss percentage uses reverse-percentage logic: divide by (1 − loss rate), never subtract the loss directly.

Successive percentage changes multiply their factors rather than adding — a 20% decrease followed by a 25% increase returns exactly to the original value, illustrating why the two changes must never be simply added.

Simple interest accrues a fixed amount every period on the original principal; compound interest accrues on the growing principal-plus-interest each period. For exactly 2 years, CI − SI = P × (R/100)² is a direct shortcut that avoids computing both values from scratch.