Profit & Loss – Quick Study Notes
(Chandigarh Police Sub‑Inspector – Quantitative Aptitude)
1. Simple Explanation
- Profit = Selling Price (SP) – Cost Price (CP).
- Loss = Cost Price (CP) – Selling Price (SP).
- Percentage profit/loss is calculated on the Cost Price.
Why CP is the base?
The profit or loss is the extra amount added to or deducted from the original investment (CP).
2. Key Points to Remember
| Concept | Formula | Notes |
|---|---|---|
| Profit/Loss | ( \text{Profit} = SP - CP ) ( \text{Loss} = CP - SP ) |
Positive value → Profit, Negative → Loss |
| Profit/Loss % | ( \text{Profit%} = \frac{SP-CP}{CP}\times 100% ) ( \text{Loss%} = \frac{CP-SP}{CP}\times 100% ) |
Always divide by CP. |
| Selling Price from % | ( SP = CP \times \left(1 \pm \frac{\text{%}}{100}\right) ) | “+” for profit, “–” for loss. |
| Cost Price from % | ( CP = SP \times \left(1 \pm \frac{\text{%}}{100}\right)^{-1} ) | “+” for profit, “–” for loss. |
| Multiple Transactions | Add all CPs → Total CP; add all SPs → Total SP; then use above formulas. | Treat each transaction separately before combining. |
| Common Mistake | Using selling price as the base for percentage calculations. | ALWAYS use CP. |
| Quick Check | If SP > CP → Profit; if SP < CP → Loss. | Simple sanity check before calculating % |
3. Worked Example
Problem
A shopkeeper buys 100 units of a product at ₹12 each.
He sells them at a 25 % profit.
How many units must he sell at a 15 % loss on the same cost to break even?
Step‑by‑Step Solution
-
Compute CP and SP for the first part
- CP of 100 units = (100 \times 12 = ₹1,200).
- Profit % = 25 % → (SP = CP \times (1 + 0.25) = 1,200 \times 1.25 = ₹1,500).
- Profit = (SP - CP = 1,500 - 1,200 = ₹300).
-
Determine break‑even condition
- Break even means total profit = total loss.
- Let (x) be the number of units sold at a 15 % loss.
-
Compute loss per unit
- Loss % = 15 % → Loss per unit = (12 \times 0.15 = ₹1.80).
- Total loss from (x) units = (1.80x).
-
Set total profit = total loss
[ 300 = 1.80x \implies x = \frac{300}{1.80} = 166.\overline{6} ] -
Round up to whole units
(x = 167) units (since partial units cannot be sold).
Answer: The shopkeeper must sell 167 units at a 15 % loss to break even.
Verification
- Loss from 167 units = (1.80 \times 167 = ₹300.60).
- Profit from 100 units = ₹300.
- Net loss ≈ ₹0.60 → practically zero (difference due to rounding).
4. Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Profit | (SP = CP(1 + \frac{P}{100})) | Profit % added to CP |
| Loss | (SP = CP(1 - \frac{L}{100})) | Loss % subtracted from CP |
| Profit % | (\frac{SP-CP}{CP}\times100%) | Always divide by CP |
| Loss % | (\frac{CP-SP}{CP}\times100%) | Same base |
| Multiple sales | Sum CPs → Total CP; Sum SPs → Total SP | Then apply profit/loss % |
| Break‑even | Total profit = Total loss | Solve for unknown units or price |
Practice Tip:
Always write down CP and SP first. If a percentage is given, convert it to a decimal (e.g., 25 % → 0.25) before multiplying. This reduces errors and speeds up calculation—essential for the SI exam.