Uttar Pradesh Police Constable – Quantitative Aptitude
Topic: Profit & Loss
Goal: Quickly understand the core concepts, formulas, and common problem types so you can solve any Profit & Loss question in the exam.
1. Simple Explanation
| Concept | What it means |
|---|---|
| Cost Price (CP) | The price at which an item is purchased. |
| Selling Price (SP) | The price at which an item is sold. |
| Profit | SP – CP when SP > CP. |
| Loss | CP – SP when SP < CP. |
| Profit Percentage | (Profit / CP) × 100. |
| Loss Percentage | (Loss / CP) × 100. |
Key idea: Profit/Loss is always measured relative to the cost price.
2. Key Points & Formulas
| Symbol | Meaning | Formula |
|---|---|---|
| CP | Cost Price | – |
| SP | Selling Price | – |
| P | Profit amount | SP – CP |
| L | Loss amount | CP – SP |
| % Profit | Profit percent | P / CP × 100 |
| % Loss | Loss percent | L / CP × 100 |
| Selling Price for given % Profit | CP × (1 + %/100) |
|
| Selling Price for given % Loss | CP × (1 – %/100) |
|
| Cost Price for given % Loss | SP / (1 – %/100) |
|
| Profit after multiple transactions | Add profits, subtract losses | Sum of individual profits – Sum of individual losses |
| Average Price | When buying multiple items at different prices | Total Cost / Total Quantity |
Tip: Always convert percentages to decimals (e.g., 20 % → 0.20) before multiplying.
3. Worked Example (Step‑by‑Step)
Problem:
A shopkeeper bought 50 packets of biscuits at ₹38 each. He sold them at a profit of 15 %. What is the shopkeeper’s total profit and the selling price per packet?
Step 1 – Find Cost Price (CP) of one packet
- CP per packet = ₹38
- Total CP = 50 × 38 = ₹1,900
Step 2 – Calculate Profit per packet
- Profit % = 15 % → 0.15
- Profit per packet = CP × 0.15 = 38 × 0.15 = ₹5.70
Step 3 – Find Selling Price (SP) per packet
- SP = CP + Profit = 38 + 5.70 = ₹43.70
Step 4 – Total Profit for all 50 packets
- Total Profit = Profit per packet × 50 = 5.70 × 50 = ₹285
Result
- Selling price per packet: ₹43.70
- Total profit: ₹285
4. Common Problem Types
| Type | Typical Question | Quick Strategy |
|---|---|---|
| CP → SP with % profit/loss | “Buy at ₹X, sell at Y% profit.” | Use SP = CP × (1 + %/100) |
| SP → CP with % profit/loss | “Sell at ₹Y, lose 10 %.” | Use CP = SP / (1 – %/100) |
| Multiple transactions | “Bought 3 items, sold 2 items at different prices.” | Compute individual profits/losses, then sum. |
| Average price | “Buy 3 items at ₹10, ₹12, ₹14 and sell all at 20 % profit.” | Find average CP, then apply profit. |
| Profit & Loss in series | “Buy at 10 % loss, then sell at 20 % profit.” | Treat each stage separately or use combined percentage: New CP = Old CP × (1 – 0.10) then SP = New CP × (1 + 0.20) |
5. Quick Revision Summary
| Point | Description |
|---|---|
| Always relate to CP | Profit & loss percentages are taken against the cost price. |
| Profit = SP – CP | Loss = CP – SP. |
| Profit % = (Profit / CP) × 100 | Loss % = (Loss / CP) × 100. |
| Key formulas | SP = CP × (1 ± %/100) |
| Multiple stages | Compute successive CPs and SPs or add/subtract individual profits/losses. |
| Conversion | 15 % → 0.15; 10 % loss → divide by 0.90. |
| Checking | After SP calculation, verify that (SP – CP) / CP equals the stated profit percentage. |
Remember: In competitive exams, the quickest way is to convert percentages to decimals, use the direct SP or CP formulas, and then finish with a simple multiplication or division.
Practice Question (Try before you check!)
A trader bought a quantity of goods for ₹12,000. He sold them at a 12 % loss. Later he bought the same quantity again at ₹13,000 and sold them at a 10 % profit. What was his overall profit or loss?
(Use the methods above to solve.)
Good luck with your preparation!