Profit & Loss – Study Notes
(Uttar Pradesh Forest Guard – Quantitative Aptitude)
1. Simple Explanation
- Profit – The amount earned over the cost price.
- Loss – The amount lost relative to the cost price.
- Cost Price (CP) – How much the item costs to buy/produce.
- Selling Price (SP) – The price at which the item is sold.
Profit or loss is usually expressed as a percent of the cost price.
2. Key Points / Definitions
| Term | Symbol | Formula / Value | Notes |
|---|---|---|---|
| Cost Price | CP | – | Price paid to acquire the item. |
| Selling Price | SP | – | Price at which the item is sold. |
| Profit | P | SP – CP |
Positive if SP > CP. |
| Loss | L | CP – SP |
Positive if CP > SP. |
| Profit % | %P |
[(SP – CP) / CP] × 100 |
Ratio of profit to cost. |
| Loss % | %L |
[(CP – SP) / CP] × 100 |
Ratio of loss to cost. |
| Mark‑up | – | SP – CP |
Same as profit (in absolute terms). |
| Mark‑down | – | CP – SP |
Same as loss (in absolute terms). |
Quick formula cheat‑sheet
| Quantity | Formula |
|---|---|
| SP | CP + Profit |
| CP | SP – Profit |
| Profit | CP × %P / 100 |
| Loss | CP × %L / 100 |
| CP from SP & % | SP ÷ (1 + %P/100) |
| SP from CP & % | CP × (1 + %P/100) |
3. Worked Example
Problem
A shopkeeper bought 10 identical items for ₹120 each.
He sold them at a profit of 20 % on each item.
What was his total profit? What was the total selling price?
Step‑by‑step solution
-
Cost Price of one item
[ CP_{\text{one}} = ₹120 ] -
Profit per item (20 %)
[ \text{Profit}{\text{one}} = CP{\text{one}} \times \frac{20}{100} = 120 \times 0.20 = ₹24 ] -
Selling Price of one item
[ SP_{\text{one}} = CP_{\text{one}} + \text{Profit}_{\text{one}} = 120 + 24 = ₹144 ] -
Total Cost Price (10 items)
[ CP_{\text{total}} = 10 \times 120 = ₹1200 ] -
Total Profit
[ \text{Profit}_{\text{total}} = 10 \times 24 = ₹240 ] -
Total Selling Price
[ SP_{\text{total}} = 10 \times 144 = ₹1440 ]
Answer
- Total profit = ₹240
- Total selling price = ₹1440
4. Common Problem Types
| Type | Typical Question | Strategy |
|---|---|---|
| Single item | “Buy at ₹X, sell at Y, find profit/loss %.” | Use SP – CP and divide by CP. |
| Multiple items | “Buy 20 items at ₹A each, sell at ₹B each.” | Compute total CP & SP. |
| Mixed items | “Buy 5 items at ₹A each and 3 at ₹B each; sell all at ₹C each.” | Sum CPs, then compute profit/loss. |
| Cost price from profit % | “Selling price is ₹C and profit is 15 %.” | CP = SP ÷ 1.15. |
| Profit % from CP & SP | “CP = ₹150, SP = ₹180.” | (SP–CP)/CP × 100. |
| Discount problems | “Item marked ₹200, discounted 25 %.” | SP = 200 × (1 – 0.25). |
| Compound discounts | “Two successive discounts of 10 % and 20 %.” | Apply sequentially: SP = 200 × 0.9 × 0.8. |
| Break‑even point | “Selling price ₹S, variable cost ₹V, fixed cost ₹F.” | S × Q = V × Q + F → Q = F/(S–V). |
5. Tips & Tricks
-
Quick mental percentage
- 10 % of a number = divide by 10.
- 20 % = divide by 5.
- 25 % = quarter of the number.
-
Work backwards – If you know the selling price and profit % → find cost price:
[ CP = \frac{SP}{1 + \frac{%P}{100}}