Profit & Loss – Study Notes (UGC NET – Quantitative Aptitude)
1. Simple Explanation
- Cost Price (CP) – price at which an item is bought.
- Selling Price (SP) – price at which an item is sold.
- Profit – (SP > CP).
- Loss – (SP < CP).
The profit or loss is measured as a percentage of the cost price (or sometimes of the selling price).
2. Key Points & Formulas
| Concept | Formula | Notes |
|---|---|---|
| Profit / Loss amount | ( \text{Profit} = SP - CP ) ( \text{Loss} = CP - SP ) |
Positive value → profit, negative → loss |
| Profit % | ( \displaystyle %P = \frac{SP-CP}{CP}\times 100 ) | Often called “margin of profit” |
| Loss % | ( \displaystyle %L = \frac{CP-SP}{CP}\times 100 ) | |
| Selling Price in terms of CP & % | ( SP = CP\left(1 + \frac{%P}{100}\right) ) ( SP = CP\left(1 - \frac{%L}{100}\right) ) |
|
| Cost Price in terms of SP & % | ( CP = \frac{SP}{1 + \frac{%P}{100}} ) ( CP = \frac{SP}{1 - \frac{%L}{100}} ) |
|
| Profit / Loss for k items | ( \text{Total Profit} = k \times \text{Profit per item} ) | |
| Weighted average for multiple items | If items 1…n have CPs (C_i) and sold at SPs (S_i) then total profit = (\sum(S_i-C_i)). | |
| Inverse relation | For a fixed profit (P) with CP = (C): ( SP = C + P ). Hence ( %P = \frac{P}{C}\times 100). | |
| Discount & Profit | If an item is sold at discounted price (D) from its list price (L): ( D = L(1-d/100) ). Profit depends on CP. |
|
| Break‑even point | CP = SP (i.e., 0% profit/loss). |
Common Pitfalls
| Mistake | Correct Approach |
|---|---|
| Using SP instead of CP in the denominator of %P or %L | Always use CP as denominator. |
| Adding percentage values directly (e.g., 20% + 30% = 50%) | Percentages are not additive unless they refer to the same base. |
| Assuming profit % = loss % for equal absolute amounts | Profit % and loss % are relative to CP, not absolute. |
3. Worked Example
Question
100 items are sold at a profit of 10% each.
Then 120 items are sold at a loss of 15% each.
What is the net profit or loss percentage on the entire batch?
Step‑by‑Step Solution
-
Choose a convenient unit for CP
Let the CP of each item be ₹100 (any other value works; the ratio stays same). -
Compute SP for each type
| Item | CP (₹) | Profit/Loss | Percent | SP (₹) |
|---|---|---|---|---|
| A – 100 items | 100 | +10% | 10 | (100 \times 1.10 = 110) |
| B – 120 items | 100 | –15% | 15 | (100 \times 0.85 = 85) |
- Total CP & Total SP
- Total CP = ( (100+120) \times 100 = 220 \times 100 = ₹22{,}000)
- Total SP = ( 100 \times 110 + 120 \times 85 = 11{,}000 + 10{,}200 = ₹21{,}200)
- Net Profit / Loss
- Net Profit = Total SP – Total CP = (21{,}200 - 22{,}000 = -₹800)
(negative → loss)
- Net Loss %
[ %L = \frac{800}{22{,}000} \times 100 = 3.636% \approx 3.64% ]
Answer: The entire batch incurs a loss of about 3.64 %.
4. Quick Revision Summary
- Profit/Loss = (SP - CP).
- Profit % = ((SP-CP)/CP \times 100).
- Loss % = ((CP-SP)/CP \times 100).
- SP can be expressed as (CP(1 \pm %/100)).
- For multiple items, total profit = sum of individual profits.
- Always use CP as the base for percentages.
- Break‑even when SP = CP (0 % profit/loss).
- Remember: 20% profit on ₹100 = ₹20; 20% loss on ₹100 = ₹20; but 20% profit on ₹200 = ₹40, 20% loss on ₹200 = ₹40 – the percentages are relative to the same base.
Keep practicing with mixed‑type problems (profit on some items, loss on others) and check your answer by verifying that total CP > total SP for a loss, or vice‑versa. Happy studying!