Profit & Loss – Quick Study Notes
Program: Goa Police Sub‑Inspector (SI)
Subject: Quantitative Aptitude
1. Simple Explanation
Profit (or loss) occurs when the selling price (SP) differs from the cost price (CP) of an item.
- Profit = SP – CP (positive value)
- Loss = CP – SP (positive value)
The profit or loss rate is expressed as a percentage of the cost price.
[ \text{Profit %} = \frac{\text{Profit}}{\text{CP}} \times 100% ] [ \text{Loss %} = \frac{\text{Loss}}{\text{CP}} \times 100% ]
2. Key Points (Quick Reference)
| Concept | Formula | Note |
|---|---|---|
| Profit/Loss Amount | (P = SP - CP) | Positive ⇨ profit, negative ⇨ loss |
| Profit/Loss % | (\frac{P}{CP} \times 100%) | Always use CP as denominator |
| Selling Price | (SP = CP + P) | Or (SP = CP \times \left(1 + \frac{P%}{100}\right)) |
| Cost Price | (CP = SP - P) | Or (CP = SP \times \left(1 - \frac{L%}{100}\right)) |
| Discount (if any) | (Discount = \frac{D% \times SP}{100}) | Reduce SP before selling |
| Multiple items | Treat per unit, then total | Avoid mixing units |
| Profit after discount | (SP_{\text{after}} = SP \times \left(1 - \frac{D%}{100}\right)) | Then compute profit |
| Reverse problems | Given any two of {CP, SP, P, P%} | Find the third |
| Percentage of increase/decrease | (\frac{\text{New} - \text{Old}}{\text{Old}} \times 100%) | Same as profit/loss % if Old = CP |
3. Common Problem Types
| Problem Type | What to look for | Typical trick |
|---|---|---|
| Simple Profit/Loss | CP & SP given | Compute difference |
| Profit/Loss % | CP and % given | Convert % to decimal |
| Discount + Profit | CP, discount %, profit % | Apply discount before profit |
| Multiple items | Quantity & SP/CP per unit | Multiply by quantity |
| Average cost | Two batches with different CP | Weighted average |
| Profit after tax / commission | Extra deductions | Subtract from SP before profit |
| Rate of return | Income/Investment | Treat as profit on investment |
4. Worked Example
Problem Statement
A shopkeeper bought 100 shirts at a cost price of ₹1,200 each.
He sold them at a discount of 20 % on the marked price.
The shopkeeper made a total profit of 12 % on the whole batch.
Find the marked price (MP) of each shirt.
Step‑by‑Step
| Step | Calculation | Reason |
|---|---|---|
| 1 | Cost price per shirt (CP) = ₹1,200 | Given |
| 2 | Total CP for 100 shirts = 100 × ₹1,200 = ₹120,000 | Total cost |
| 3 | Total profit = 12 % of total CP = 0.12 × ₹120,000 = ₹14,400 | Profit % |
| 4 | Total selling price (SP) = Total CP + Total profit = ₹120,000 + ₹14,400 = ₹134,400 | SP = CP + Profit |
| 5 | Discount amount per shirt = 20 % of MP = 0.20 × MP | Discount definition |
| 6 | Selling price per shirt = MP – 0.20 MP = 0.80 MP | Apply discount |
| 7 | Total SP = 100 × (0.80 MP) = 80 MP | Because 0.80 MP per shirt |
| 8 | Set equal to total SP found: 80 MP = ₹134,400 | Solve for MP |
| 9 | MP = ₹134,400 ÷ 80 = ₹1,680 | Marked price per shirt |
| 10 | Check: 20 % discount → 0.80 × ₹1,680 = ₹1,344 (SP per shirt) | 100 × ₹1,344 = ₹134,400 |
| 11 | Profit per shirt = ₹1,344 – ₹1,200 = ₹144 | 12 % of ₹1,200 = ₹144 |
Answer: The marked price of each shirt is ₹1,680.
5. Quick Revision Summary
| Topic | Formula | Example |
|---|---|---|
| Profit/Loss Amount | (P = SP - CP) | SP = ₹1,344, CP = ₹1,200 → P = ₹144 |
| Profit % | (\frac{P}{CP} \times 100%) | (144/1200 × 100 = 12%) |
| Selling Price after Discount | (SP = MP × (1 - D%)) | MP = ₹1,680, D = 20 % → SP = ₹1,344 |
| Total Profit | Total = Total CP × Profit % | ₹120,000 × 12 % = ₹14,400 |
| Average Cost | (\frac{Total,CP}{Total,Quantity}) | Use weighted average if batches differ |
| Reverse Problem | Given any two of {CP, SP, P, P%} ⇒ find third | Solve linear equations |
Tips for the Exam
- Read carefully – identify whether the percentage is of CP or SP.
- Use decimals early to avoid rounding errors.
- Check units – always keep track of per‑unit vs total values.
- Work backwards in reverse problems – easier