📌 Percentages – Quick Study Guide
Program: Punjab Police Constable
Subject: Quantitative Aptitude
Topic: Percentages
1️⃣ Simple Explanation
Percentage = part of a whole expressed in hundredths
1% = 1 part out of 100 parts.
When you have a number X and you want p% of X, you multiply X × p ÷ 100.
When you know the part P and the whole H, the percentage is (P ÷ H) × 100.
2️⃣ Key Points
| Concept | What to remember | Quick check |
|---|---|---|
| Percent vs. Fraction | 1% = 1/100. | 25% = 25/100 = 1/4 |
| Conversion | % → decimal → multiply by 100. | 12.5% → 0.125 |
| Finding percentage of a number | value = number × (percentage ÷ 100) |
30% of 200 = 200 × 0.30 = 60 |
| Finding number from its percentage | number = value ÷ (percentage ÷ 100) |
60 is 30% of what? → 60 ÷ 0.30 = 200 |
| Percentage change | change = (new – old) ÷ old × 100 |
80 → 100: (20 ÷ 80) × 100 = 25% ↑ |
| Percentage increase/decrease | Add/subtract percent of the base. | 100 + 20% = 100 + (20/100 × 100) = 120 |
| Profit / Loss | Profit% = (Profit ÷ Cost Price) × 100 | Loss% = (Loss ÷ Cost Price) × 100 |
| Discount | Discount% = (Discount ÷ Original Price) × 100 | Sale Price = Original Price – Discount |
| Markup | Markup% = (Markup ÷ Cost Price) × 100 | Selling Price = Cost Price + Markup |
| Real‑World | Always keep units consistent (₹, km, kg, etc.) | Check if “percent of” refers to a whole or a part |
3️⃣ Essential Formulas
| Scenario | Formula | Example |
|---|---|---|
| Percent of a number | P = N × (p ÷ 100) |
25% of 80 → 80 × 0.25 = 20 |
| Number from its percent | N = P ÷ (p ÷ 100) |
20 is 25% of what? → 20 ÷ 0.25 = 80 |
| Percentage change | Δ% = ((New – Old) ÷ Old) × 100 |
75 → 90: ((90‑75)/75)×100 = 20% ↑ |
| Profit% | Profit% = (Profit ÷ Cost Price) × 100 |
Cost = 200, Profit = 50 → 25% |
| Loss% | Loss% = (Loss ÷ Cost Price) × 100 |
Cost = 200, Loss = 30 → 15% |
| Discount% | Discount% = (Discount ÷ Original Price) × 100 |
Original = 500, Discount = 80 → 16% |
| Markup% | Markup% = (Markup ÷ Cost Price) × 100 |
Cost = 120, Markup = 30 → 25% |
| Effective rate after successive changes | Final = Base × (1 ± r1/100) × (1 ± r2/100)… |
100 → +10% → -20% → 100×1.10×0.80 = 88 |
4️⃣ Worked Example (Full Problem)
Problem:
A shopkeeper bought a batch of 50 identical items for ₹6,000. He sold 30 items for ₹200 each. The remaining 20 items were sold later at a 10% profit on the cost price.
What is the overall profit percentage?
Solution Steps
-
Cost Price (CP) of all items
[ CP = ₹6,000 ] -
Selling Price (SP) of first 30 items
[ SP_1 = 30 \times 200 = ₹6,000 ] -
Cost per item
[ \text{Cost per item} = \frac{6000}{50} = ₹120 ] -
Selling price of remaining 20 items at 10% profit
- Profit on one item = (120 \times 0.10 = ₹12)
- SP per item = (120 + 12 = ₹132)
- Total SP for 20 items = (20 \times 132 = ₹2,640)
-
Total Selling Price
[ SP_{\text{total}} = 6000 + 2640 = ₹9,640 ] -
Total Profit
[ \text{Profit} = SP_{\text{total}} - CP = 9640 - 6000 = ₹3,640 ] -
Overall Profit %
[ \text{Profit%} = \frac{3640}{6000} \times 100 \approx 60.67% ]
Answer: The shopkeeper’s overall profit is ≈ 60.7 %.
5️⃣ Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Percent of a number | P = N × p/100 |
Use 0.01 × p to convert percent to decimal. |
| Number from its percent | N = P × 100/p |
Multiply by 100 first, then divide. |
| Percentage change | Δ% = ((New – Old)/Old) × 100 |
Positive = increase, negative = decrease |