Percentages – Quick Study Notes
(For Uttarakhand Police Constable – Quantitative Aptitude)
1. Simple Explanation
A percentage is a way of expressing a number as a part of 100.
- Formula:
[ \text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100% ] - “Part” can be any quantity (e.g., a subset of a group, a discount, a profit).
- “Whole” is the total quantity from which the part is taken.
2. Key Points & Quick Tips
| Topic | What to Remember | Quick Tip |
|---|---|---|
| Converting percentages to fractions / decimals | 1 % = 1/100 = 0.01 | Multiply the percentage by 0.01 |
| Finding “Part” | Part = (Percentage × Whole) ÷ 100 | Think “x% of Whole” |
| Finding “Whole” | Whole = (Part × 100) ÷ Percentage | Reverse the “Part” equation |
| Finding “Percentage” | Percentage = (Part ÷ Whole) × 100 | “x out of 100” |
| Increase / Decrease | New Value = Original ± (Percentage × Original ÷ 100) | Use the “Part” formula twice |
| Profit / Loss | Profit % = (Profit ÷ Cost) × 100 Loss % = (Loss ÷ Cost) × 100 |
Profit = Selling Price – Cost Price |
| Rate of change | % Change = ((New – Old) ÷ Old) × 100 | Positive = increase, negative = decrease |
| Compound percentages | Use successive calculations (e.g., 10% increase then 20% decrease) | Re‑apply the “New Value” formula step‑by‑step |
| Common errors | 1) Forgetting the “÷ 100” term 2) Mixing up “Part” and “Whole” |
Double‑check the order of operations |
3. Worked Example
Problem
A police station received 240 cases last month. This month it received 40% more cases than last month.
- How many cases did it receive this month?
- What is the percentage increase in total cases compared to the previous month?
Step‑by‑Step
-
Find the increase in cases
[ \text{Increase} = 40% \text{ of } 240 = \frac{40}{100}\times240 = 0.4\times240 = 96 ] -
Find the total cases this month
[ \text{Total} = 240 + 96 = 336 ] -
Verify the percentage increase
[ \text{% Increase} = \frac{96}{240}\times100 = 40% ]
Answer
- This month’s cases: 336
- Percentage increase: 40 %
4. Quick Revision Summary
| Item | Formula | Example |
|---|---|---|
| Percentage of a number | (\text{P% of } N = \dfrac{P}{100}\times N) | 25 % of 200 = 50 |
| Part given whole & percentage | ( \text{Part} = \dfrac{P}{100}\times \text{Whole}) | 30 % of 500 = 150 |
| Whole given part & percentage | ( \text{Whole} = \dfrac{\text{Part}\times100}{P}) | 50 is 20 % of whole → Whole = 250 |
| Percentage given part & whole | ( P = \dfrac{\text{Part}}{\text{Whole}}\times100) | 20/200 = 10 % |
| Increase/Decrease | New = Old ± (Old × P ÷ 100) | 120 + 15 % = 138 |
| Profit/Loss | Profit % = (Profit ÷ Cost) × 100 | (120 – 80)/80 × 100 = 50 % profit |
| Change in value | % Change = ((New – Old) ÷ Old) × 100 | (150 – 100)/100 × 100 = 50 % increase |
Quick Practice Questions (Try before checking the answers)
- A shop sold 75% of its stock of 200 items. How many items were sold?
- A vehicle’s speed increased from 60 km/h to 72 km/h. What is the percentage increase?
- If a 30 % discount is given on ₹250, what is the discount amount?
- The profit on a sale is ₹80. If the cost price was ₹400, what is the profit percentage?
(Answers after the exam: 1) 150 items, 2) 20 % increase, 3) ₹75 discount, 4) 20 % profit)
Final Tip
For every percentage problem, always write down the question in terms of the basic formula.
- Identify What (part, whole, percentage).
- Apply the correct formula.
- Check units (percent, decimal, fraction).
Good luck with your preparation!