📚 UPSC EPFO – Quantitative Aptitude
Topic: Percentages
Percent = “Per cent” → per 100.
1 % = 1/100 = 0.01
1. Simple Explanation
| Concept | Symbol | Meaning |
|---|---|---|
| Percentage of a number | (p%) of (x) | (\displaystyle \frac{p}{100}\times x) |
| Percent change | (\frac{\text{New} - \text{Old}}{\text{Old}}\times100% ) | Increase or decrease relative to the old value |
| Percent of a percent | (a%) of (b%) | (\displaystyle \frac{a}{100}\times \frac{b}{100}= \frac{ab}{10,000}) |
| Complement of a percent | (100% - p%) | The remaining part |
Rule of thumb: Always convert the percent to a decimal (divide by 100) before multiplying.
2. Key Points to Remember
| # | Point | Why it matters |
|---|---|---|
| 1 | Conversion – p% = p/100 |
Avoids mistakes in calculation |
| 2 | Percentage of a number – p% of x = (p/100)x |
Most common type in exams |
| 3 | Percent increase/decrease – ((New-Old)/Old)*100 |
Required for growth/decline questions |
| 4 | Percentage of a percent – a% of b% = (a×b)/10,000 |
Often used in compound interest or nested percentages |
| 5 | Percent of a total – Part/Total×100 |
Used in ratio, distribution, and statistical problems |
| 6 | Working backwards – When “percentage” is given and we need the original value (x), use x = Part ÷ (p/100) |
Helps solve “what was the original amount” questions |
| 7 | Percentage and ratio interchange | 20 % = 1:5, 50 % = 1:1, 75 % = 3:1 etc. Useful for quick mental math |
| 8 | Common pitfalls | Forgetting to divide by 100; mixing up increase vs. decrease; rounding errors in multi‑step problems |
3. Worked Example
Question (UPSC‑style)
In 2023, the EPFO reported that the total number of members increased from 2,800,000 to 3,080,000.
(a) Calculate the percentage increase in membership.
(b) If the increase itself was 10 % of the 2022 membership, verify the 2023 figure.
Step‑by‑Step Solution
(a) Percentage Increase
-
Old value (O) = 2,800,000
New value (N) = 3,080,000 -
Compute the difference:
[ D = N - O = 3,080,000 - 2,800,000 = 280,000 ] -
Percentage increase:
[ \text{Percent} = \frac{D}{O}\times100 = \frac{280,000}{2,800,000}\times100 = 0.10\times100 = 10% ]
Answer (a): 10 % increase.
(b) Verify with 10 % of 2022 Membership
-
10 % of 2022 membership (old value) =
[ 0.10 \times 2,800,000 = 280,000 ] -
Add this increase to the old value:
[ 2,800,000 + 280,000 = 3,080,000 ] -
The computed 2023 figure matches the given value. ✔️
Answer (b): Verified – the 2023 figure is indeed 10 % higher than 2022.
Quick Mental‑Math Tips
- 10 % of any number = shift decimal one place left.
- 20 % = half of 10 % (i.e., divide by 5).
- 25 % = quarter of the number.
- 50 % = half of the number.
- 75 % = 3⁄4 of the number (or 100 % – 25 %).
These shortcuts help in speed rounds.
4. Quick Revision Summary
| Concept | Formula | Example |
|---|---|---|
| Percentage of a number | (\frac{p}{100}\times x) | (25%) of 200 = 50 |
| Percent change | (\frac{\text{New} - \text{Old}}{\text{Old}}\times100) | 3,080k vs 2,800k → 10 % |
| Percent of a percent | (\frac{a}{100}\times\frac{b}{100}) | (20%) of (30%) = 0.06 = 6 % |
| Complement | (100% - p%) | 30 % complement = 70 % |
| Original value from part | (x = \frac{\text{Part}}{p/100}) | 150 is 15 % of (x) → (x = 150/0.15 = 1,000) |
Remember:
- Always convert to decimal first.
- Check units – if the problem asks for “percentage” vs “percent change”.
- Cross‑check by reversing the calculation (e.g., verify that adding 10 % to the old value gives the new value).
Use this cheat‑sheet to tackle any percentage‑related query in the UPSC EPFO exam—whether it’s a straightforward “20 % of 500” or a multi‑step growth‑rate problem. Happy studying