📊 Percentages – Study Notes
Program: Mizoram Forest Guard
Subject: Quantitative Aptitude
Topic: Percentages
Percentages are a convenient way to express a part of a whole as a fraction of 100. They are used in almost every problem involving rates, discounts, increases, taxes, and performance analysis.
1️⃣ Simple Explanation
| Concept | What it means | Quick visual |
|---|---|---|
| Percent | “Per cent” → per hundred. It is a way to say “out of 100”. | 25% = 25 out of 100 = ¼ |
| Percentage Point | A change in a percentage value. E.g., 5 % → 10 % is a 5‑percentage‑point increase. | ❌ |
| Percentage Change | How much a value has increased or decreased relative to its original value. | 10 → 12 is a 20 % increase: ((2/10)×100 = 20%) |
Formula
[
\text{Percentage} ; (p) = \frac{\text{Part}}{\text{Whole}} \times 100
]
2️⃣ Key Points to Remember
| # | Point | Why it matters |
|---|---|---|
| 1 | Convert to Decimal – For calculations, convert percentage to decimal by dividing by 100. | Easier multiplication/division. |
| 2 | Inverse Relationship – If (p%) of a number is known, the whole number can be found by (\frac{\text{Part}}{p/100}). | Quick “reverse” problems. |
| 3 | Multiple Percentages – When a number is multiplied by several percentages (e.g., discount + tax), convert each to decimal and multiply together. | Handles compound effects. |
| 4 | Percentage Increase/Decrease – Use the difference between new and old values, divided by the old value, times 100. | Standard for comparing growth. |
| 5 | Cross‑Multiplication – Useful when two percentages are proportional to two numbers. | Avoids fractions. |
| 6 | Check for “% of” vs “% Increase/Decrease” – The wording tells you which formula to use. | Prevents misinterpretation. |
| 7 | Common Tricky Cases | • “10% of 20% of 100” = 2. • “20% more than 30% of 50” = 20% of 50 = 10; 20% more → 12. |
| 8 | Use of “pp” – In some contexts, “pp” means percentage points (not percent). | Avoid confusion. |
3️⃣ Worked Example
Problem
A forest guard receives a monthly stipend of ₹12,000. He gets a 12 % increment for the next month. However, due to a national tax of 18 % on the new stipend, he only receives the net amount. What is the net stipend after tax?
Step‑by‑Step
-
Compute the increment
[ \text{Increment} = 12,% \text{ of } 12{,}000 = 0.12 \times 12{,}000 = 1{,}440 ] -
New gross stipend
[ 12{,}000 + 1{,}440 = 13{,}440 ] -
Compute tax
[ \text{Tax} = 18,% \text{ of } 13{,}440 = 0.18 \times 13{,}440 = 2{,}419.20 ] -
Net stipend
[ 13{,}440 - 2{,}419.20 = 11{,}020.80 ]
Answer: ₹11,020.80.
Quick Checklist
- Convert percentages to decimals.
- Calculate increment separately.
- Apply tax on the new gross amount.
- Subtract tax to get net.
4️⃣ Quick Revision Summary
| Topic | Key Formula / Idea | Quick Example |
|---|---|---|
| Basic % | (p% = \dfrac{p}{100}) | 25 % of 200 = (0.25 \times 200 = 50) |
| Reverse % | Whole = (\dfrac{\text{Part}}{p/100}) | 15 % of 200 = 30 → Whole = (30 ÷ 0.15 = 200) |
| % Increase | (\text{New} = \text{Old} \times (1 + p/100)) | 10 % increase on 50 → (50 \times 1.10 = 55) |
| % Decrease | (\text{New} = \text{Old} \times (1 - p/100)) | 20 % decrease on 80 → (80 \times 0.80 = 64) |
| Compound % | Multiply decimals: e.g., 10 % discount + 5 % tax → (0.90 \times 1.05 = 0.945) → 94.5 % of original. | If price ₹200 → Net = (200 × 0.945 = 189) |
| Percentage Point Change | (\Delta p = p_{\text{new}} - p_{\text{old}}) | 5 % → 8 % → Δp = 3 pp |
| Cross‑Multiplication | (\dfrac{a}{b} = \dfrac{c}{d}) → (ad = bc) | If 30 % of (x) = 15, then (x = \dfrac{15}{0.30} = 50) |
Final Tips
- Always write the problem in your own words – “12 % of ₹12,000” → “One‑tenth of the stipend plus a bit more”.
- Double‑check units – A 12 % increase is not the same as a 12 percentage‑point increase.
- Practice with real‑life data – Use forest‑guard salaries, land taxes,