5.3 Percentages
Key Takeaways
- Percent means “per hundred”: p% = p/100 = p ÷ 100 as a decimal.
- Core skill: find p% of a number using (p/100) × whole; reverse-find the whole when a percent amount is given.
- Percent increase multiplies by (1 + r); percent decrease multiplies by (1 − r); successive percents are not plain sums when applied one after another.
- Percent change = (new − old) / old × 100%; always divide by the original base named in the question.
- Practise with prices, tax language, discounts, test scores, and population-style counts—not firefighting technical percentages.
5.3 Percentages
Quick Answer: p% of N = (p/100) × N. Convert p% → decimal by dividing by 100 (15% = 0.15). For increase by r%, multiply by (1 + r/100); for decrease, multiply by (1 − r/100). Always identify the base (the “of” amount) before calculating.
Percentages dominate practical arithmetic on civil-service papers: discounts, tax-style add-ons in word problems, exam scores, and “what percent remain.” Keep every example in ordinary civilian life.
Meaning of percent
Percent means per hundred. The whole is 100%.
| Percent | Fraction | Decimal |
|---|---|---|
| 1% | 1/100 | 0.01 |
| 5% | 1/20 | 0.05 |
| 10% | 1/10 | 0.1 |
| 12.5% | 1/8 | 0.125 |
| 20% | 1/5 | 0.2 |
| 25% | 1/4 | 0.25 |
| 50% | 1/2 | 0.5 |
| 75% | 3/4 | 0.75 |
| 100% | 1 | 1 |
| 150% | 3/2 | 1.5 |
Quick mental tools.
- 10% of a number: move the decimal one place left (10% of 360 = 36).
- 5% = half of 10%; 1% = move decimal two places left; 15% = 10% + 5%.
- 25% = quarter; 50% = half; 75% = half + quarter.
Finding a percent of a number
Formula: (percent ÷ 100) × whole.
Worked example A — simple “of”
Find 15% of $240.
- 15% = 0.15
- 0.15 × 240 = 36
- So 15% of $240 is $36.
Mental path: 10% = $24; 5% = $12; total $36.
Worked example B — score interpretation
A practice quiz has 40 questions; a candidate answers 70% correctly. Correct count: 0.70 × 40 = 28. Incorrect: 40 − 28 = 12, which is 30% of 40.
Worked example C — “what percent is A of B?”
What percent is 18 of 24?
- Fraction: 18/24 = 3/4
- 3/4 = 0.75 = 75%
Formula: (part ÷ whole) × 100% = (18 ÷ 24) × 100% = 75%.
Percent increase and decrease
Increase by r%: new = original × (1 + r/100).
Decrease by r%: new = original × (1 − r/100).
You can also compute the change amount first: change = (r/100) × original; then add or subtract.
Worked example D — price increase
A bus fare of $8.00 increases by 25%.
- Increase amount: 0.25 × 8 = $2
- New fare: 8 + 2 = $10
- Or: 8 × 1.25 = $10
Worked example E — discount
A shirt marked $120 is sold at 30% off.
- Discount: 0.30 × 120 = $36
- Sale price: 120 − 36 = $84
- Or: 120 × 0.70 = $84 (paying 70% of the mark)
Trap: subtracting 30 from 120 to get 90 confuses percentage points with dollars.
Worked example F — percent change between two numbers
A weekly grocery spend rises from $200 to $250.
- Change: 250 − 200 = 50
- Percent increase: (50 ÷ 200) × 100% = 25%
Trap: dividing by the new amount 250 → 20% uses the wrong base. Unless the stem says otherwise, percent change uses the original value.
Worked example G — decrease then check reverse
A laptop falls from $4,000 to $3,400.
- Drop: 600
- Percent decrease: (600 ÷ 4,000) × 100% = 15%
Reverse check: 4,000 × 0.85 = 3,400. Good.
Reverse percentage problems
Sometimes you know the result after a percent change and must recover the original.
Worked example H — original price after discount
After a 20% discount, a blender costs $160. What was the original price?
- Sale price is 80% of original: 0.80 × original = 160
- Original = 160 ÷ 0.80 = $200
Trap: increasing $160 by 20% → 160 × 1.20 = $192, which is not the reverse of a 20% discount. You must divide by 0.80, not multiply by 1.20.
Worked example I — finding the whole from a part
12 students are 15% of a club. How many students are in the club?
- 0.15 × whole = 12
- whole = 12 ÷ 0.15 = 80
Successive percentages
Applying 10% then another 10% is not the same as 20% once, when each percent acts on a new base.
Worked example J — successive discounts
Price $100. First 10% off → $90. Then another 10% off the new price → 0.10 × 90 = $9; pay $81. Single 20% off $100 would be $80. Successive 10% + 10% → 19% total effective discount from original, not 20%.
Worked example K — increase then decrease
Value $200. Up 10% → $220. Then down 10%: 0.10 × 220 = $22; end at $198, not back to $200. Order and changing bases matter.
Expressing one quantity as a percent of another (comparison)
Example. District A has 45 community volunteers; District B has 60. A is what percent of B?
(45 ÷ 60) × 100% = 75%.
B is what percent of A? (60 ÷ 45) × 100% = 133 1/3% (or about 133.3%).
Read “A as a percent of B” as (A/B) × 100%, not the reverse.
Percent of percent and “percentage points”
If support rises from 40% to 50%, that is a rise of 10 percentage points, but the relative percent increase in the rate is (10 ÷ 40) × 100% = 25%. Exam stems usually ask one or the other clearly—match the wording.
Multi-step civilian budget example
A family allocates monthly income of $8,000 as follows: 25% rent, 15% transport, 10% savings; the rest is other expenses.
- Rent: 0.25 × 8,000 = $2,000
- Transport: 0.15 × 8,000 = $1,200
- Savings: 0.10 × 8,000 = $800
- Sum of these three: 2,000 + 1,200 + 800 = $4,000 → 50%
- Other expenses: 100% − 50% = 50% → $4,000
If next month income rises 5%, new income = 8,000 × 1.05 = $8,400. If rent stays fixed at $2,000, rent is then (2,000 ÷ 8,400) × 100% ≈ 23.8% of income—not still exactly 25% unless the stem keeps percentages fixed.
Common trap table (percentages)
| Trap | Example of error | Correct idea |
|---|---|---|
| Wrong base | % change using new value as denominator | Divide change by original |
| % as raw subtract | 20% off $80 → $60 by doing 80 − 20 | 20% of 80 is 16; pay $64 |
| Reverse discount error | Add % back instead of dividing by (1 − r) | Sale = (1 − r) × original |
| Successive = sum | Two 10% discounts = 20% | Compound on new base |
| Of vs more than | “20% more than 50” as 20% of 50 only | 50 × 1.20 = 60 |
| Mixing fraction forms | 0.5% treated as 50% | 0.5% = 0.005 |
| 100% + r% written wrong | Increase 100 by 10% → 110% of original = 110, not +10 only if original ≠ 100 | Scale carefully |
Exam strategy for percent MCQs
- Underline of what (the base).
- Convert the percent to a decimal or easy fraction.
- Estimate: 19% of 400 is about 0.2 × 400 = 80; exact 76—options near 800 or 8 are place-value disasters.
- For increase/decrease language, decide whether the question wants new amount, change amount, or percent change.
- Reject any solution path that needs firefighting pump or pressure formulas—this subject is general Mathematics.
Section checkpoint
You can convert percent forms, compute p% of N, reverse to find wholes, handle increase/decrease, and avoid base and successive-percent traps. The final foundations section links these skills to ratios, rates, and proportions—another way of writing and solving “parts of a whole” relationships.
What is 20% of 350?
A phone priced at $500 is reduced by 15%. What is the sale price?
After a 25% discount, a fan costs $180. What was the original price?
A savings balance rises from $800 to $920. What is the percent increase?