Percentages
Per cent means per hundred. Convert to fractions, use the multiplying factor, and remember that successive changes are not simply added.
Fractions to learn as percentages
| Fraction | Per cent | Fraction | Per cent |
|---|---|---|---|
| 1/2 | 50% | 1/8 | 12½% |
| 1/3 | 33⅓% | 1/9 | 11 1/9 % |
| 1/4 | 25% | 1/10 | 10% |
| 1/5 | 20% | 1/11 | 9 1/11 % |
| 1/6 | 16⅔% | 1/12 | 8⅓% |
| 1/7 | 14 2/7 % | 3/8 | 37½% |
Formula sheet
| Situation | Formula |
|---|---|
| What per cent is a of b | a ÷ b × 100 |
| Percentage change | Change ÷ original value × 100 |
| Successive changes a% and b% | a + b + ab/100 (use a minus sign for a fall) |
| A is r% more than B; B is less than A by | r ÷ (100 + r) × 100 % |
| A is r% less than B; B is more than A by | r ÷ (100 − r) × 100 % |
| Price rises r%; cut in use to keep spending the same | r ÷ (100 + r) × 100 % |
| Population P growing r% a year for n years | P × (1 + r/100)^n |
Worked example 1: successive changes
The price of rice rises by 20% and then falls by 10%. If it was ₹50 a kg at first, what is it now, and what is the net percentage change?
- Standard method: after the rise, 50 × 1.20 = ₹60; after the fall, 60 × 0.90 = ₹54. Change = ₹4 on ₹50 = 8% rise.
- Shortcut (net change formula): a + b + ab/100 = 20 + (−10) + (20 × −10)/100 = 20 − 10 − 2 = 8% rise.
- Check: 50 × 1.08 = ₹54.
₹54 a kg; a net rise of 8%.
Worked example 2: price rise and consumption
The price of sugar rises by 25%. By what percentage must a family cut its sugar use so that its spending does not change?
- Standard method: suppose sugar cost ₹40 a kg and the family used 5 kg, spending ₹200. The new price is ₹50 a kg, so ₹200 now buys 4 kg. The cut is 1 kg out of 5 kg = 20%.
- Shortcut: cut = r ÷ (100 + r) × 100 = 25 ÷ 125 × 100 = 20%.
- Check: new price factor × new use factor = 1.25 × 0.80 = 1.00, so spending is unchanged.
20%.
Worked example 3: pass marks
A student needs 35% of the maximum marks to pass. Riya scores 210 marks and fails by 14 marks. Find the maximum marks.
- Standard method: pass mark = 210 + 14 = 224, and this is 35% of the maximum. Maximum = 224 × 100 ÷ 35 = 640.
- Shortcut (unitary): 35% ↔ 224, so 5% ↔ 224 ÷ 7 = 32, and 100% = 32 × 20 = 640.
- Check: 35% of 640 = 224, and 224 − 210 = 14.
640 marks.
Every bank, SSC, railway and state PSC paper uses percentages, both as direct sums (successive change, pass marks, elections, population growth) and inside profit and loss, interest and data interpretation. Placement tests reward knowing fraction equivalents such as 1/8 = 12½% by heart.
2 marks: “What per cent of 2 kg is 250 g?” Use the same unit: 250 ÷ 2,000 × 100 = 12.5%.
5 marks: “A's salary is 25% more than B's. By what per cent is B's salary less than A's?” r ÷ (100 + r) × 100 = 25 ÷ 125 × 100 = 20%. Check with B = ₹100: A = ₹125, and 25 is 20% of 125.
10 marks: “(a) A price rises 20% and then falls 10%: find the net change. (b) A student needs 35% to pass, scores 210 and fails by 14: find the maximum marks. (c) A town of 20,000 grows 10% a year: find its population after 2 years.” (a) 8% rise. (b) 640. (c) 20,000 × 1.1 × 1.1 = 24,200. Show the formula and one check line for each.
Two changes: a + b + ab/100. 'More than' and 'less than' are not mirror images: 25% more ↔ 20% less. Learn 1/8 = 12½% and its friends.