Percentages
CSAT · Quantitative Aptitude
Introduction
Percentages are the language of CSAT arithmetic: profit, interest, data interpretation and growth questions all reduce to them. Fluency with fraction equivalents and the successive-change formula saves minutes across the paper.
Key concepts
Fraction equivalents to memorise
| % | Fraction | % | Fraction |
|---|---|---|---|
| 50 | 1/2 | 16.67 | 1/6 |
| 33.33 | 1/3 | 14.29 | 1/7 |
| 25 | 1/4 | 12.5 | 1/8 |
| 20 | 1/5 | 11.11 | 1/9 |
| 37.5 | 3/8 | 62.5 | 5/8 |
Percentage change
Change % = (new − old) ÷ old × 100. The base is always the original value. A rise from 80 to 100 is 25%, but a fall from 100 to 80 is 20%.
Successive changes
Two changes of x% and y% give a net change of x + y + xy/100 (use negative signs for decreases). A 20% rise followed by a 20% fall is a net 4% fall, not zero.
More than / less than
If A is r% more than B, then B is r/(100 + r) × 100 % less than A. If the price of a good rises by r%, consumption must fall by r/(100 + r) × 100 % to keep spending unchanged.
Population and depreciation
After n years at r% growth: P(1 + r/100)ⁿ; depreciation: P(1 − r/100)ⁿ. Different rates in different years are multiplied one after another.
Exam marks
If the pass mark is p% and a candidate scoring m fails by f marks, then p% of the maximum = m + f.
- Change % = (new − old)/old × 100
- Net of x% and y% = x + y + xy/100
- Reduction to offset a rise of r% = r/(100 + r) × 100
- Growth: P(1 + r/100)ⁿ
Solved examples
Example 1
The price of an article is first increased by 50% and then decreased by 10%. What is the net effect on the price?
(a) decrease of 35% (b) increase of 35% (c) increase of 37% (d) increase of 40%
- Net % change = x + y + xy/100.
- = (50) + (-10) + (50)(-10)/100 = 35%.
- So the net effect is an increase of 35%.
Answer: (b) increase of 35%
Example 2
The price of an article is first increased by 10% and then decreased by 30%. What is the net effect on the price?
(a) decrease of 23% (b) increase of 23% (c) decrease of 25% (d) decrease of 20%
- Net % change = x + y + xy/100.
- = (10) + (-30) + (10)(-30)/100 = -23%.
- So the net effect is a decrease of 23%.
Answer: (a) decrease of 23%
Example 3
To pass an exam a student needs 40% of the maximum marks. A student who scored 215 marks failed by 25 marks. What are the maximum marks?
(a) 600 (b) 700 (c) 480 (d) 550
- Pass mark = 215 + 25 = 240.
- 40% of maximum = 240, so maximum = 240 × 100 ÷ 40 = 600.
Answer: (a) 600
Example 4
A's salary is 100% more than B's. By what percentage is B's salary less than A's?
(a) 60% (b) 54% (c) 50% (d) 100%
- Let B = 100, so A = 200.
- B is less than A by 100 on a base of 200.
- 100/200 × 100 = 50%.
Answer: (c) 50%
Example 5
The population of a town is 40,000. It grows by 10% in the first year and 5% in the second year. What is the population after two years?
(a) 44,000 (b) 46,600 (c) 46,000 (d) 46,200
- After year 1: 40,000 × 110/100 = 44,000.
- After year 2: × 105/100 = 46,200.
Answer: (d) 46,200
Example 6
The price of sugar rises by 20%. By what percentage must a family reduce its consumption so that its expenditure on sugar does not change?
(a) 20% (b) 25% (c) 21 2/3% (d) 16 2/3%
- Expenditure = price × quantity must stay the same.
- Reduction = r/(100 + r) × 100 = 20/120 × 100 = 16 2/3%.
Answer: (d) 16 2/3%
Example 7
A's salary is 60% more than B's. By what percentage is B's salary less than A's?
(a) 37 1/2% (b) 60% (c) 41 1/2% (d) 150%
- Let B = 100, so A = 160.
- B is less than A by 60 on a base of 160.
- 60/160 × 100 = 37 1/2%.
Answer: (a) 37 1/2%
Example 8
The population of a town is 40,000. It grows by 5% in the first year and 20% in the second year. What is the population after two years?
(a) 42,000 (b) 50,400 (c) 50,800 (d) 50,000
- After year 1: 40,000 × 105/100 = 42,000.
- After year 2: × 120/100 = 50,400.
Answer: (b) 50,400
- Use fraction equivalents: 37.5% of 640 = 3/8 × 640 = 240.
- x% of y = y% of x (8% of 50 = 50% of 8 = 4).
- For a series of changes, multiply the factors: +10% then +20% = 1.1 × 1.2 = 1.32 → +32%.
- Taking the new value as the base.
- Adding successive percentages directly.
- Confusing 'percentage points' (difference of two percentages) with 'per cent' change.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. The price of an article is first increased by 30% and then decreased by 40%. What is the net effect on the price?
(a) decrease of 22% (b) decrease of 10% (c) decrease of 24% (d) increase of 22%
Q2. The population of a town is 50,000. It grows by 10% in the first year and 20% in the second year. What is the population after two years?
(a) 66,500 (b) 65,000 (c) 55,000 (d) 66,000
Q3. A's salary is 50% more than B's. By what percentage is B's salary less than A's?
(a) 100% (b) 33 1/3% (c) 37 1/3% (d) 50%
Q4. The price of sugar rises by 25%. By what percentage must a family reduce its consumption so that its expenditure on sugar does not change?
(a) 25% (b) 20% (c) 33 1/3% (d) 22%
Q5. What is 35% of 800?
(a) 280 (b) 2285 5/7 (c) 288 (d) 560
Q6. The price of an article is first increased by 50% and then decreased by 30%. What is the net effect on the price?
(a) increase of 20% (b) decrease of 5% (c) increase of 5% (d) increase of 7%
Q7. To pass an exam a student needs 40% of the maximum marks. A student who scored 208 marks failed by 32 marks. What are the maximum marks?
(a) 480 (b) 600 (c) 550 (d) 700
Q8. The price of an article is first increased by 50% and then decreased by 40%. What is the net effect on the price?
(a) increase of 10% (b) decrease of 11% (c) decrease of 10% (d) decrease of 12%
Q9. To pass an exam a student needs 33% of the maximum marks. A student who scored 66 marks failed by 33 marks. What are the maximum marks?
(a) 300 (b) 250 (c) 400 (d) 198
Q10. What is 62.5% of 1200?
(a) 1500 (b) 1920 (c) 762 (d) 750
Q11. What is 37.5% of 720?
(a) 277.2 (b) 540 (c) 270 (d) 1920
Q12. The price of an article is first decreased by 20% and then increased by 20%. What is the net effect on the price?
(a) increase of 4% (b) no change (c) decrease of 6% (d) decrease of 4%
Q1 – (a) decrease of 22%. Net % change = x + y + xy/100. = (30) + (-40) + (30)(-40)/100 = -22%. So the net effect is a decrease of 22%.
Q2 – (d) 66,000. After year 1: 50,000 × 110/100 = 55,000. After year 2: × 120/100 = 66,000.
Q3 – (b) 33 1/3%. Let B = 100, so A = 150. B is less than A by 50 on a base of 150. 50/150 × 100 = 33 1/3%.
Q4 – (b) 20%. Expenditure = price × quantity must stay the same. Reduction = r/(100 + r) × 100 = 25/125 × 100 = 20%.
Q5 – (a) 280. 35% of 800 = 800 × 35 ÷ 100 = 280.
Q6 – (c) increase of 5%. Net % change = x + y + xy/100. = (50) + (-30) + (50)(-30)/100 = 5%. So the net effect is an increase of 5%.
Q7 – (b) 600. Pass mark = 208 + 32 = 240. 40% of maximum = 240, so maximum = 240 × 100 ÷ 40 = 600.
Q8 – (c) decrease of 10%. Net % change = x + y + xy/100. = (50) + (-40) + (50)(-40)/100 = -10%. So the net effect is a decrease of 10%.
Q9 – (a) 300. Pass mark = 66 + 33 = 99. 33% of maximum = 99, so maximum = 99 × 100 ÷ 33 = 300.
Q10 – (d) 750. 62.5% of 1200 = 1200 × 62.5 ÷ 100 = 750.
Q11 – (c) 270. 37.5% of 720 = 720 × 37.5 ÷ 100 = 270.
Q12 – (d) decrease of 4%. Net % change = x + y + xy/100. = (-20) + (20) + (-20)(20)/100 = -4%. So the net effect is a decrease of 4%.
- Base = original value
- Successive: x + y + xy/100
- r% more ⇒ the other is r/(100+r) less
- Learn 1/6, 1/7, 1/8, 1/9 as %
