Percentages for SSC CHSL

Percentage is the most useful single topic in SSC CHSL maths. It appears as questions of its own in both tiers, and it hides inside profit and loss, discount, interest, mixtures and every data interpretation set. Get fast here and you get faster everywhere else.

The habit that makes the difference: stop treating a percentage as "divide by 100" and start treating it as a fraction and a multiplier. Nearly every shortcut in this guide is one of those two ideas in disguise.

A percentage is a fraction

"Per cent" means "per hundred", so r% is simply r/100. That has two quick consequences.

• Fraction equivalents turn percentages into division. 16⅔% of 360 is 360 ÷ 6 = 60. No multiplication by 16.67 needed.
• x% of y = y% of x. Both equal xy/100. So 8% of 25 is the same as 25% of 8, which is 2.

Learn multiples too: 37½% = 3/8, 62½% = 5/8, 87½% = 7/8, 66⅔% = 2/3, 83⅓% = 5/6. Once these are automatic, a question like "37½% of 480" is just 480 × 3/8 = 180.

Percentage change and the multiplier

Percentage change = change ÷ original × 100. The base is always the original value, the one you started from.

The multiplier turns every change into a single multiplication:

• An increase of r% means multiply by (1 + r/100). A 15% rise is × 1.15.
• A decrease of r% means multiply by (1 − r/100). A 20% fall is × 0.8.

Why it helps: you never compute the change separately and then add it. A salary of ₹24,000 rising by 15% is 24,000 × 1.15 = ₹27,600 in one step. It also works backwards: if a value after a 20% rise is 600, the original was 600 ÷ 1.2 = 500.

Successive changes

A change of a% followed by b% gives a net change of a + b + ab/100. Use negative values for decreases.

Why: the two multipliers are (1 + a/100) and (1 + b/100). Multiply them and you get 1 + (a + b)/100 + ab/10,000. Written as a percentage, that is a + b + ab/100. The ab/100 term is the "change on the change".

• +10% then +20%: 10 + 20 + 200/100 = +32%.
• −20% then +25%: −20 + 25 − 500/100 = 0. The value is back where it started.
• +20% length and −10% breadth of a rectangle: 20 − 10 − 2 = +8% in area.

More than, less than

If A is r% more than B, then B is r/(100 + r) × 100% less than A.
If A is r% less than B, then B is r/(100 − r) × 100% more than A.

Why: the base changes. "A is 25% more than B" uses B as the base, so take B = 100 and A = 125. Now "B is less than A by how much?" uses A as the base: 25 out of 125, which is 20%.

Price and consumption

If the price of an item rises by r%, a household must cut consumption by r/(100 + r) × 100% to keep spending the same. If the price falls by r%, it can raise consumption by r/(100 − r) × 100%.

Why: expenditure = price × quantity. If spending is fixed and price becomes 1.2 times, quantity must become 1/1.2 = 5/6 of what it was, a cut of 1/6, or 16⅔%. This is the same reversal as "more than, less than", applied to a product.

Six worked questions

Q1. A's income is 25% more than B's. By what percentage is B's income less than A's?
Take B = 100, so A = 125. B is 25 less on a base of 125: 25/125 × 100 = 20%.

Q2. The price of sugar rises by 20%. By how much must a family reduce consumption so that its spending on sugar does not change?
20/120 × 100 = 16⅔%.

Q3. A person spends 75% of their income. Income rises by 20% and expenditure by 10%. By what percentage do savings rise?
Take income = 100: expenditure 75, savings 25. New income 120, new expenditure 82.5, new savings 37.5. The rise is 12.5 on 25, which is 50%.

Q4. A student needs 36% to pass. They score 190 marks and fail by 26 marks. Find the maximum marks.
The pass mark is 190 + 26 = 216, and that is 36% of the total. Total = 216 ÷ 0.36 = 600.

Q5. In an election between two candidates, 10% of the votes cast were invalid. The winner got 60% of the valid votes and won by 1,800 votes. How many votes were cast?
The loser got 40% of valid votes, so the margin is 20% of valid votes = 1,800. Valid votes = 9,000. These are 90% of all votes, so the total is 9,000 ÷ 0.9 = 10,000.

Q6. A town of 8,000 people grows by 10% in the first year and falls by 5% in the second. Find the population after two years.
8,000 × 1.1 × 0.95 = 8,360. Check with the formula: 10 − 5 − 0.5 = 4.5%, and 4.5% of 8,000 is 360.

Common mistakes

Practice

1. Find 37½% of 480.
2. A number is decreased by 20% and then increased by 25%. What is the net change?
3. A's salary is 25% less than B's. By what percentage is B's salary more than A's?
4. A student scoring 40% fails by 12 marks. Another scoring 45% gets 18 marks more than the pass mark. Find the maximum marks.
5. A town's population of 20,000 falls by 10% in one year and rises by 10% the next. Find the population after two years.
6. The price of petrol rises by 25%. By what percentage should a driver reduce consumption to keep spending the same?
7. If 30% of a number is 72, what is 45% of it?
8. The length of a rectangle increases by 20% and its breadth decreases by 10%. Find the percentage change in its area.

Answers:

1. 180. 37½% = 3/8, and 480 × 3/8 = 180.
2. No change. −20 + 25 − 5 = 0.
3. 33⅓%. Take B = 100 and A = 75. B is 25 more on a base of 75.
4. 600. The gap between the two scores is 5% of the total = 12 + 18 = 30 marks, so the total is 600. Check: pass mark 252, and 45% of 600 = 270 = 252 + 18.
5. 19,800. 20,000 × 0.9 × 1.1.
6. 20%. 25/125 × 100.
7. 108. 72 × 45/30 = 108. There is no need to find the number (240) first.
8. +8%. 20 − 10 − 200/100 = 8.

What to do next

• Write out the fraction table from memory until you can do it in under two minutes.
• Solve 20 successive-change questions using only multipliers, no long division.
• Practise the "more than, less than" reversal with 100 as the base until it takes ten seconds.
• Carry the same ideas into profit and loss and discounts, and then simple and compound interest, where the multiplier does most of the work.