SSC MTS & Havaldar · Quantitative Aptitude
Percentage
Conversion between fractions and percentages and their application in real-life expenditure, population and change problems.
Six concepts, and they are six because that is what the chapter honestly holds — every percentage question in every paper is one of these six ideas, usually with the base moved somewhere you did not look. Percentage is not hard arithmetic. It is arithmetic where the denominator is hidden in the wording.
- SSC MTS & Havaldar
- Medium level
- 6 concepts
- 106 practice questions
1A percent is a fraction with 100 underneath
x% means x/100, and nothing else. The whole practical skill in this chapter is refusing to compute with the percentage and converting it to a fraction first: 37.5% of 1024 is a three-second division once you see 3/8, and a long multiplication otherwise.
The equivalents worth knowing by heart are the awkward ones — 16⅔% = 1/6, 8⅓% = 1/12, 9 1/11 % = 1/11, 14 2/7 % = 1/7 — because the exam writes a number in that form precisely when the fraction makes the arithmetic collapse. A percentage with a fraction inside it is a hint, not a cruelty.
Figure. Both bars are the same share of a whole. The top bar counts hundredths (37.5 of 100); the bottom counts eighths (3 of 8). Converting the percent to the fraction is what collapses 37.5\% of 1024 into a three-second 3/8 multiplication — the rest of the lookup table in this concept works the same way.
How it works
- Spot the fractionRead 37.5% as 3/8 and 83⅓% as 5/6 before touching the number it acts on.
- Divide, don't multiply1/6 of a number is one division; 16.666% of it is a multiplication and a rounding error.
- Chain the fractionsTwo percentages applied in turn multiply as fractions: 3/8 then 1/6 is 1/16.
| Percentage | Fraction | Percentage | Fraction |
|---|---|---|---|
| 50% | 1/2 | 12½% | 1/8 |
| 33⅓% | 1/3 | 11 1/9 % | 1/9 |
| 25% | 1/4 | 10% | 1/10 |
| 20% | 1/5 | 9 1/11 % | 1/11 |
| 16⅔% | 1/6 | 8⅓% | 1/12 |
| 14 2/7 % | 1/7 | 6¼% | 1/16 |
| 37.5% | 3/8 | 62.5% | 5/8 |
Two awkward percentages in a row
Find 37.5% of 1024, then 16⅔% of the result, and express the answer as a percentage of the original number.
- 37.5% = 3/8, so 3/8 × 1024384
- 16⅔% = 1/6, so 384 ÷ 664
- 64 as a fraction of 10241/16
- 1/16 as a percentage6.25%
Pro tip. You could have written the last row first: 3/8 × 1/6 = 1/16 = 6.25%, without ever computing 384 or 64. Whenever a question applies two percentages in turn and asks only for the overall effect, multiply the fractions and leave the intermediate values alone — they are there to be calculated by candidates who did not notice.
37.5% of a number is 384. The number is
- 1024
- 144
- 640
You are given the part and asked for the whole, so divide: 384 ÷ (3/8) = 384 × 8/3 = 1024. Multiplying instead gives 144, which is 37.5% of 384 — the right operation in the wrong direction, and by far the commonest way this is dropped. 640 comes from reading 37.5% as 3/5.
2Percentage change, and the base it is taken on
Percentage change is (new − old) / old × 100. The numerator is easy and nobody gets it wrong; the denominator is the whole question. It is always the value you are changing from, never the one you are changing to, and never the larger of the two.
That single choice makes the relationship asymmetric. A rise from 4500 to 5175 is a 15% increase, but the fall from 5175 back to 4500 is only a 13.04% decrease — the same 675 measured against a bigger base. Any question that walks a quantity up and then back down is testing this and nothing else.
Figure. The short bracket is the change and the tall one is the base. Percentage change is the short bracket divided by the tall one, and the only decision in the question is which bar the tall bracket is drawn against. Move it to the right-hand bar and the same 675 becomes 13.04%.
How it works
- Name the baseUnderline the word after "more than", "less than" or "compared with" — that is the base.
- Take the differenceNew minus old, keeping the sign; a negative answer is a decrease, not an error.
- Divide by the base, then × 100Divide by the base you named, not by whichever number happens to be handy.
The same 675, two different answers
A town's population rises from 4500 to 5175. Find the percentage increase, and then the percentage decrease if it were to fall back to 4500.
- Difference = 5175 − 4500675
- Rise: base is 4500, so 675 / 4500 × 10015% increase
- Fall: base is now 5175, so 675 / 5175 × 10013.04% decrease
- Same numerator, larger denominatorsmaller percentage
Pro tip. This is why "the population rose 15% and then fell 15%" does not return you to 4500 — the second 15% is charged on 5175 and takes off 776.25, not 675. Every up-then-down question in this chapter is a consequence of rows two and three, and if you can state which number is on the bottom you have already done the hard part.
A is 25% of B. Then B is what percent of A?
- 75%
- 125%
- 400%
A = B/4, so B = 4A, which is 400% of A. The base has switched from B to A and the small number is now underneath, which is exactly why the answer is above 100%. 75% is the part of B that A is not, and 125% treats 25% as a change rather than a share.
3Two changes in a row
Changes of a% and then b% do not add. The second acts on a quantity the first has already moved, so the net effect is a + b + ab/100 — with increases positive and decreases negative. The ab/100 term is the interaction, and it is the entire content of the formula.
The special case worth memorising is a rise of x% followed by a fall of x%: the net is x − x − x²/100 = −x²/100, always a loss and never zero. Up 10% then down 10% leaves you 1% short; up 20% then down 20% leaves you 4% short. The order makes no difference — 0.9 × 1.1 and 1.1 × 0.9 are the same number — which is often the second half of the question.
Figure. The horizontal line marks where the quantity started. The rise lifts the bar above it by 10 units; the fall then removes 10% of 110, which is 11 units, and drops the bar one unit below the line it began on. The two arrows look symmetric and are not, because they are measured against different bars.
How it works
- Sign each changeA 20% discount is a = −20, not 20. Getting a sign wrong flips the interaction term too.
- Add, then add the product over 100a + b + ab/100. Two decreases make ab positive, which is why 28% and not 30%.
- Read the sign of the answerNegative means a net fall. Quote it as a decrease rather than a negative increase.
Successive discounts of 20% and 10%
A shop takes 20% off the marked price and then a further 10% off what remains. What single discount is that?
- a = −20, b = −10; net = a + b + ab/100−20 − 10 + 2 = −28%
- Check on a marked price of ₹1000: 1000 × 0.80₹800
- Then 800 × 0.90₹720
- Total taken off: 280 out of 100028%, not 30%
Pro tip. The +2 in row one is why the customer pays more than the advertised 30% suggests: the second discount is charged on the reduced price, so 10% of 800 is only 80 rather than 100. If you prefer factors to the formula, 0.8 × 0.9 = 0.72 gives the same answer in one step and never has a sign to lose — and it scales, since three discounts are just three factors.
A price rises by 10% and then falls by 10%. Compared with where it began, it is now
- Unchanged, the two cancelling
- 1% lower
- 1% higher
10 − 10 + (10)(−10)/100 = −1, or equivalently 1.1 × 0.9 = 0.99. The fall is charged on the raised price, so it removes more than the rise put on. The result is always a loss of x²/100 whichever way round the two changes come, which is why "unchanged" is never the answer to this question.
4Undoing a change costs more than it saved
To reverse a fall of x% you need a rise of x/(100 − x) × 100 percent; to reverse a rise of x% you need a fall of x/(100 + x) × 100. The two formulas are the same idea seen twice: the amount to be moved is unchanged, but the base has shrunk in one case and grown in the other.
The same pair answers the comparison questions. If A is x% more than B, then B is x/(100 + x) × 100 percent less than A — so 25% more one way is 20% less the other. Reading those two numbers as if they must match is the single most common percentage error there is.
Figure. The two brackets are the same length — the same 20 units come off and go back on. They are different percentages only because the bar they are measured against is different: 100 on the way down, 80 on the way up. Nothing else is going on in this concept.
How it works
- Set the original to 100Percentages are ratios, so you may choose the starting value; 100 makes every step arithmetic.
- Apply the changeA 20% cut takes 100 to 80. Write the new value down; it is the new base.
- Measure the return against itYou must climb 20, but from 80, and 20/80 is 25% — not the 20% you came down.
A 20% pay cut, restored
A salary is cut by 20%. By what percent must the reduced salary be raised to restore the original figure?
- Take the salary as 100; after a 20% cut80
- To reach 100 again you must add20
- But 20 is now measured against 80: 20/80 × 10025%
- Formula check: x/(100 − x) × 100 with x = 2025%
Pro tip. Notice the pairing this produces: −20% is undone by +25%, and +25% is undone by −20%. That reciprocal pair, along with −25% ↔ +33⅓% and −50% ↔ +100%, covers most of what is asked. The rule behind all of them is that the return percentage is always the larger of the two, because you are always climbing from the smaller number.
A's income is 25% more than B's. B's income is less than A's by
- 25%
- 20%
- 33⅓%
Take B = 100, so A = 125. The gap of 25 is now measured against A: 25/125 × 100 = 20%. Answering 25% keeps the base on B when the question has moved it to A. 33⅓% is the answer to the opposite question — if A were 25% less than B, B would be 33⅓% more than A.
5Holding a product constant
Expenditure is price times consumption; area is length times breadth; distance is speed times time. Whenever the product is to be held fixed and one factor rises by x%, the other must fall by x/(100 + x) × 100 percent — the reversing rule from the previous concept, wearing a different hat.
So a 25% rise in price needs a 20% cut in consumption, and a 20% fall in price allows a 25% rise. The pairs are the same reciprocal pairs, and the mistake is the same too: cutting consumption by the 25% the price went up by overshoots and leaves you spending less than before.
Figure. Every point on this curve spends the same money. Walking from the first marked point to the second raises the price by a quarter and lowers the consumption by a fifth — unequal percentages, equal products. The curve flattens as you go right, which is the geometric statement that the same percentage rise in price costs less and less consumption once the price is already high.
How it works
- Name the productWrite down what is being held fixed — the bill, the area, the distance. Everything follows from it.
- Use the new baseThe reduction is measured against the raised price, so the denominator is 100 + x.
- Check by multiplying125 × 80 = 10000 = 100 × 100. If the two products differ, the percentage is wrong.
Sugar goes up 25%
Sugar costs ₹40 a kilogram and a family spends ₹1000 a month on it. The price rises 25%. By what percent must the family cut its consumption to keep the bill at ₹1000?
- Present consumption: 1000 ÷ 4025 kg
- New price: 40 × 1.25₹50 per kg
- What ₹1000 now buys: 1000 ÷ 5020 kg
- Cut = 5/25 × 10020%
Pro tip. The concrete route above and the formula 25/125 × 100 give the same 20%, and the concrete route is worth doing once so you believe the formula. After that, use the formula — and sanity-check it against the fact that the answer must be less than the rise, because you are dividing by more than 100.
The price of a commodity falls by 20%. To spend exactly the same amount, consumption must rise by
- 20%
- 25%
- 16⅔%
The price is now 80, so the rise is measured against 80: 20/80 × 100 = 25%. Answering 20% keeps the base at the old price. 16⅔% is 20/120 × 100, the answer to the mirror question where the price had risen by 20% instead.
6Points and percent are different units
When the quantity being measured is itself a percentage, there are two different ways to describe a change and they give different numbers. A pass rate moving from 40% to 50% has risen by 10 percentage points — an absolute difference — and by 25 percent, which is that difference measured against the 40 it started from.
Exams exploit the gap deliberately, and they can make it worse by changing the total at the same time. A rate can rise while the count falls, or the other way round, and a question that gives you both is asking whether you noticed which one it wants.
Figure. The rate cluster is the raw move from 40\% to 50\%. The second cluster is the two legal readings of that move: +10 percentage points (absolute) versus +25\% (relative to the starting 40). Same change, different units — exams pick whichever wording traps the other reading.
In 2023, 40% of 2000 candidates passed. In 2024, 50% of 2400 passed.
| Question | Computation | Answer |
|---|---|---|
| By how many percentage points did the rate rise? | 50 − 40 | 10 points |
| By what percent did the rate rise? | (50 − 40)/40 × 100 | 25% |
| By what percent did the number of passes rise? | (1200 − 800)/800 × 100 | 50% |
| By what percent did the number of failures rise? | (1200 − 1200)/1200 × 100 | 0% |
Rate up, failures flat
In 2023, 40% of 2000 candidates passed an examination. In 2024, 50% of 2400 passed. Describe the change in every way the question might ask for.
- Passes in 2023: 40% of 2000800
- Passes in 2024: 50% of 24001200
- Rate: 40% to 50%10 points, a 25% rise
- Failures: 1200 in 2023, 1200 in 2024unchanged
Pro tip. Four defensible numbers — 10, 25, 50 and 0 — all describing the same two years, and the wording alone picks between them. Before computing anything, decide two things: is the quantity a rate or a count, and is the change wanted as a difference or as a proportion of the start? The last row is the one to notice, because a pass rate improving by a quarter while not a single extra candidate avoids failing is exactly the sort of fact these questions are built around.
A team's win rate rises from 60% to 66%. The percentage increase in its win rate is
- 6%
- 10%
- 66%
(66 − 60)/60 × 100 = 10%. The 6 is the rise in percentage points, which is what most people read off and is a different unit — it is the numerator, not the answer. Because the quantity is itself a percentage, both numbers are quotable and the wording decides; "percentage increase" always means the second one.
Notes
- Percentage as a Fraction: x\% means \frac{x}{100}, so 25\% = \frac{1}{4}; memorising fraction equivalents (e.g. 12.5\%=\frac{1}{8}, 16\tfrac{2}{3}\%=\frac{1}{6}) turns percentage problems into fast fraction work.
- Successive Percentage Change: Two changes of a\% and b\% combine to a single change of \left(a+b+\frac{ab}{100}\right)\%, where increases are positive and decreases negative.
- Net Effect of Equal Rise and Fall: Increasing a quantity by x\% then decreasing by x\% (or vice versa) always gives a net decrease of \frac{x^2}{100}\%, never zero.
- Product Constancy: If A \times B is fixed and A rises by x\%, then B must fall by \frac{x}{100+x}\times100\% to keep the product constant — key for price/consumption and length/breadth problems.
- Percentage Point vs Percent: A rise from 40\% to 50\% is a 10 percentage-point rise but a 25\% relative increase; exams exploit this distinction as a trap.
Formulas
- Percentage change: \%\text{ change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100
- Successive change: net \% = a + b + \frac{ab}{100}
- Equal x\% rise then fall: net change = -\frac{x^2}{100}\%
- If A is x\% more than B, then B is less than A by \frac{x}{100+x}\times100\%
- If A is x\% less than B, then B is more than A by \frac{x}{100-x}\times100\%
- Value after change: \text{New} = \text{Old}\times\left(1 \pm \frac{x}{100}\right)
Exam traps & shortcuts
- Convert awkward percentages into their fraction form (33\tfrac{1}{3}\%=\frac{1}{3}, 9\tfrac{1}{11}\%=\frac{1}{11}) so 33\tfrac13\% of desk price becomes a one-step division.
- For 'price increased by x\%, how much less to consume' use \frac{x}{100+x}\times100 directly instead of assuming a base amount.
- For two successive changes just plug into a+b+\frac{ab}{100} with correct signs; a single formula avoids two multiplication steps.
Reference tables
The same table answers three different questions, because they are the same question. Read a row as: a fall of the first undone by a rise of the second; A being x% more than B making B y% less than A; a price rise of x% needing a consumption cut of y%.
| x% | x/(100 − x) × 100 | x/(100 + x) × 100 |
|---|---|---|
| 10% | 11 1/9 % | 9 1/11 % |
| 20% | 25% | 16⅔% |
| 25% | 33⅓% | 20% |
| 33⅓% | 50% | 25% |
| 50% | 100% | 33⅓% |
Every line here should be reconstructible from the concept above it, not merely recalled.
| Quantity | Relation | Watch for |
|---|---|---|
| Percentage of a number | x% of N = (x/100) × N | Convert to a fraction before multiplying |
| Percentage change | (New − Old)/Old × 100 | The base is always the old value |
| Value after a change | New = Old × (1 ± x/100) | Chain factors for several changes |
| Two successive changes | a + b + ab/100 | Signs first; two falls make ab positive |
| Equal rise then fall | net = −x²/100 | A loss, never zero, either order |
| Undoing a fall of x% | x/(100 − x) × 100 | Always bigger than x |
| Undoing a rise of x% | x/(100 + x) × 100 | Always smaller than x |
| Product held constant | one up x% means the other down x/(100 + x) × 100 | Multiply back to check |
| Percentage points | a difference of two rates | Not the same as the percentage increase |
Recap
Read only this the night before.
- Fractions
- Convert before computing. 12½ = 1/8, 16⅔ = 1/6, 8⅓ = 1/12, 9 1/11 = 1/11, 14 2/7 = 1/7. A fraction inside a percentage is a hint.
- The base
- Percentage change divides by the value you came from. Change the base and the same difference becomes a different percentage.
- Successive
- a + b + ab/100, or just multiply the factors. Equal up-then-down loses x²/100, in either order.
- Reversal
- −20 ↔ +25, −25 ↔ +33⅓, −50 ↔ +100. Going back always costs a bigger percentage than coming down.
- Constant product
- Price up x% means consumption down x/(100 + x) × 100. Multiply the two back and check you get the same bill.
- Points
- 40% to 50% is 10 points and 25 percent. When the quantity is itself a rate, read the wording before computing.
Practise Percentage
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