Percentage is the single most rewarding chapter in CDS Elementary Mathematics — it feeds directly into Profit & Loss, Interest, Data Interpretation and Ratio. The word literally means “per hundred”, so every percentage is just a fraction with denominator 100. Master a handful of fraction equivalents and three core formulas, and you will solve most questions in seconds without long division.
Why Percentage is a Scoring Topic
In the CDS & OTA written paper, Percentage rarely appears alone — it is woven into Profit & Loss, Simple and Compound Interest, Ratio, Mixtures, Allegation and Data Interpretation. A candidate who is weak in percentage quietly loses marks across nearly half the arithmetic section, often without realising why.
The good news is that the concept itself is small and self-contained. Once you can move fluently between a percentage, a fraction and a decimal, the rest of the chapter is simply pattern recognition. The CDS examiner usually tests speed rather than depth here, so a candidate who has memorised shortcuts will finish a percentage question in well under a minute, while someone relying on long multiplication may take three times as long.
Because the Elementary Mathematics paper carries 100 questions in 120 minutes, every saved second matters. Investing two days now to truly master percentage conversions pays back across the entire arithmetic portion of the syllabus. Treat this chapter as the foundation on which Profit & Loss and Interest are later built.
Percentage problems are ratio problems in disguise. Whenever you see “% of”, mentally read it as “multiply by that fraction”, and the question usually collapses into one short calculation.
The Core Definition
A percentage is a fraction whose denominator is fixed at 100. The symbol % simply stands for “÷ 100”.
x% = x⁄100. So 25% = 25⁄100 = 1⁄4, and 0.6 = 60⁄100 = 60%.
Two everyday conversions you will use constantly:
- Fraction → %: multiply by 100. For example, 3⁄5 × 100 = 60%.
- % → fraction: divide by 100. For example, 35% = 35⁄100 = 7⁄20.
- Decimal ↔ %: shift the decimal point two places. 0.045 = 4.5% and 120% = 1.2.
To find x% of a number N, compute (x⁄100) × N. To express one quantity A as a percentage of B, compute (A⁄B) × 100. These two operations cover the majority of direct questions in the paper.
It also helps to remember that percentage is commutative in a useful way: x% of y equals y% of x. So 16% of 25 is the same as 25% of 16, which is just one quarter of 16, namely 4. Spotting this symmetry can convert an ugly computation into a one-line answer. Similarly, 18% of 50 equals 50% of 18, which is simply 9.
The Must-Memorise Fraction Table
The fastest CDS aspirants never divide by awkward percentages — they recall the equivalent fraction. Learn this short table cold.
- 1⁄2 = 50% • 1⁄3 = 33⅓% • 1⁄4 = 25%
- 1⁄5 = 20% • 1⁄6 = 16⅔% • 1⁄8 = 12½%
- 1⁄9 = 11&frac19;% • 1⁄10 = 10% • 1⁄11 = 9&frac1;⁄11%
- 1⁄12 = 8⅓% • 1⁄16 = 6¼% • 1⁄20 = 5%
To find 37.5% of 64, read 37.5% as 3⁄8. Then 3⁄8 × 64 = 24 — instant, no multiplication of decimals.
The multiples follow naturally once the unit fraction is known. Since 1⁄8 = 12½%, you immediately get 3⁄8 = 37½%, 5⁄8 = 62½% and 7⁄8 = 87½%. Likewise, from 1⁄6 = 16⅔% you can read 5⁄6 = 83⅓%. Building these by multiplication, rather than memorising each separately, keeps the table small and reliable under exam pressure.
Practise the reverse direction too. When a question gives you a percentage such as 14&frac27;% or 9&frac1;⁄11%, train yourself to recognise them instantly as 1⁄7 and 1⁄11. The moment you replace the awkward percentage with a clean fraction, the arithmetic becomes trivial.
Percentage Increase and Decrease
When a quantity changes, the change is always measured on the original value, not the new one.
% change = (Change ÷ Original value) × 100.
New value after r% increase = Original × (1 + r⁄100).
New value after r% decrease = Original × (1 − r⁄100).
For example, a salary of ₹40,000 raised by 15% becomes 40000 × 1.15 = ₹46,000. The same salary cut by 15% becomes 40000 × 0.85 = ₹34,000. Notice how the multiplier method avoids two separate steps of finding the change and then adding or subtracting it — one multiplication does both.
This multiplier idea scales to repeated changes as well. If a stock falls 20% and then rises 25%, multiply 0.80 × 1.25 = 1.00, meaning the stock is back exactly where it began. Thinking in multipliers turns several percentage chains into a single line of arithmetic and is the habit that separates fast candidates from slow ones.
A 20% rise followed by a 20% fall does not return you to the start. Increase and decrease act on different bases, so you end lower than the original (see the successive-change rule below).
Successive Percentage Change
When two percentage changes happen one after another, do not add them. Combine them with a single formula.
Net % change for two changes a% and b% = a + b + (a × b)⁄100.
Use a positive sign for an increase and a negative sign for a decrease.
So a 20% increase followed by a 20% decrease gives: 20 + (−20) + (20 × −20)⁄100 = −4%. The quantity is 4% lower than where it started. The formula extends to three changes by applying it twice: first combine a and b, then combine that result with c.
This rule is exactly what powers compound interest, where the rate is added repeatedly each period, and population growth over several years. Whenever a question describes a quantity changing in stages, reach for the successive-change formula rather than naively adding the percentages, which is the error the examiner is hoping you will make.
If a value first rises and then falls by the same r%, the net effect is always a fall of (r²⁄100)%. For r = 10, that is a 1% drop.
The “A is x% More/Less than B” Trap
CDS loves to test the difference between “A is more than B” and “B is less than A”. The base flips, so the percentages differ.
If A is x% more than B, then B is [x⁄(100 + x)] × 100 % less than A.
If A is x% less than B, then B is [x⁄(100 − x)] × 100 % more than A.
For example, if A is 25% more than B, then B is 25⁄125 × 100 = 20% less than A. The numbers are not symmetric — always note which quantity is the base.
Pick B = 100 to keep arithmetic clean. If A is 25% more, A = 125; then B is short of A by 25 out of 125 = 20%. This trick avoids the formula entirely.
Worked Example: Exam Marks
A student scores 30% in an exam and fails by 18 marks. Another student scores 45% and gets 27 marks more than the pass mark. Find the maximum marks.
Pass mark = 30% of M + 18 = 0.30M + 18
Pass mark = 45% of M − 27 = 0.45M − 27
0.30M + 18 = 0.45M − 27
18 + 27 = 0.45M − 0.30M
45 = 0.15M
M = 45 ÷ 0.15 = 300
Maximum marks = 300, and the pass mark = 0.30 × 300 + 18 = 108.
Worked Example: Population Growth
Compound percentage growth, such as population over years, uses the same multiplier idea as compound interest.
A town’s population is 50,000 and grows 10% in the first year and 20% in the second year. Find the population after two years.
After year 2 = 55000 × (1 + 20⁄100) = 55000 × 1.20 = 66000
Check by net change = 10 + 20 + (10 × 20)⁄100 = 32%
50000 × 1.32 = 66000
Final population = 66,000, a clean 32% overall rise.
The same logic answers reverse questions. If a population is 66,000 after two years of 10% and 20% growth, the original is found by dividing, not subtracting: 66000 ÷ 1.32 = 50,000. Students who try to subtract 32% from 66,000 get the wrong figure, because the 32% was calculated on the smaller starting value, not on the final one.
Common Mistakes to Avoid
- Wrong base: percentage change is always on the original, never the new value.
- Adding successive changes: 10% then 10% is not 20%; it is 21% because of compounding.
- Confusing “of” and “more”: “20% of 50” = 10, but “20% more than 50” = 60.
- Point vs per cent: if a rate rises from 5% to 7%, that is a 2 percentage-point rise but a 40% relative rise.
Reducing a fraction wrongly: 0.5% is 0.005, not 0.5. Always remember the % sign hides a division by 100.
Previous-Year Style Question
Q. The price of sugar is increased by 25%. By what percentage must a family reduce its consumption so that the expenditure on sugar remains unchanged?
Answer: Expenditure = Price × Consumption. To keep it fixed after a 25% price rise, consumption must fall by 25⁄(100 + 25) × 100 = 25⁄125 × 100 = 20%. Shortcut: a price rise of 1⁄4 needs a consumption cut of 1⁄5.
The Fixed-Expenditure Shortcut
The sugar question above is a recurring CDS pattern: a price changes but spending stays constant. Memorise the direct rule.
If price rises by r%, required reduction in consumption = [r⁄(100 + r)] × 100 %.
If price falls by r%, possible increase in consumption = [r⁄(100 − r)] × 100 %.
This is the very same “more/less” base-flip formula from earlier — recognising that connection saves you from memorising a separate rule. The numerator stays as the original change, and the denominator becomes the new total after the change.
The reverse situation is just as common. If the price of an item falls, a family can buy more of it for the same money. A 20% fall in price lets consumption rise by 20⁄(100 − 20) × 100 = 25%. Reading these as fractions makes them memorable: a one-fifth price drop allows a one-quarter rise in quantity.
Keep the fraction equivalents handy: a 50% rise needs a 1⁄3 (33⅓%) cut, a 100% rise needs a 1⁄2 (50%) cut, and a 25% rise needs a 1⁄5 (20%) cut.
Quick Revision
- x% = x⁄100; to find x% of N, multiply N by x⁄100.
- Memorise fraction equivalents (1⁄8 = 12½%, 3⁄8 = 37½%) to skip division.
- % change is measured on the original value.
- Successive changes: net = a + b + ab⁄100, with signs.
- Equal rise then fall of r% gives a net fall of r²⁄100 %.
- Fixed expenditure: price up r% → cut consumption by r⁄(100 + r) × 100 %.
Drill ten mixed questions daily from a topic-wise CDS PYQ set, and percentage will become one of your fastest, most reliable scoring areas.
Frequently asked questions
How many percentage questions appear in the CDS Maths paper?
Direct percentage questions are usually two to four, but the concept also underpins Profit & Loss, Interest and Data Interpretation, so it indirectly influences ten or more marks. It is among the highest-yield arithmetic topics.
What is the fastest way to calculate percentages without a calculator?
Convert the percentage into its equivalent fraction and multiply. For example, 37.5% becomes 3⁄8, so 37.5% of 64 is simply 3⁄8 × 64 = 24. Memorising the fraction table is the single biggest time-saver.
Why does a 10% increase then a 10% decrease not return the original value?
Because the two changes act on different bases. The increase is on the original amount, but the decrease is on the larger amount, so you end 1% (that is r²⁄100) below where you started.
What is the difference between percentage points and per cent?
A move from 5% to 7% is a rise of 2 percentage points but a 40% relative increase (2 out of 5). CDS examiners often test this distinction, so read the wording carefully.
Which books should I use for CDS percentage practice?
Build the concept from NCERT Class 7 and 8, then drill speed using a 38-year topic-wise CDS Maths PYQ book and a dedicated Percentage PYQ set. At The Cavalier we pair each rule with timed PYQ practice.
Related CDS / OTA Maths topics
Want a teacher to walk you through CDS / OTA Maths?
Cavalier's CDS / OTA batches break every topic into classroom sessions with daily practice, tests and doubt-clearing.