Percentage is the backbone of AFCAT Numerical Ability. It feeds directly into Profit-Loss, Simple and Compound Interest, Data Interpretation and population sums, so every minute you spend here pays off across the paper. This Cavalier guide starts from the meaning of ‘per cent’, hands you the fraction-to-percentage table every topper memorises, and then turns the tricky parts — successive change and net effect — into one-line shortcuts.
What a percentage really is
The word per cent simply means ‘out of a hundred’. So 37% is nothing more than the fraction 37/100, or the decimal 0.37. Every percentage question, no matter how it is dressed up, reduces to comparing some quantity against a base of 100.
Examiners build entire sub-topics on this one idea: marks, discounts, interest, salary hikes, election votes and exam pass rates are all percentages in disguise. For AFCAT this means a single solid grip on Percentage unlocks at least three or four other Numerical Ability areas at once.
At The Cavalier we tell our defence aspirants to treat Percentage as the ‘parent topic’ of the quantitative section. The arithmetic is light, the formulas are few, and almost everything can be done mentally once you stop converting fractions the long way. The only real enemies are the base trap — using the wrong number as 100% — and the careless reading of ‘increase’ versus ‘more than’.
To turn any percentage into a usable number, divide by 100. To turn a fraction or decimal into a percentage, multiply by 100. Keep this two-way street clear and half the topic solves itself.
The fraction-to-percentage table you must memorise
The single biggest speed boost in this topic is knowing the common fraction values by heart. When you see 12.5% you should instantly think ‘one-eighth’ and multiply by the base directly instead of doing 12.5 ÷ 100.
- 1/2 = 50%, 1/3 = 33⅓%, 2/3 = 66⅔%
- 1/4 = 25%, 3/4 = 75%, 1/5 = 20%
- 1/6 = 16⅔%, 1/8 = 12.5%, 3/8 = 37.5%
- 1/9 = 11⅛%, 1/11 = 9.09%, 1/12 = 8⅓%
- 1/16 = 6.25%, 1/20 = 5%, 1/25 = 4%
Why does this help? Because most AFCAT percentages are clean fractions. ‘Find 37.5% of 64’ looks fiddly, but as 3/8 × 64 it is just 24 — done mentally. The table converts division into multiplication, which is always faster and less error-prone.
Find 16⅔% of 540.
Whenever a percentage looks ugly, ask: ‘Is this a known fraction?’ If yes, switch to the fraction and multiply. You will rarely need long division on AFCAT day.
Finding the part, the percentage or the whole
Every basic percentage question is one of three forms, and all three come from the same relationship.
Part = (Percentage ÷ 100) × Whole. Rearranged:
- Percentage = (Part ÷ Whole) × 100
- Whole = (Part × 100) ÷ Percentage
Always identify which quantity is missing before you touch the numbers. If the question gives the percentage and the whole, you want the part. If it gives the part and the whole, you want the percentage. If it gives the part and the percentage, you want the whole — this last one, back-calculating the base, is where most candidates slip.
32 is 8% of which number?
‘A is what per cent of B’ always divides by B, the quantity after ‘of’. Reversing this is the most common single error in the whole topic.
Percentage increase and decrease
To increase a number by p%, multiply it by (1 + p/100). To decrease it by p%, multiply by (1 − p/100). Using the multiplying factor directly is far faster than finding the change and adding it back.
- Increase by 20% ⇒ × 1.20
- Decrease by 15% ⇒ × 0.85
- Increase by 5% ⇒ × 1.05
A salary of 25,000 is raised by 12%. Find the new salary.
Percentage change between two values uses the original as the base. If a quantity moves from an old value to a new one, the change is measured against where it started, never against where it ended. This is the heart of dozens of AFCAT questions on price rise, population and production.
Percentage change = (New − Old) ÷ Old × 100. A positive result is an increase; a negative result is a decrease.
Successive percentage change in one step
When a quantity changes by a%, then changes again by b%, you cannot simply add a and b. The second change acts on an already-changed value. The single formula below combines both shifts at once.
Net change = a + b + (ab ÷ 100) per cent, using signs: + for increase, − for decrease.
A price rises 20% and then rises a further 10%. Find the net change.
So a 20% rise followed by a 10% rise is a 32% rise, not 30%. The extra 2% is the second increase acting on the first. The same formula handles mixed cases: for a 20% rise then a 20% fall, use a = +20 and b = −20.
A number is increased by 20% and then decreased by 20%. Find the net change.
An increase then an equal decrease never returns you to the start — you always end up lower. A 20% up and 20% down leaves a 4% net loss every time.
Reversing a change and the net-effect trick
Two patterns appear almost every cycle. First, if a quantity is increased by p%, the percentage by which the new value must be decreased to get back to the original is not p%.
To reverse a p% increase: required decrease = p ÷ (100 + p) × 100 per cent. To reverse a p% decrease: required increase = p ÷ (100 − p) × 100 per cent.
A's income is 25% more than B's. By what per cent is B's income less than A's?
So although A is 25% more than B, B is only 20% less than A — because the two statements use different bases. Spotting which quantity is the base is the whole game in these comparison questions.
Fraction shortcut: 25% more means the ratio 5:4, so the reverse gap is 1 out of 5 = 20%. Converting the percentage to a fraction makes the base flip obvious.
Price, consumption and expenditure problems
A favourite AFCAT setup: a commodity's price changes and you must keep expenditure the same by adjusting consumption. Since Expenditure = Price × Consumption, if expenditure is fixed, price and consumption move inversely.
If price rises by p%, consumption must fall by p ÷ (100 + p) × 100 per cent to keep expenditure unchanged.
The price of sugar rises by 25%. By what per cent must a family cut its consumption so the sugar bill stays the same?
Notice this is the same reversal formula as before — the price increase must be undone on the consumption side. Recognising that two different-looking questions share one formula is exactly the pattern-spotting that AFCAT rewards.
Population and repeated-growth problems
When a quantity grows by the same percentage every year, the growth compounds. After n years the multiplying factor is applied n times — this is the same engine that drives Compound Interest.
Final value = Initial value × (1 + r/100)n for growth, or (1 − r/100)n for decline, where r is the annual rate and n the number of years.
A town's population is 10,000 and grows 10% per year. Find the population after 2 years.
For two years the answer (12,100) is higher than a flat 20% rise (12,000) would give, because the second year's growth acts on the larger first-year figure. To find an earlier population, divide by the factor instead of multiplying.
Growth and decline can mix: a population that rises 10% one year and falls 10% the next ends below where it started — use the successive-change formula to get the net −1%.
Marks, pass percentage and the failed-candidate trap
Exam-based percentage questions are common and easy marks if you read carefully. The classic version gives a candidate's marks, the pass mark as a percentage, and the shortfall, then asks for the maximum marks.
If a student scores S marks and fails by F marks at a pass percentage p%, then the pass mark is S + F, and Maximum marks = (S + F) × 100 ÷ p.
A student scores 180 marks and fails by 20 marks. The pass mark is 40%. Find the maximum marks.
Do not apply the 40% to the student's score. The pass percentage is always of the maximum marks, which is the unknown you are solving for.
Previous-year style question
Q. In an election between two candidates, one got 55% of the total valid votes. 20% of the total votes were invalid. If the total number of votes was 7,500, how many valid votes did the other candidate get?
Answer: Valid votes = 80% of 7500 = 0.80 × 7500 = 6000. The winner took 55% of valid votes, so the other candidate took 45%. Other candidate's votes = 45% of 6000 = 0.45 × 6000 = 2700 votes.
Traps and time-savers to keep in mind
- Adding two successive percentages instead of using a + b + ab/100.
- Assuming a p% rise is undone by a p% fall — it never is.
- Using the wrong base: ‘A is more than B’ vs ‘B is less than A’ have different bases.
- Applying the pass percentage to the score rather than the maximum marks.
Convert every common percentage to its fraction the instant you read it — 12.5% to 1/8, 16⅔% to 1/6, 37.5% to 3/8. Multiplication by a small fraction beats division by an ugly decimal every time.
Choose 100 as your base
When a question is purely about percentages with no actual values given, assume the starting quantity is 100. Every change then becomes an easy whole number, and the final answer is read off directly as a percentage. This single habit turns abstract ‘by what per cent’ questions into simple arithmetic and is one of the most reliable time-savers in the whole Numerical Ability section.
Quick revision
- Per cent means ‘out of 100’: divide by 100 to use, multiply by 100 to express.
- Memorise the fraction table — 1/8 = 12.5%, 1/6 = 16⅔%, 3/8 = 37.5%.
- Increase by p% means × (1 + p/100); decrease means × (1 − p/100).
- Successive change = a + b + ab/100, with correct signs.
- Reverse a p% rise with p/(100 + p) × 100; this also fixes price-consumption sums.
- Repeated yearly growth uses (1 + r/100)n, just like compound interest.
Frequently asked questions
How important is Percentage for AFCAT?
It is the single most important Numerical Ability topic because Profit-Loss, Interest, Data Interpretation and population growth all rest on it. Mastering Percentage lifts your score across several other areas at once.
Why doesn't a 20% increase cancel a 20% decrease?
Because the decrease acts on the larger, already-increased value. Using the successive-change formula, 20% up then 20% down gives a net 4% loss, never zero.
How do I quickly find a percentage like 37.5% of a number?
Convert it to a fraction. 37.5% is 3/8, so 37.5% of 64 is just 3/8 times 64 = 24. The fraction table turns slow division into fast mental multiplication.
If A is 25% more than B, why is B not 25% less than A?
The two statements use different bases. A is measured against B, but the reverse gap is measured against A. The reversal formula gives B as 25/125 times 100 = 20% less than A.
What is the fastest way to handle pure percentage questions with no numbers?
Assume the starting quantity is 100. Every change becomes an easy whole number, and the final figure can be read straight off as the answer in per cent.
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