Percentage is simply a fraction with denominator 100. The word comes from the Latin "per centum," meaning "per hundred." When you say 35%, you mean 35 out of every 100 parts. That is all it ever means — nothing more exotic.
Here is the analogy that makes it stick: think of a percentage as a universal ruler. You can compare salaries, election results, discount rates, and population growth on the same scale only because percentages normalize everything to base 100. A shopkeeper's 40% markup and a city's 10% population growth are immediately comparable even though the raw numbers are in completely different units.
The three core moves you will use in every AFCAT question:
Move 1 — Convert to a multiplier. Instead of writing "increase by 25%," write "multiply by 1.25." Instead of "decrease by 15%," write "multiply by 0.85." This single habit eliminates most calculation errors, especially in multi-step problems like compound growth or successive discounts.
Move 2 — Anchor on base 100. When a question asks "what percentage of X is Y," fix X as 100 and scale proportionally. This is not a formal algebraic step — it is a mental shortcut that bypasses equation setup.
Move 3 — Fraction-percent equivalents. A handful of fractions appear constantly in AFCAT: 1/4 = 25%, 1/5 = 20%, 1/3 ≈ 33.33%, 3/4 = 75%, 1/8 = 12.5%, 2/3 ≈ 66.67%. Recognizing these on sight saves you the multiplication step entirely.
Look — AFCAT is not a paper that throws bizarre percentage problems at you. The patterns repeat: mark-up and discount (percentage of percentage), savings as a residual (100% minus expenses), set theory with two categories (union formula), and successive change (multiplicative model). Once you recognize which template a question belongs to, you are 80% done.
If a quantity changes by r%, the new quantity is the original multiplied by (1 + r/100) for an increase or (1 - r/100) for a decrease.
Formal notation: New = Original × (1 ± r/100)
This becomes powerful when changes are chained. For two successive changes of a% and b%:
Net multiplier = (1 ± a/100)(1 ± b/100)
Expanding this gives the net percentage change formula:
Net % = a + b + (ab/100) (use appropriate signs)
So a 25% increase followed by a 20% decrease:
Net % = 25 + (-20) + (25 × (-20)/100) = 25 - 20 - 5 = 0%
That is zero net change — a fact that appears directly in one of the PYQs below.
This is the most common disguised percentage problem in AFCAT. The trap is that mark-up percentage is on Cost Price (CP) while discount percentage is on Marked Price (MP). They are not the same base.
If you mark up by m%: MP = CP × (1 + m/100)
If you then give discount of d%: SP = MP × (1 - d/100)
Combining: SP = CP × (1 + m/100) × (1 - d/100)
Profit % = [(SP - CP) / CP] × 100 = [(1 + m/100)(1 - d/100) - 1] × 100
For the standard AFCAT question (mark up 40%, discount 15%):
SP/CP = 1.40 × 0.85 = 1.19 → Profit = 19%
These are trivially easy once you see the structure. If someone spends a% on food and b% on rent, savings = (100 - a - b)% of income. Set that equal to the given amount and divide. No algebra needed.
For any two sets A and B within a universal set:
|A ∪ B| = |A| + |B| - |A ∩ B|
In percentage terms:
%( A or B ) = %A + %B - %(A and B)
%(Neither) = 100% - %(A or B)
This is the union formula, and it handles every "survey" or "preference" problem in AFCAT.
For compound growth at rate r% per period over n periods:
Final = Initial × (1 + r/100)^n
For depreciation: replace + with -.
The key insight for reverse depreciation problems (the machine question): if you are given the final value and asked for the original, divide by the multiplier rather than multiplying.
Original = Final / (multiplier)^n
For the specific case: machine depreciates 15% per year, value after 2 years = 68000.
Original = 68000 / (0.85)^2 = 68000 / 0.7225 ≈ 94118
Do not try to add back 15% twice — that is wrong because 15% of 68000 is a different amount than 15% of the original.
When a question says "25% of x added to 75 equals x," set it up as:
0.25x + 75 = x
75 = 0.75x
x = 100
The speedup: recognize that 0.75 = 3/4, so x = 75 × 4/3 = 100. Fraction arithmetic is faster than decimal arithmetic here.
When the difference in votes between two candidates corresponds to a known percentage difference, equate that percentage to the given absolute number and solve for total.
(% difference) × Total = Absolute difference
If A has 45% and B has 35%, the gap is 10%. If 10% of total = 12000, then total = 120000. Direct division — no equations needed.
Convert every percentage change to a decimal multiplier immediately. Chain multiplications instead of computing step by step.
Example: 40% markup then 15% discount. Standard method: Compute MP = 140, then SP = 140 × 0.85 = 119. Two separate multiplications. ~40 seconds. Shortcut: Write 1.40 × 0.85 in one step. Note 1.4 × 0.85 = 1.4 × (1 - 0.15) = 1.4 - 0.21 = 1.19. Done in ~12 seconds. The product minus 1 gives you the profit/loss fraction directly. 0.19 = 19% profit.
For two successive percentage changes a and b (with signs), the net change is a + b + ab/100.
Example: +25% then -20%.
Standard method: 100 → 125 → 100. Three steps, two multiplications. ~35 seconds.
Shortcut: 25 + (-20) + (25 × (-20)/100) = 5 - 5 = 0. Mental arithmetic in ~8 seconds.
The ab/100 term is the second-order correction. When both are positive, it adds; when signs differ, it subtracts. This is the term most people miss in exam conditions.
Never set up a full equation for income problems. Identify the savings percentage directly by subtracting all expense percentages from 100.
Example: 60% on food, 20% on rent, saves Rs. 4000. Standard method: Let income = x, set up 0.20x = 4000. ~25 seconds. Shortcut: Savings % = 100 - 60 - 20 = 20%. Income = 4000 / 0.20 = 4000 × 5 = 20000. Mental division by a unit fraction. ~10 seconds. Works for any number of expense categories. Just subtract them all from 100 and divide savings by that fraction.
%(Neither) = 100 - %A - %B + %(both).
Example: 45% like tea, 35% like coffee, 20% like both. Standard method: Draw a Venn diagram, fill in regions. ~40 seconds. Shortcut: Neither = 100 - 45 - 35 + 20 = 40. Straight arithmetic, 4 numbers, ~8 seconds. The critical step people get wrong: they subtract the overlap twice (once from each set) and forget to add it back. The formula automatically corrects for that.
To find original value before n periods of r% depreciation, divide the current value by (1 - r/100)^n. Do not add the percentage back to the current value.
Example: 15% annual depreciation, current value 68000, find value 2 years ago.
Wrong approach: 68000 + 15% + 15% ≈ 90000. This is incorrect because the percentages apply to different bases.
Correct shortcut: (0.85)^2 = 0.7225. Then 68000 / 0.7225. Note 0.7225 = 5776/8000 — or just compute 68000 / 0.7225 ≈ 94118.
Standard method (trial-and-error with large numbers): ~90 seconds. Division method: ~30 seconds once you know 0.85^2.
Read the question and immediately classify it into one of five templates:
Template 1 — Equation type ("X% of N + constant = N"): Isolate N algebraically, use fraction equivalents for speed.
Template 2 — Residual type (income/savings/expenses): Add up all given percentages, subtract from 100, divide the given amount by that residual fraction.
Template 3 — Mark-up + discount type: Multiply (1 + markup/100) × (1 - discount/100), subtract 1, convert to percentage.
Template 4 — Set theory type (survey/preference with "both" and "neither"): Apply Neither = 100 - A - B + Both. Done in one line.
Template 5 — Growth/decay type: Apply Final = Initial × (multiplier)^n. For reverse problems, divide Final by (multiplier)^n.
Template 6 — Difference type (elections, comparisons): Express the difference as a percentage of total, equate to the given absolute difference, divide.
If a question feels hard, you have probably mis-classified it. Re-read the first and last sentences — the first sentence gives you the template, the last sentence tells you what you are solving for.
Why this question — This is the canonical mark-up-and-discount question. The trap is applying both percentages to the same base.
Solving path: Fix CP = 100. MP = 100 × 1.40 = 140. SP = 140 × 0.85 = 119. Profit = 19 on a base of 100 → 19%. The shortcut: 1.40 × 0.85 = 1.19, profit = 19%, directly from the multiplier.
Why this question — Tests whether you can set up and solve a single-variable percentage equation without over-complicating it.
Solving path: 0.25x + 75 = x → 75 = 0.75x → x = 75 × (4/3) = 100. Recognize 0.75 = 3/4, so multiply by its reciprocal 4/3. This is faster than decimal division.
Why this question — Classic residual savings problem. Gets people who try to set up a full equation instead of reading off the savings percentage.
Solving path: Expenses = 60% + 20% = 80%. Savings = 20%. So 20% of income = 4000. Income = 4000 × 5 = 20000. That multiplication by 5 is because 20% = 1/5, so the whole = 5 times the part.
Why this question — Tests the union formula. Many candidates add 45 + 35 = 80 and subtract from 100, forgetting to account for the overlap correctly.
Solving path: People who like tea or coffee = 45 + 35 - 20 = 60%. Neither = 100 - 60 = 40%. The subtraction of 20% removes the double-counting of people who like both.
Why this question — Election percentage difference problem. The key is recognizing that the 12000-vote margin corresponds directly to the percentage gap, not to valid votes only.
Solving path: A got 45%, B got 35% of total votes polled. Margin = 10% of total = 12000. Total = 12000 / 0.10 = 120000. Note: the 20% invalid votes are a distractor — you are already working with percentages of total votes, so invalid votes do not affect the difference calculation between A and B.
Why this question — Compound growth over two periods. Tests whether you apply growth multiplicatively or additively (a very common mistake).
Solving path: Year 1: 50000 × 1.10 = 55000. Year 2: 55000 × 1.10 = 60500. The common wrong answer is 60000, which comes from 50000 × 1.20 (adding 10% twice instead of compounding). The 500-rupee difference is the interest-on-interest term.
Why this question — Reverse depreciation. Tests whether you understand that percentages apply to the value at the start of each period, not the current value.
Solving path: V × (0.85)^2 = 68000. V = 68000 / 0.7225 ≈ 94118. To compute 0.7225: (0.85)^2 = 0.85 × 0.85. Think of it as (1 - 0.15)^2 = 1 - 0.30 + 0.0225 = 0.7225. Then 68000 / 0.7225 — the answer 94118 matches option A directly, so stop once you confirm the denominator is around 0.72.
Why this question — Successive percentage change. The net result of 0% is counterintuitive enough that it appears as an AFCAT question precisely to catch students who add percentages.
Solving path: Use the formula: 25 + (-20) + (25 × (-20)/100) = 25 - 20 - 5 = 0. Or use multipliers: 1.25 × 0.80 = 1.00. Either way, the net change is exactly 0%. Do not add 25 and -20 to get 5% — that ignores the second-order term.
Applying discount to Cost Price instead of Marked Price. In mark-up-discount problems, the discount is always on MP. The profit is always calculated relative to CP. Confusing the two bases is the single most common error in this category.
Adding successive percentages instead of multiplying multipliers. A 10% growth followed by another 10% growth is not 20% total growth. It is 1.10 × 1.10 = 1.21, which is 21%. The 1% difference comes from 10% applied to the already-grown quantity.
Forgetting to subtract the overlap in set problems. %A + %B double-counts the people in both categories. You must subtract %(A and B) once. The formula is |A ∪ B| = |A| + |B| - |A ∩ B|, not |A| + |B|.
Adding depreciation back to the current value to find original value. If a machine is worth Rs. 68000 after two years of 15% depreciation, the original value is not 68000 + 15% + 15%. Each year's percentage applies to the value at the beginning of that year, not the final value. Always divide by the multiplier.
Treating "20% votes were invalid" as a separate deduction in election problems. When candidates' vote shares are given as percentages of total votes polled, the margin between them is simply the percentage difference of total votes. Invalid votes are already embedded in the total; they do not change the A-minus-B calculation.
Using (1 + r/100)^n for simple growth instead of compound growth (and vice versa). Population growth and compound interest are multiplicative — use the exponent form. Flat percentage problems ("salary increased by 10%") are single-period — use × 1.10 once. Misidentifying the model leads to the wrong formula even when the arithmetic is correct.