In this guide
This sheet is for the last few weeks before AFCAT, when you need to see every formula at once and check that you can still use it. Each line has a small example you can redo in your head. If one of them makes you pause, that topic needs another practice set.
The official syllabus puts Numerical Ability at Class 10 level. Alongside the usual arithmetic, it names mensuration, probability, statistics, heights and distances, and exponents, so those are included here too.
Percentage, profit and discount
| Topic | Formula | Example |
|---|---|---|
| Percentage change | (new − old) ÷ old × 100 | 80 → 100: 25% rise |
| Successive changes | a + b + ab/100 | +10% then +20%: 32% |
| x is r% more than y | y is r/(100 + r) × 100 % less than x | r = 25 → y is 20% less |
| Profit % | profit ÷ CP × 100 | CP 400, SP 500: 25% |
| CP from SP | CP = SP × 100 ÷ (100 + profit %) | SP 660 at 10%: CP 600 |
| Successive discounts | a + b − ab/100 | 20% and 10%: 28% |
| Marked price | SP = MP × (100 − discount %) ÷ 100 | MP 800, 15% off: SP 680 |
Why a + b + ab/100 works: multiplying by (1 + a/100)(1 + b/100) gives 1 + a/100 + b/100 + ab/10,000. The last term, written as a percentage, is ab/100. For discounts, a and b are negative, which gives a + b − ab/100.
Full guides: percentage and profit and loss.
Interest
| Topic | Formula | Example |
|---|---|---|
| Simple interest | SI = P × R × T ÷ 100 | ₹5,000, 8%, 3 years: ₹1,200 |
| Compound amount | A = P(1 + R/100)ᵀ | ₹10,000, 10%, 2 years: ₹12,100 |
| CI − SI, 2 years | P(R/100)² | ₹5,000, 10%: ₹50 |
| Half-yearly compounding | Halve R, double T | 10% a year for 1 year → 5% for 2 half-years |
Ratio, averages, mixtures and partnership
| Topic | Formula | Example |
|---|---|---|
| Share from a ratio | part ÷ total parts × amount | 3 : 5 of 64: 24 and 40 |
| Combining ratios | Make the common term equal | A : B = 2 : 3, B : C = 4 : 5 → 8 : 12 : 15 |
| Average | sum ÷ count | 4, 8, 15, 23: 12.5 |
| Replacement | new − old = change in average × count | Average of 10 rises by 2 when a 60 kg person is replaced: new person 80 kg |
| Alligation | cheaper : dearer = (d − m) : (m − c) | c = 30, d = 40, m = 36: 2 : 3 |
| Partnership | profit ratio = capital × time | ₹2,000 for 12 months vs ₹3,000 for 8 months: 1 : 1 |
Time, work, speed and distance
| Topic | Formula | Example |
|---|---|---|
| Work together | ab ÷ (a + b) days | 10 and 15 days: 6 days |
| Fill and empty | ab ÷ (b − a) | Fills in 6 h, empties in 9 h: 18 h |
| Men–days–hours | M₁D₁H₁/W₁ = M₂D₂H₂/W₂ | 12 × 8 × 10 = 16 × 6 × D: D = 10 |
| km/h → m/s | × 5/18 | 72 → 20 |
| Average speed, equal distances | 2ab ÷ (a + b) | 40 and 60: 48 |
| Train passes a pole | train length ÷ speed | 150 m at 20 m/s: 7.5 s |
| Train passes a platform | (train + platform) ÷ speed | 500 m total at 20 m/s: 25 s |
| Boat and stream | boat = (down + up) ÷ 2, stream = (down − up) ÷ 2 | 12 and 8: boat 10, stream 2 |
Number system, HCF and LCM
| Topic | Rule | Example |
|---|---|---|
| HCF × LCM | = product of the two numbers | 30, 36: 6 × 180 = 1,080 = 30 × 36 |
| Divisible by 3 or 9 | Digit sum divisible by 3 or 9 | 7,236: digit sum 18 |
| Divisible by 4 | Last two digits divisible by 4 | 3,516: 16 |
| Divisible by 11 | Alternate digit sums differ by 0 or a multiple of 11 | 2,728: (2 + 2) − (7 + 8) = −11 |
| HCF of fractions | HCF of numerators ÷ LCM of denominators | 2/3, 4/9: 2/9 |
| LCM of fractions | LCM of numerators ÷ HCF of denominators | 2/3, 4/9: 4/3 |
Mensuration
| Shape | Area or surface | Volume |
|---|---|---|
| Rectangle | l × b; perimeter 2(l + b) | — |
| Triangle | ½ × base × height | — |
| Equilateral triangle | (√3/4) a² | — |
| Circle | πr²; circumference 2πr | — |
| Cube | 6a² | a³ |
| Cuboid | 2(lb + bh + hl) | l × b × h |
| Cylinder | curved 2πrh; total 2πr(r + h) | πr²h |
| Cone | curved πrl, where l = √(r² + h²) | ⅓ πr²h |
| Sphere | 4πr² | (4/3) πr³ |
| Hemisphere | curved 2πr²; total 3πr² | (2/3) πr³ |
Examples. A cylinder with r = 7 cm and h = 10 cm has volume 22/7 × 49 × 10 = 1,540 cm³. A cone with r = 3 and h = 4 has slant height √(9 + 16) = 5, so its curved surface is π × 3 × 5 = 15π.
Heights and distances
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Almost every AFCAT-level question uses tan θ = height ÷ distance. If the angle of elevation of a tower's top is 60° from a point 20 m from its foot, the height is 20 × √3 ≈ 34.64 m. At 45°, height equals distance.
Probability and statistics
- Probability = favourable outcomes ÷ total outcomes, between 0 and 1. Two dice give 36 outcomes; a sum of 7 comes from 6 of them, so P = 6/36 = 1/6.
- Mean = sum ÷ count.
- Median is the middle value once the data are sorted. For an even count, average the two middle values. For 3, 7, 8, 12, 15, 20 the median is (8 + 12) ÷ 2 = 10.
- Mode is the most frequent value.
Exponents
| Law | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2⁵ × 2³ = 2⁸ = 256 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁸ ÷ 2⁶ = 2² = 4 |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| a⁰ = 1 | 7⁰ = 1 |
| a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| a to the power 1/n = ⁿ√a | 8 to the power 1/3 = ∛8 = 2 |
Self-test
Try these in 10 minutes without looking at the tables above.
- A price rises from ₹240 to ₹300. What is the percentage increase?
- An article costing ₹800 is sold for ₹920. Find the profit percentage.
- Find the simple interest on ₹6,000 at 7.5% a year for 4 years.
- Find the compound interest on ₹8,000 at 10% a year for 2 years.
- A train 180 m long runs at 54 km/h. How long does it take to pass a pole?
- From a point 25 m from the foot of a tower, the angle of elevation of the top is 45°. How tall is the tower?
- One card is drawn from a standard pack of 52. What is the probability that it is a king?
- Simplify 3⁴ × 3² ÷ 3⁵.
Answers:
- 25%. Rise of 60 on 240.
- 15%. Profit 120 on CP 800.
- ₹1,800. 6,000 × 7.5 × 4 ÷ 100.
- ₹1,680. 8,000 × 1.21 = 9,680.
- 12 seconds. 54 km/h = 15 m/s, and 180 ÷ 15 = 12.
- 25 m. tan 45° = 1.
- 1/13. 4 kings out of 52 cards.
- 3. 3⁴⁺²⁻⁵ = 3¹.
How to use this sheet
- Read one section a day in the last two weeks and redo each example without looking.
- Mark any line you could not redo, and do a 15-question set on that topic the same day.
- Read the whole sheet once more the day before the exam, then stop.
- If the same kind of slip keeps appearing, read common numerical mistakes.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Indian Air Force website .
Get the next AFCAT guide by email
New guides every week. No spam, unsubscribe any time.