Number System
Types of Numbers
Natural Numbers (N)
1, 2, 3, 4 ... (counting numbers, starts at 1)
Whole Numbers (W)
0, 1, 2, 3 ... (natural + zero)
Integers (Z)
...-3, -2, -1, 0, 1, 2, 3... (positive + negative + 0)
Rational Numbers (Q)
p/q form where q≠0. Includes all integers, fractions, terminating & recurring decimals.
Irrational Numbers
√2, √3, π, e — cannot be expressed as p/q. Non-terminating, non-recurring.
Real Numbers (R)
Rational ∪ Irrational. All numbers on the number line.
Prime & Composite Numbers
Prime: Exactly 2 factors (1 and itself). 2 is the only even prime. 1 is neither prime nor composite.
Primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 — 25 primes total.
Divisibility Rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit even (0,2,4,6,8) | 348 ✓ |
| 3 | Sum of digits divisible by 3 | 123 → 1+2+3=6 ✓ |
| 4 | Last 2 digits divisible by 4 | 1324 → 24÷4 ✓ |
| 5 | Last digit 0 or 5 | 745 ✓ |
| 6 | Divisible by both 2 and 3 | 312 ✓ |
| 7 | Double last digit, subtract from rest; repeat if needed | 161 → 16-(2×1)=14 ✓ |
| 8 | Last 3 digits divisible by 8 | 1256 → 256÷8 ✓ |
| 9 | Sum of digits divisible by 9 | 729 → 7+2+9=18 ✓ |
| 11 | (Sum of odd-position digits) – (Sum of even-position digits) = 0 or multiple of 11 | 1331 → (1+3)–(3+1)=0 ✓ |
| 25 | Last 2 digits divisible by 25 | 1475 → 75÷25 ✓ |
HCF & LCM
HCF × LCM = Product of two numbers
HCF (GCD)
- Largest number that divides all given numbers
- HCF of fractions = HCF of numerators / LCM of denominators
LCM
- Smallest number divisible by all given numbers
- LCM of fractions = LCM of numerators / HCF of denominators
For "largest number dividing a, b, c leaving same remainder r" → HCF of (a−b), (b−c), (a−c). For "leaving remainders r₁, r₂ …" → HCF of (a−r₁), (b−r₂), …
Unit Digit Cyclicity
| Base | Cycle | Pattern |
|---|---|---|
| 2 | 4 | 2,4,8,6,2,4,8,6... |
| 3 | 4 | 3,9,7,1,3,9,7,1... |
| 4 | 2 | 4,6,4,6... |
| 7 | 4 | 7,9,3,1,7,9,3,1... |
| 8 | 4 | 8,4,2,6,8,4,2,6... |
| 0,1,5,6 | 1 | Always same digit |
| 9 | 2 | 9,1,9,1... (odd→9, even→1) |
To find unit digit of a^n: find (n mod cycle). If remainder=0, use last value in cycle. Example: 7^53 → 53 mod 4 = 1 → unit digit = 7¹ = 7
The HCF of two numbers is 11 and their LCM is 693. If one number is 77, what is the other?SSC CGL 2019 pattern
What is the unit digit of 3^75?Direct application of cyclicity
Percentage
Core Formulas
x% of y = (x × y) / 100
% Change = [(New − Old) / Old] × 100
Net % change = a + b + (ab/100) [use –b for decrease]
Fraction ↔ Percentage Conversions (Must memorise)
| Fraction | % | Fraction | % |
|---|---|---|---|
| 1/2 | 50% | 1/7 | 14.28% |
| 1/3 | 33.33% | 1/8 | 12.5% |
| 1/4 | 25% | 1/9 | 11.11% |
| 1/5 | 20% | 1/10 | 10% |
| 1/6 | 16.67% | 1/11 | 9.09% |
| 2/3 | 66.67% | 3/4 | 75% |
| 2/5 | 40% | 3/5 | 60% |
| 4/5 | 80% | 5/6 | 83.33% |
Key Relationships
If A is x% more than B → B is [x/(100+x)] × 100 % less than A
If A is x% less than B → B is [x/(100−x)] × 100 % more than A
Two successive discounts of a% and b% = Net discount = a + b − ab/100
Example: 20% then 10% = 20+10−2 = 28% net discount
A number is increased by 20% and then decreased by 20%. Net change?Successive % application
Profit & Loss
Fundamental Formulas
Profit% = (Profit / CP) × 100
SP = CP × (100 + P%) / 100
Loss% = (Loss / CP) × 100
SP = CP × (100 − L%) / 100
CP = SP × 100 / (100 + P%) OR CP = SP × 100 / (100 − L%)
Discount Formulas
Discount = Marked Price (MP) − Selling Price (SP)
Discount% = (Discount / MP) × 100
SP = MP × (100 − D%) / 100
P% = [(MP − CP) / CP] × 100
Dishonest Dealer / False Weight
Profit% = [(True weight − False weight) / False weight] × 100
If a dealer uses 900g weight instead of 1kg: P% = (1000−900)/900 × 100 = 11.11%
Two Items — Same CP/SP Situations
Selling 2 items at same SP, one at x% profit and other at x% loss → Always a loss of (x/10)² % = x²/100 %
A shopkeeper marks his goods 40% above cost price and gives 25% discount. Profit %?MP and discount problem
Two articles sold at Rs 990 each. One at 10% profit, other at 10% loss. Net result?Classic SSC trap
Simple & Compound Interest
Simple Interest
SI = (P × R × T) / 100
A = P + SI = P(1 + RT/100)
Compound Interest
A = P × (1 + R/100)ⁿ
CI = A − P = P[(1 + R/100)ⁿ − 1]
Half-Yearly & Quarterly Compounding
| Type | Rate becomes | Time becomes |
|---|---|---|
| Half-yearly | R/2 | 2n |
| Quarterly | R/4 | 4n |
| Monthly | R/12 | 12n |
Key SI vs CI Differences
For same P, R, T:
CI − SI (2 years) = P(R/100)²
CI − SI (3 years) = P(R/100)²(3 + R/100)
10% CI for 2 years = 10+10+(10×10)/100 = 21% effective rate. Use the successive % formula.
At what rate % CI, Rs 800 amounts to Rs 882 in 2 years?
Ratio & Proportion
Key Concepts
a:b = c:d → ad = bc (product of means = product of extremes)
Types of Ratios
Duplicate Ratio
a²:b² (square of ratio a:b)
Sub-duplicate
√a:√b
Triplicate
a³:b³
Componendo
(a+b):(a−b) from a:b
Dividendo
(a−b):b
Componendo-Dividendo
(a+b)/(a−b) = (c+d)/(c−d)
Partnership
Simple partnership: Profit ratio = ratio of capitals (same time)
Compound partnership: Profit ratio = Capital × Time
A invests ₹12000 for 6 months, B invests ₹8000 for 12 months. Profit ratio = (12000×6):(8000×12) = 72000:96000 = 3:4
Time & Work
Core Approach
If A can do work in n days, A's 1-day work = 1/n. Assume total work = LCM of days for easier calculation.
A+B together finish in = (a×b)/(a+b) days
Pipes & Cisterns
Inlet pipe fills → positive work. Outlet pipe drains → negative work.
Net rate = Sum of all inlet rates − Sum of all outlet rates.
Work & Wages
Wages are distributed in the ratio of work done. If A:B work ratio is 3:2, wages are also 3:2.
Problem: A can finish in 12 days, B in 15 days, C in 20 days. A and B work for 3 days, then C joins. How many more days?
Solution: LCM(12,15,20) = 60 = total work.
A's rate = 5/day, B's rate = 4/day, C's rate = 3/day
Work by A+B in 3 days = (5+4)×3 = 27
Remaining = 60−27 = 33
Rate of A+B+C = 5+4+3 = 12/day
More days = 33/12 = 2.75 days
A pipe fills a tank in 6 hours. A leak drains it in 12 hours. How long to fill with both?
Speed, Distance & Time
Core Formulas
Unit Conversions
km/h → m/s: multiply by 5/18
m/s → km/h: multiply by 18/5
Trains
| Situation | Distance to cover |
|---|---|
| Train crosses a pole/man | Length of train |
| Train crosses a platform/bridge | Length of train + Length of platform |
| Two trains same direction | Sum of lengths / |S1−S2| |
| Two trains opposite direction | Sum of lengths / (S1+S2) |
Boats & Streams
Speed upstream = Boat speed − Stream speed
Speed downstream = Boat speed + Stream speed
Boat speed = (downstream + upstream) / 2
Stream speed = (downstream − upstream) / 2
Average Speed
Avg Speed = 2ab/(a+b) [harmonic mean, NOT arithmetic mean!]
Going 60 km/h, returning 40 km/h → Average ≠ 50. Average = 2×60×40/100 = 48 km/h
Geometry
Lines & Angles
Supplementary Angles
Sum = 180°
Complementary Angles
Sum = 90°
Vertically Opposite
Always equal
Transversal
Alternate interior = equal (parallel lines). Co-interior = 180°.
Triangles
Sum of angles = 180°. Exterior angle = sum of two non-adjacent interior angles.
Important Triangle Properties
- Centroid (G): Intersection of medians. Divides each median in 2:1 from vertex.
- Incentre (I): Intersection of angle bisectors. Equidistant from all three sides.
- Circumcentre (O): Intersection of perpendicular bisectors. Equidistant from all vertices.
- Orthocentre (H): Intersection of altitudes.
- For equilateral triangle: G, I, O, H all coincide.
Pythagoras Theorem & Triples
a² + b² = c² (c = hypotenuse)
Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41) — memorise multiples too.
Circles
| Property | Statement |
|---|---|
| Angle at centre | Twice the angle at circumference on same arc |
| Angles in same segment | Equal |
| Angle in semicircle | Always 90° |
| Cyclic quadrilateral | Opposite angles sum to 180° |
| Tangent-radius | Always perpendicular |
| Tangents from external point | Equal length |
| Chord-tangent angle | = Angle in alternate segment (Tangent-Chord theorem) |
Mensuration
2D Shapes
| Shape | Area | Perimeter |
|---|---|---|
| Square (side a) | a² | 4a |
| Rectangle (l×b) | l×b | 2(l+b) |
| Triangle | ½ × base × height | a+b+c |
| Equilateral △ (side a) | (√3/4)a² | 3a |
| Circle (radius r) | πr² | 2πr |
| Trapezium | ½(a+b)×h | a+b+c+d |
| Rhombus (diagonals d1,d2) | ½×d1×d2 | 4×side |
| Parallelogram | base × height | 2(a+b) |
3D Shapes
| Shape | Volume | Lateral SA | Total SA |
|---|---|---|---|
| Cube (a) | a³ | 4a² | 6a² |
| Cuboid (l,b,h) | lbh | 2h(l+b) | 2(lb+bh+hl) |
| Cylinder (r,h) | πr²h | 2πrh | 2πr(r+h) |
| Cone (r,h,l) | ⅓πr²h | πrl | πr(r+l) |
| Sphere (r) | ⁴⁄₃πr³ | — | 4πr² |
| Hemisphere (r) | ⅔πr³ | 2πr² | 3πr² |
Slant height of cone: l = √(r² + h²)
Trigonometry
Trigonometric Ratios
sin θ = P/H · cos θ = B/H · tan θ = P/B
cosec θ = H/P · sec θ = H/B · cot θ = B/P
Standard Angle Values
| θ | sin | cos | tan | cosec | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | ∞ | 1 | ∞ |
| 30° | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | ∞ | 1 | ∞ | 0 |
sin 0°, 30°, 45°, 60°, 90° = √(0/4), √(1/4), √(2/4), √(3/4), √(4/4) = 0, 1/2, 1/√2, √3/2, 1
Key Identities
Heights & Distances
Angle of Elevation: Looking UP. tan θ = height / horizontal distance
Angle of Depression: Looking DOWN. Angle of depression = angle of elevation (alternate angles)
Algebra
Important Identities
If a+b+c = 0, then a³+b³+c³ = 3abc
Key SSC CGL Algebra Questions
If x + 1/x = 3:
x² + 1/x² = (x+1/x)² − 2 = 9−2 = 7
x³ + 1/x³ = (x+1/x)³ − 3(x+1/x) = 27−9 = 18
x⁴ + 1/x⁴ = (x²+1/x²)² − 2 = 49−2 = 47
Mixture & Alligation
Rule of Alligation
Cheaper : Dearer = (Mean price − Cheaper price) : (Dearer price − Mean price)
Draw a cross. Write cheaper (c) at top-left, dearer (d) at top-right, mean (m) in centre.
Ratio = (d−m) : (m−c)
Example: Mix ₹60/kg and ₹90/kg to get ₹80/kg
Ratio = (90−80):(80−60) = 10:20 = 1:2
Repeated Dilution
Remaining pure = P × (1 − r/V)ⁿ
(P=initial, r=removed each time, V=total volume)
Data Interpretation
Types of DI Questions
Bar Charts
Compare values, find % change between years, ratio between categories.
Line Graphs
Find trends, % increase/decrease, max/min rate of change.
Pie Charts
Central angle = (value/total)×360. % share = (value/total)×100.
Tables
Row/column sums, averages, percentage comparisons across rows/columns.
DI Approach
- Read all questions first, then analyse the chart
- Approximate: 33% ≈ ⅓, 25% = ¼ — use fractions for speed
- For % change: (New−Old)/Old×100. Check sign (increase/decrease)
- For ratios: simplify before calculating
- Don't calculate exact values if options are far apart — estimate
For Pie chart with total = 1200: 30° slice = (30/360)×1200 = 100. Always: (degrees/360) × total.