Master Maths: Essential Revision Notes & Strategies for SSC, UPSC, Bank, Railway & More

The Ultimate Mains Revision Guide

Every Concept Covered for All Competitive Exams

🚀 Quick Revision Index

1. Number System
2. HCF & LCM
3. Simplification
4. Percentages
5. Profit & Loss
6. Ratio & Proportion
7. Time, Speed & Distance
8. Time & Work
9. SI & CI
10. Algebra
11. Geometry
12. Mensuration
13. Data Interpretation
14. Trigonometry
15. Statistics
16. Permutations, Combinations & Probability
17. Logarithms
18. Bonus Study Tips

Share on WhatsApp Share on Telegram Share on Facebook

🔢

1. Number System

Fundamental Concepts

Types of Numbers

• Natural Numbers (N): Counting numbers. {1, 2, 3, ...}
• Whole Numbers (W): Natural numbers including zero. {0, 1, 2, 3, ...}
• Integers (Z): Positive and negative whole numbers. {..., -2, -1, 0, 1, 2, ...}
• Rational Numbers (Q): Any number that can be written as p/q, where q ≠ 0.
• Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., √2, π).
• Prime Numbers: Numbers > 1 with only two factors: 1 and itself (e.g., 2, 3, 5, 11).
• Composite Numbers: Numbers > 1 with more than two factors (e.g., 4, 6, 8, 9).
• Co-prime Numbers: Two numbers whose HCF is 1 (e.g., 8 and 9).

Place Value & Face Value

In 573, the Face Value of 7 is 7. The Place Value of 7 is 70.

Exam-Oriented Concepts & Shortcuts

Divisibility Rules

• by 2, 4, 8: Check if last 1, 2, or 3 digits are divisible by 2, 4, or 8 respectively.
• by 3, 9: Check if the sum of digits is divisible by 3 or 9 respectively.
• by 6: Check divisibility by both 2 and 3.
• by 11: (Sum of odd position digits) − (Sum of even position digits) is 0 or divisible by 11.

Unit Digit of Powers

Use cycles. For 7²³, the cycle of 7 is {7, 9, 3, 1}. Remainder of 23÷4 is 3. The 3rd digit is 3.

Remainder Theorem

Dividend = (Divisor × Quotient) + Remainder. A key application is finding remainders of polynomials.

Progression Formulas

• AP n^th term: a + (n-1)d
• AP Sum of n terms: (n/2) [2a + (n-1)d]
• GP n^th term: ar^n-1
• GP Sum of n terms: a(rⁿ-1)/(r-1)

🔢

2. HCF & LCM

Fundamental Concepts

• HCF (Highest Common Factor): The largest number that divides two or more given numbers perfectly.
• LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more given numbers.

Exam-Oriented Concepts & Shortcuts

The Product Rule

For two numbers 'a' and 'b': HCF(a, b) × LCM(a, b) = a × b

HCF and LCM of Fractions

• HCF of Fractions = HCF of Numerators / LCM of Denominators
• LCM of Fractions = LCM of Numerators / HCF of Denominators

🔍

3. Simplification & Approximation

Fundamental Concepts

Order of Operations (VBODMAS)

The correct sequence: Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction.

Laws of Indices (Exponents)

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• a⁰ = 1

Exam-Oriented Concepts & Shortcuts

Square Root Approximation

√(a² + b) ≈ a + b/(2a) (for small b)

📊

4. Percentages

Fundamental Concepts

A percentage is a fraction out of 100.

Exam-Oriented Concepts & Shortcuts

Memorizing Key Fractions

10%=1/10, 12.5%=1/8, 20%=1/5, 25%=1/4, 33.33%=1/3, 50%=1/2, 75%=3/4

Successive Percentage Change

Net % Change = A + B + (AB/100). If a value increases and decreases by the same x%, net change is a loss of `-x²/100`.

🛒

5. Profit, Loss, Discount

Fundamental Concepts

• CP: Buying price. SP: Selling price. MP: Label price.

Exam-Oriented Concepts & Shortcuts

SP = MP × (100 - Discount%)/100 = CP × (100 + Gain%)/100

Successive Discounts

Single equivalent discount for A% and B% is `A + B - (AB/100)`.

When CP of X articles = SP of Y articles

Profit/Loss % = `[(X-Y)/Y] × 100`.

Previous Year Question Spotlight

(SSC CGL) A shopkeeper sells an item for ₹1,440 after giving a 20% discount on the marked price. Had he not given any discount, the profit would have been 50%. What is the cost price of the item?

Solution Approach: First, find the Marked Price (MP) using the discount. MP = 1440 / (1 - 0.20) = ₹1800. If no discount is given, SP = MP = ₹1800. This SP gives a 50% profit. So, CP = 1800 / 1.50 = ₹1200.

🤝

6. Ratio & Proportion

Fundamental Concepts

• Ratio (a:b): Compares two similar quantities.
• Proportion (a:b::c:d): An equality of two ratios.
• Partnership: Profit is shared in the ratio of (Investment × Time).

Exam-Oriented Concepts & Shortcuts

Combining Ratios

If A:B=m:n and B:C=p:q, then A:B:C = mp : np : nq.

Mean/Third/Fourth Proportional

Mean proportional to a,b is √ab. Third proportional to a,b is b²/a. Fourth proportional to a,b,c is (b×c)/a.

🕰️

7. Time, Speed & Distance

Fundamental Concepts

• Core Formula: Distance = Speed × Time.
• Relative Speed (Opposite): Speeds Add (S1+S2).
• Relative Speed (Same): Speeds Subtract (|S1-S2|).
• Boats & Streams: Downstream Speed (u) = B+S; Upstream Speed (v) = B-S.

Exam-Oriented Concepts & Shortcuts

Average Speed (for equal distances)

Formula: `(2xy) / (x+y)`

Problems on Trains

Time to cross a pole = Length of Train / Speed.
Time to cross a platform = (Length of Train + Platform) / Speed.

👷

8. Time & Work

Fundamental Concepts

Total Work = Efficiency × Time.

Exam-Oriented Concepts & Shortcuts

LCM Method

Example: A=12 days, B=18 days. Together? Work = LCM(12,18)=36. Efficiencies: A=3, B=2. Combined=5. Time = 36/5 = 7.2 days.

M-D-H Formula

`(M1×D1×H1)/W1 = (M2×D2×H2)/W2`

Previous Year Question Spotlight

(IBPS PO) A can do a piece of work in 20 days and B can do it in 30 days. They work together for 7 days and then both leave the work. Then C alone finishes the remaining work in 10 days. In how many days will C finish the full work?

Solution Approach: Use the LCM method. Total Work = LCM(20, 30) = 60 units. A's efficiency = 3, B's = 2. Together their efficiency is 5. Work done in 7 days = 5 * 7 = 35 units. Remaining work = 60 - 35 = 25 units. C does 25 units in 10 days, so C's efficiency = 25/10 = 2.5 units/day. Time for C to do the full 60 units = 60 / 2.5 = 24 days.

💰

9. SI & CI

Fundamental Concepts

• Simple Interest (SI): Interest on original principal only. `SI = (P×R×T)/100`.
• Compound Interest (CI): Interest on principal + accumulated interest.

Exam-Oriented Concepts & Shortcuts

Amount Formulas

Amount (A) = Principal (P) + Interest (I)
For CI, `A = P(1 + R/100)<sup>T</sup>`. Then `CI = A - P`.

Compounding Variations

If compounded half-yearly: Rate = R/2, Time = 2T.
If compounded quarterly: Rate = R/4, Time = 4T.

CI-SI Difference Formulas

For 2 years: `P(R/100)²`
For 3 years: `P(R/100)² (3 + R/100)`
Example: Difference on ₹1000 for 2 years at 10% p.a. = 1000(10/100)² = ₹10.

Installments

For CI: `Amount = x/(1+R/100) + x/(1+R/100)² + ...` where x is each installment.

🧮

10. Algebra

Fundamental Concepts

• Linear Equation: ax+b=0.
• Quadratic Equation (ax²+bx+c=0): Roots = `[-b ± √(b²-4ac)] / 2a`. Sum of roots = -b/a. Product of roots = c/a.
• Algebraic Expressions: Combinations of variables and constants with algebraic operations (e.g., 3x² + 2y - 5).

Exam-Oriented Identities

(a+b)² = a²+b²+2ab
(a-b)² = a²+b²-2ab
a²-b² = (a+b)(a-b)
(a+b+c)² = a²+b²+c²+2(ab+bc+ca)
a³+b³ = (a+b)(a²-ab+b²)
a³-b³ = (a-b)(a²+ab+b²)
a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)
If a+b+c=0, then a³+b³+c³ = 3abc

Direct Application

If x + 1/x = k, then x²+1/x² = k²-2, and x³+1/x³ = k³-3k.

Solving Equations and Inequalities

• Solving Equations: The goal is to isolate the variable. For linear equations, use basic arithmetic. For quadratic equations, use factorization or the quadratic formula.
• Inequalities: Solved like equations, but remember to reverse the inequality sign if you multiply or divide by a negative number. Example: `5 - 2x > 1` → `-2x > -4` → `x < 2`.

📐

11. Geometry

Fundamental Concepts

• Triangles: Sum of angles=180°.
• Triangle Centers: Centroid, Incenter, Circumcenter, Orthocenter.
• Circles: Properties of chords, tangents, and secants.

Exam-Oriented Theorems & Formulas

Key Theorems

Apollonius', Angle Bisector, Alternate Segment, Mid-Point Theorems.

Triangle Rules

Sine Rule: a/sinA = b/sinB = c/sinC.
Cosine Rule: cosA = (b²+c²-a²)/2bc.

Polygons

Sum of Interior Angles = (n-2)×180°. Diagonals = n(n-3)/2.

📦

12. Mensuration

Formulas for 2D Shapes

Shape	Area	Perimeter
Square	a²	4a
Rectangle	l×b	2(l+b)
Circle	πr²	2πr
Equilateral Triangle	(√3/4)a²	3a

Formulas for 3D Shapes

Shape	Volume	Curved SA	Total SA
Cube	a³	4a²	6a²
Cuboid	l×b×h	2h(l+b)	2(lb+bh+hl)
Cylinder	πr²h	2πrh	2πr(r+h)
Cone	⅓πr²h	πrl	πr(r+l)
Sphere	(4/3)πr³	4πr²	4πr²

📊

13. Data Interpretation

Fundamental Concepts

Reading data from Bar Graphs, Line Graphs, Pie Charts, and Tables.

Exam-Oriented Strategy

1. Read the Question First.
2. Approximate.
3. Master Percentages & Averages.

Analyzing Data: A Practical Example

Scenario: A pie chart shows the monthly expenditure of a family. The total monthly income is ₹36,000. The sectors are: Food (30%), Rent (20%), Education (15%), Savings (15%), and Others (20%).

Typical Question: What is the central angle for the 'Education' sector?

Analysis & Solution:
A full circle is 360°. The 'Education' sector represents 15% of the total.
Central Angle = 15% of 360° = (15/100) × 360° = 54°.
To find the amount spent on food: 30% of ₹36,000 = (30/100) × 36000 = ₹10,800.

📏

14. Trigonometry

Fundamental Concepts & Values Table

θ°	sinθ	cosθ	tanθ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

Exam-Oriented Formulas

Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Max & Min Values

Max/min value of `a sinθ + b cosθ` is `±√(a²+b²)`.

Height & Distance

This is an application of trigonometry to find the height of objects or distance between them, which are not easily measurable.

• Angle of Elevation: The angle formed when an observer looks up at an object.
• Angle of Depression: The angle formed when an observer looks down at an object.

Example: The angle of elevation of the top of a tower from a point 100m away is 30°. The height of the tower is `h = distance × tan(θ)` = 100 × tan(30°) = 100 × (1/√3) = 100/√3 m.

📈

15. Statistics

Fundamental Concepts

• Mean: The average.
• Median: The middle value of sorted data.
• Mode: The most frequent value.

Exam-Oriented Concepts & Shortcuts

Empirical Relationship

For moderately skewed data: `Mode ≈ 3(Median) - 2(Mean)`.

🎲

16. Permutations, Combinations & Probability

Fundamental Concepts

• Factorial (n!): The product of all positive integers up to n. `n! = n × (n-1) × ... × 1`. (0! = 1)
• Permutation (ⁿP_r): Arrangement. The number of ways to arrange 'r' items from a set of 'n' items. Order matters. `<sup>n</sup>P<sub>r</sub> = n! / (n-r)!`.
• Combination (ⁿC_r): Selection. The number of ways to choose 'r' items from a set of 'n' items. Order does not matter. `<sup>n</sup>C<sub>r</sub> = n! / [r! (n-r)!]`.
• Probability P(E): The likelihood of an event 'E' occurring. `P(E) = (Favorable Outcomes) / (Total Outcomes)`.

Exam-Oriented Concepts

• Key Probability Rule: For events A and B, `P(A or B) = P(A) + P(B) - P(A and B)`.
• Deck of Cards: 52 cards total, 4 suits (Hearts, Diamonds, Clubs, Spades), 13 cards per suit, 12 face cards (J, Q, K), 4 Aces.
• Dice Problems: For a single die, total outcomes = 6. For two dice, total outcomes = 6×6 = 36.

🪵

17. Logarithms

Fundamental Concepts

A logarithm answers the question "what exponent do we need to raise a base to, to get another number?". If `b<sup>c</sup> = a`, then `log<sub>b</sub>(a) = c`.

• Common Logarithm: Base 10, written as log(x).
• Natural Logarithm: Base e (≈2.718), written as ln(x).

Key Properties & Formulas

• Product Rule: `log<sub>b</sub>(mn) = log<sub>b</sub>(m) + log<sub>b</sub>(n)`
• Quotient Rule: `log<sub>b</sub>(m/n) = log<sub>b</sub>(m) - log<sub>b</sub>(n)`
• Power Rule: `log<sub>b</sub>(m<sup>p</sup>) = p × log<sub>b</sub>(m)`
• Change of Base Rule: `log<sub>b</sub>(a) = log<sub>c</sub>(a) / log<sub>c</sub>(b)`
• Important Values: `log<sub>b</sub>(1) = 0`, `log<sub>b</sub>(b) = 1`

🎓

18. Bonus Study Tips

Pro-Tips to Ace the Quant Section

• Master the Basics First: Strong fundamentals make shortcuts intuitive.
• Daily Formula Revision: Spend 15 minutes every morning just reading this guide.
• Practice with a Timer: Solve questions under exam-like time constraints.
• Analyze Your Mocks: Focus on why you made a mistake, not just that you made one.
• Solve Previous Year Questions (PYQs): This is the best way to understand the exam pattern, question difficulty, and important topics.