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🔢 Complete Mathematics Guide

Master all mathematics topics with shortcuts for JKSSB & Competitive Exams

📑 Topics Covered

1. 🔢 Number System

📌 Types of Numbers

Natural Numbers: 1, 2, 3, 4, ... (Positive integers) Whole Numbers: 0, 1, 2, 3, ... (Natural + 0) Integers: ..., -2, -1, 0, 1, 2, ... (Positive, negative, zero) Prime Numbers: 2, 3, 5, 7, 11, 13, ... (Divisible only by 1 and itself) Composite Numbers: 4, 6, 8, 9, 10, ... (Has more than 2 factors)

🎯 Quick Tricks

Divisibility Rules:

• Divisible by 2: Last digit is even

• Divisible by 3: Sum of digits is divisible by 3

• Divisible by 5: Last digit is 0 or 5

• Divisible by 9: Sum of digits is divisible by 9

• Divisible by 11: Alternating sum of digits is divisible by 11

Example:

Is 1234 divisible by 3?

Solution:

Sum of digits = 1+2+3+4 = 10

10 is not divisible by 3

Answer: NO

✍️ Practice Questions

Q1: Is 5432 divisible by 11?

Show Answer

Answer: YES

Alternating sum = 5-4+3-2 = 2 (Not divisible by 11)

Actually: 5-4+3-2 = 2, so NO

2. 📊 LCM & HCF

📌 Definitions

HCF (Highest Common Factor): Largest number that divides both numbers LCM (Least Common Multiple): Smallest number divisible by both numbers Formula: HCF × LCM = Product of two numbers

🎯 Finding LCM & HCF

Prime Factorization Method:

• For HCF: Take common factors with lowest power

• For LCM: Take all factors with highest power

Example:

Find LCM and HCF of 12 and 18

Solution:

12 = 2² × 3

18 = 2 × 3²

HCF = 2 × 3 = 6

LCM = 2² × 3² = 36

Verify: 6 × 36 = 216 = 12 × 18 ✓

✍️ Practice Questions

Q1: Find HCF of 24 and 36

Show Answer

Answer: 12

24 = 2³ × 3, 36 = 2² × 3²

HCF = 2² × 3 = 12

Q2: Find LCM of 15 and 20

Show Answer

Answer: 60

15 = 3 × 5, 20 = 2² × 5

LCM = 2² × 3 × 5 = 60

3. 📈 Percentage

📌 Basic Formula

Percentage = (Part / Whole) × 100 Part = (Percentage / 100) × Whole Whole = (Part × 100) / Percentage

🎯 Percentage Change

Percentage Increase = ((New - Old) / Old) × 100 Percentage Decrease = ((Old - New) / Old) × 100

Example:

If price increases from 100 to 120, what is the percentage increase?

Solution:

Percentage Increase = ((120 - 100) / 100) × 100 = 20%

Answer: 20%

✍️ Practice Questions

Q1: What is 25% of 200?

Show Answer

Answer: 50

(25/100) × 200 = 50

4. 💰 Profit & Loss

📌 Basic Formulas

Profit = Selling Price - Cost Price Loss = Cost Price - Selling Price Profit % = (Profit / CP) × 100 Loss % = (Loss / CP) × 100 SP = CP + Profit

🎯 Quick Shortcuts

If profit is 20%: SP = CP × 1.20 If loss is 20%: SP = CP × 0.80 Marked Price (MP) - Discount = Selling Price

Example:

CP = 100, Profit = 20%. Find SP.

Solution:

SP = CP × (1 + 20/100) = 100 × 1.20 = 120

Answer: 120

✍️ Practice Questions

Q1: CP = 500, SP = 600. Find profit %

Show Answer

Answer: 20%

Profit = 600 - 500 = 100

Profit % = (100/500) × 100 = 20%

5. 🏦 Simple & Compound Interest

📌 Formulas

Simple Interest = (P × R × T) / 100 Amount = P + SI Compound Interest = P(1 + R/100)^T - P Amount = P(1 + R/100)^T

🎯 Where:

P = Principal (initial amount)

R = Rate of interest (% per annum)

T = Time (in years)

SI = Simple Interest

CI = Compound Interest

Example:

P = 1000, R = 10%, T = 2 years. Find SI and Amount.

Solution:

SI = (1000 × 10 × 2) / 100 = 200

Amount = 1000 + 200 = 1200

Answer: SI = 200, Amount = 1200

✍️ Practice Questions

Q1: P = 5000, R = 5%, T = 3. Find CI

Show Answer

Answer: 788.13

Amount = 5000(1.05)³ = 5788.13

CI = 5788.13 - 5000 = 788.13

6. ⏱️ Time & Work

📌 Basic Concept

Work = Rate × Time If A does work in 'a' days, A's 1 day work = 1/a If A and B work together: Combined work = 1/a + 1/b

🎯 Quick Method

If A completes work in 5 days and B in 10 days:

A's work per day = 1/5

B's work per day = 1/10

Combined work per day = 1/5 + 1/10 = 3/10

Time to complete together = 10/3 days

Example:

A completes work in 6 days, B in 12 days. How long together?

Solution:

A's work/day = 1/6, B's work/day = 1/12

Combined = 1/6 + 1/12 = 3/12 = 1/4

Time = 4 days

Answer: 4 days

✍️ Practice Questions

Q1: A does work in 8 days, B in 16 days. Together?

Show Answer

Answer: 5.33 days (or 16/3 days)

Combined = 1/8 + 1/16 = 3/16

Time = 16/3 days

7. 🚗 Time, Speed & Distance

📌 Basic Formula

Distance = Speed × Time Speed = Distance / Time Time = Distance / Speed

🎯 Unit Conversion

km/h to m/s: Multiply by 5/18 m/s to km/h: Multiply by 18/5 Example: 72 km/h = 72 × 5/18 = 20 m/s

Example:

A car travels 300 km in 5 hours. Find speed.

Solution:

Speed = Distance / Time = 300 / 5 = 60 km/h

Answer: 60 km/h

✍️ Practice Questions

Q1: Speed = 45 km/h, Time = 2 hours. Distance?

Show Answer

Answer: 90 km

Distance = 45 × 2 = 90 km

8. ⚖️ Ratio & Proportion

📌 Definitions

Ratio: Comparison of two quantities (a:b) Proportion: Equality of two ratios (a:b = c:d) If a:b = c:d, then a×d = b×c (Cross multiplication)

🎯 Quick Tricks

If ratio is 2:3 and total is 50:

First part = (2/(2+3)) × 50 = 20

Second part = (3/(2+3)) × 50 = 30

Example:

Divide 120 in ratio 3:5

Solution:

First part = (3/8) × 120 = 45

Second part = (5/8) × 120 = 75

Answer: 45 and 75

✍️ Practice Questions

Q1: If 2:3 = x:15, find x

Show Answer

Answer: 10

2 × 15 = 3 × x

30 = 3x, x = 10

9. 📊 Average

📌 Formula

Average = Sum of all values / Number of values Sum = Average × Number of values

🎯 Quick Tricks

Average of consecutive numbers = (First + Last) / 2 Average of 1 to n = (n+1) / 2 Average of 1 to 100 = 50.5

Example:

Average of 10, 20, 30, 40, 50

Solution:

Sum = 10+20+30+40+50 = 150

Average = 150 / 5 = 30

Answer: 30

✍️ Practice Questions

Q1: Average of 5 numbers is 20. What is their sum?

Show Answer

Answer: 100

Sum = 20 × 5 = 100

10. 📐 Algebra Basics

📌 Important Formulas

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² (a + b)(a - b) = a² - b² (a + b)³ = a³ + 3a²b + 3ab² + b³ a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a - b)(a² + ab + b²)

🎯 Solving Linear Equations

General form: ax + b = c

Solution: x = (c - b) / a

Example:

Solve: 2x + 5 = 15

Solution:

2x = 15 - 5 = 10

x = 10 / 2 = 5

Answer: x = 5

✍️ Practice Questions

Q1: Expand (a + 3)²

Show Answer

Answer: a² + 6a + 9

(a + 3)² = a² + 2(a)(3) + 3² = a² + 6a + 9