👉 Master GED Math — Topic-wise Lessons & Practice 🧠
55%
Algebra & Functions
45%
Other Topics
Algebra Dominates GED Math Exam.
Number & Quantities
Distributive Properties
📌 Distributive Property
The distributive property helps you multiply a number across terms inside brackets.
🔵Formula:
a(b + c) = ab + ac
🧠 How it works:
Multiply the number outside the bracket with each term inside Then simplify the expression
✏️Example 1:
3(x + 4) = 3x + 12
✏️ Example 2:
5(2x − 3) = 10x − 15
⚠️ Common Mistakes:Forgetting to multiply every termSign errors (especially with negative numbers)
💡 Tip:
Always check if you multiplied ALL terms inside the bracket.
LCM
Methods to Find LCM
The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is divisible by all the given numbers.
🔹 Example:Find LCM of 4 and 6: Multiples of 4: 4, 8, 12, 16, 20… Multiples of 6: 6, 12, 18, 24…
👉 Common multiples: 12, 24…
👉 Smallest common multiple = 12
So, LCM of 4 and 6 = 12
🔹 Methods to find LCM:
1. Listing method (write multiples)
2. Prime factorization method
3. Division method
🔹 1. Listing Method (Write Multiples)
In this method, we list the multiples of each number until we find the smallest common multiple.
Steps: Write multiples of each number Find common multiples Choose the smallest one
Example:
Multiples of 3: 3, 6, 9, 12, 15…
Multiples of 5: 5, 10, 15, 20…
👉 Common multiple = 15
So, LCM = 15
🔹 2. Prime Factorization Method
In this method, we break numbers into prime factors.
Steps: Find prime factors of each number Take highest powers of all primes Multiply them
Example:
12 = 2 × 2 × 3
18 = 2 × 3 × 3LCM = 2² × 3² = 36
🔹 3. Division Method
In this method, we divide numbers by common prime numbers step by step.
Steps: Write numbers in a row Divide by smallest prime number Continue until all become 1 Multiply all divisors
Example:
12, 18
÷2 → 6, 9
÷3 → 2, 3
÷2 → 1, 3
÷3 → 1, 1LCM = 2 × 3 × 2 × 3 = 36
✨ Summary:LCM can be found using different methods, but all methods give the same result. Choose the method that is easiest for you!
GCF
🔹 What is GCF?
The Greatest Common Factor (GCF) is the largest positive number that divides two or more numbers exactly without leaving any remainder.It is also known as the Greatest Common Divisor (GCD).
🔹Example:
Find the GCF of 12 and 18:
Factors of 12 → 1, 2, 3, 4, 6, 12
Factors of 18 → 1, 2, 3, 6, 9, 18
✔ Common factors → 1, 2, 3, 6
👉 Greatest common factor = 6
So, GCF of 12 and 18 = 6
🔹 Methods to Find GCF:
1. Listing Factors Method
Write all factors of each number Find common factors Select the greatest one
2. Prime Factorization Method
Break numbers into prime factors Multiply the common prime factors with the lowest powers
Example:
12 = 2 × 2 × 3
18 = 2 × 3 × 3Common factors = 2 × 3
👉 GCF = 6
3. Division Method
Divide numbers by common prime numbers Continue until no common division is possible Multiply all common divisors
✨ Key Point:
The GCF helps us simplify fractions and solve problems involving grouping or sharing equally.
Comparing Fraction
🔹 What are Fractions?
A fraction represents a part of a whole. It has two parts:
Numerator (top number) → parts we have
Denominator (bottom number) → total equal parts
Example: 3/4 means 3 parts out of 4 equal parts.
🔹 What does Comparing Fractions mean?Comparing fractions means finding out which fraction is:
Greater (>)
Smaller (<)
Equal (=)
🔹Methods to Compare Fractions:
1. Same Denominator Method
If denominators are same, compare numerators.
Example:
3/8 and 5/8
👉 5 > 3
So, 5/8 > 3/8
2. Same Numerator Method
If numerators are same, compare denominators.
Example:
3/4 and 3/6
👉 Smaller denominator means larger fraction
So, 3/4 > 3/63.
Cross Multiplication Method
Multiply diagonally to compare fractions.
Example:
2/3 and 3/52 × 5 = 10
3 × 3 = 9👉 10 > 9
So, 2/3 > 3/5
✨ Key Point:To compare fractions easily, make sure they have a common basis (same numerator or denominator) or use cross multiplication.
Ordering Decimal
🔹 What are Decimals?
Decimals are numbers that have a whole part and a fractional part, separated by a decimal point (.).
Example: 2.5, 0.75, 3.125
🔹 What is Ordering Decimals?
Ordering decimals means arranging decimal numbers in:
Ascending order (smallest to largest)
Descending order (largest to smallest)
🔹 How to Compare and Order Decimals:
1. Compare Whole Numbers First
Look at the digits before the decimal point.
Example:
3.4 and 5.2
👉 5.2 is greater because 5 > 32.
Equal Whole Numbers → Compare Decimal PartIf whole numbers are same, compare digits after decimal.
Example:
2.3 and 2.8
👉 2.8 > 2.33. Use Zeros to Make Comparison EasyYou can add zeros to make decimal places equal.
Example:
1.2 = 1.20
1.25 = 1.25
👉 Now compare easily
🔹 Example of Ordering:
Arrange in ascending order: 2.5, 2.05, 2.15
Step 1: Make equal decimal places
2.50, 2.05, 2.15Step 2: Compare
👉 2.05 < 2.15 < 2.50✔
Ascending order: 2.05, 2.15, 2.50
✨ Key Point:
Always compare whole numbers first, then decimal digits place by place for accurate ordering.
Law of Exponent
🔹 What are Exponents?
Exponents show how many times a number is multiplied by itself.
Example:
2³ = 2 × 2 × 2 = 8
Here, 2 is the base and 3 is the exponent.
🔹 Laws of Exponents:
1️⃣ Product Rule ✖️
When multiplying powers with the same base, add the exponents.
Rule:
aᵐ × aⁿ = aᵐ⁺ⁿ
Example:
2³ × 2² = 2⁵ = 32
2️⃣ Quotient Rule ➗
When dividing powers with the same base, subtract the exponents.Rule:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Example:
5⁵ ÷ 5² = 5³ = 125
3️⃣ Power of a Power Rule 🔥
When a power is raised to another power, multiply the exponents.Rule:
(aᵐ)ⁿ = aᵐⁿ
Example:
(3²)³ = 3⁶ = 729
4️⃣ Power of a Product Rule 📦
When a product is raised to a power, distribute the exponent.Rule:
(ab)ⁿ = aⁿ × bⁿExample:
(2 × 3)² = 2² × 3² = 36
5️⃣ Zero Exponent Rule 0️⃣
Any non-zero number raised to power 0 is 1.Rule:
a⁰ = 1
Example:
7⁰ = 1
✨ Key Point 💡
Laws of exponents help simplify large calculations by using simple rules of powers.
Squares & Cubes
🔹 What are Squares? 🟦
A square of a number means multiplying the number by itself once.
Formula:
n² = n × n
Examples:
4² = 4 × 4 = 16
7² = 7 × 7 = 49
10² = 10 × 10 = 100
👉 Squares are also called “power of 2” ⚡
🔹 What are Cubes? 🧊
A cube of a number means multiplying the number by itself three times.
Formula:
n³ = n × n × n
Examples:
3³ = 3 × 3 × 3 = 27
5³ = 5 × 5 × 5 = 125
2³ = 2 × 2 × 2 = 8
👉 Cubes are also called “power of 3” 🚀
🔹 Difference Between Squares & Cubes ⚖️
🟦 Square (n²): number multiplied twice
🧊 Cube (n³): number multiplied three times
Examples:
2² = 4
2³ = 8
👉 Cubes grow faster than squares 📈
🔹 Real-Life Uses 🌍
📏 Finding area of a square
📦 Finding volume of a cube
🧮 Solving algebra problems 🔢
Understanding number patterns
✨ Key Point
💡Squares and cubes help us understand how numbers grow quickly when multiplied repeatedly 🚀
🔹 What is a Square Root?
🔲A square root of a number is a value that, when multiplied by itself, gives the original number.
Symbol: √
Formula:
√n = number whose square is n
Examples:
√16 = 4 (because 4 × 4 = 16)
√25 = 5 (because 5 × 5 = 25)
√36 = 6 (because 6 × 6 = 36)
👉 Square root is the reverse of squaring 🔄
🔹 What is a Cube Root?
🧊A cube root of a number is a value that, when multiplied by itself three times, gives the original number.
Symbol: ∛
Formula:
∛n = number whose cube is n
Examples:
∛27 = 3 (because 3 × 3 × 3 = 27)
∛64 = 4 (because 4 × 4 × 4 = 64)
∛125 = 5 (because 5 × 5 × 5 = 125)
👉 Cube root is the reverse of cubing 🔄
🔹 Difference Between Square Root & Cube Root ⚖️
🔲 Square Root (√): reverse of square (power 2) 🧊 Cube Root (∛): reverse of cube (power 3)Examples:
√9 = 3 → because 3² = 9
∛8 = 2 → because 2³ = 8
🔹 Uses in Real Life 🌍
📏 Geometry (finding side lengths, area, volume)
🧮 Solving algebra problems
🏗️ Architecture & construction calculations 📊 Science and engineering formulas
✨ Key Point
💡Square roots and cube roots help us find the original number before squaring or cubing 🔄
Absolute Value
🔹 What is Absolute Value?
📌The absolute value of a number is its distance from zero on a number line.
👉 It is always non-negative (positive or zero) because distance can never be negative.
🔹 Symbol of Absolute Value
📊Absolute value is written using vertical bars:
| x |
Examples: |5| = 5 |-5| = 5 |0| = 0 |-12| = 12
👉 Both 5 and -5 are 5 units away from zero, so their absolute value is the same.
🔹 Absolute Value on Number Line
📍 Positive numbers → stay the same ➡️
Negative numbers → become positive⬅️➡️
Example: -3 is 3 units away from 0 +3 is also 3 units away from 0
🔹 Important Properties ✨
🔹 |a| ≥ 0 always
🔹 |a| = |-a|
🔹 |0| = 0
🔹 Real-Life Uses 🌍
📏 Measuring distance (no direction needed)
🌡️ Temperature difference
💰 Profit/loss calculations
🧮 Mathematics & science problems
✨ Key Point
💡Absolute value tells us how far a number is from zero, ignoring its sign 🔢➡️0
Scientific Notation
🔹 What is Scientific Notation?
📌Scientific notation is a way of writing very large or very small numbers in a simple form using powers of 10.
👉 It makes calculations easier and faster.
🔹 Standard Form
🧠A number in scientific notation is written as:
a × 10ⁿ Where:
🔹 a is a number between 1 and 10 (1 ≤ a < 10)
🔹 n is an integer (positive or negative)
Examples: ✏️ 5,000 = 5 × 10³ 70,000,000 = 7 × 10⁷ 0.004 = 4 × 10⁻³ 0.00056 = 5.6 × 10⁻⁴
🔹 How it Works 🔄
📈 Large numbers → positive power of 10
📉 Small numbers → negative power of 10
Example: 1,000 = 10³ 0.01 = 10⁻²
✨ Key Point
💡Scientific notation helps us write and understand extremely large or small numbers in a simple form using powers of 10 🔢✨
Percentage
🔹 What is Percentage? 📌
A percentage is a way of expressing a number as a fraction of 100.
👉 The word “percent” means “per hundred”.
🔹 Symbol of Percentage 🔣
Percentage is written using the symbol: %
Example:
50% means 50 out of 100
🔹 Converting Percentage 🧠
✔ Percentage to fraction:
50% = 50/100 = 1/2
✔ Percentage to decimal:
25% = 25 ÷ 100 = 0.25
✔ Fraction to percentage:
1/4 = 25%
Example ✏️ 10% = 10 out of 100
75% = 75 out of 100
100% = whole amount
0% = nothing
🔹 How to Calculate Percentage 📊
Formula:
Percentage = (Part / Whole) × 100
Example:
If you score 40 out of 50:
= (40/50) × 100
= 80% 🎉
🏷️ Discount 💸
🔹 What is Discount?
A discount is a reduction in the marked price of an item. It is given by sellers to attract customers.
🔹 Important Terms 🧠
🏷️ Marked Price (MP): Price written on the product
💰 Selling Price (SP): Price at which product is sold
🎁 Discount: Amount reduced from marked price
🔹 Formula ✏️
Discount = Marked Price − Selling Price
Discount % = (Discount / Marked Price) × 100
🔹 Example 🛒
Marked Price = ₹1000
Discount = ₹200
👉 Selling Price = ₹800
👉 Discount % = (200/1000) × 100 = 20%
📈 Profit and Loss 💰
🔹 What is Profit?
😊When Selling Price > Cost Price, it is called Profit.
🔹 What is Loss?
😢When Cost Price > Selling Price, it is called Loss.
✨ Key Point 💡
🎁 Discount = price reduction 📈 Profit = gain 📉 Loss = loss in value
All are calculated using percentages (%)
Rate (Speed)
🔹 What is Rate? 📌
A rate tells us how one quantity changes with respect to another.
👉 It compares two different units (like distance/time, price/kg, etc.)
What is Speed? 🚀
Speed is a special type of rate that tells us how fast an object moves.
👉 It is the distance covered in a given time.
🔹 Formula of Speed 🧠
Speed= Distance/Time
🔹 Units of Speed 📏
🚗 km/h (kilometres per hour)
🚶 m/s (metres per second)
🔹 Example ✏️
If a car covers 100 km in 2 hours:
Speed = 100 ÷ 2 = 50 km/h 🚗
✨ Key Point 💡
Speed helps us understand how quickly distance is covered in time 🚀⏱️
Undefined Numerical Expression
🔹 What is an Undefined Numerical Expression? 📌
An undefined numerical expression is a mathematical expression that does not have a valid or meaningful value.
👉 It means the result cannot be calculated using normal rules of mathematics.
🔹 When does an expression become undefined?
⚠️An expression becomes undefined in cases like:
🚫 1. Division by Zero
Any number divided by zero is undefined.
Example:
5 ÷ 0 = ❌ Undefined
👉 Because we cannot divide something into zero parts.
🚫 2. Zero divided by Zero
0 ÷ 0 is also undefined.
👉 Because it can give multiple possible answers.
🔹 Important Examples ✏️
10 ÷ 0 = Undefined 🚫
0 ÷ 0 = Undefined 🚫
7 ÷ 0 = Undefined 🚫
🔹 Why is it Undefined? 🧠
Because: Division has no meaning when divisor is zero It breaks basic rules of arithmetic It leads to inconsistency in answers
✨ Key Point 💡
Any expression involving division by zero is undefined and has no mathematical value.
Algebra & Function
Multiplication & Division
✨ Understanding Signs
In mathematics, numbers can be:
- ➕ Positive numbers → greater than zero (e.g., 2, 5, 10)
- ➖ Negative numbers → less than zero (e.g., -2, -5, -10)
👉 The sign tells us the direction and value of a number.
⚡ Basic Sign Rules
➕ × ➕ = ➕
Positive × Positive = Positive
➖ × ➖ = ➕
Negative × Negative = Positive
➕ × ➖ = ➖
Positive × Negative = Negative
➖ × ➕ = ➖
Negative × Positive = Negative
🔥 Power Rule for Signs (Very Important)
- Even power → result becomes positive
- Odd power → sign stays same as base
📌 Example:
(−a)3=−a3(-a)^3 = -a^3
👉 Negative number raised to an odd power stays negative.
🧠 Example Question
If n is a positive number, will the solution to the following problem be positive or negative?
📌 Problem:
(–n)³ · n
✏️ Step-by-step Thinking:
- Since n is positive, then –n is negative
- A negative number raised to an odd power stays negative:
(−n)3<0(-n)^3 < 0
- Now multiply:
- negative × positive = negative
🎯 Final Answer:
👉 The result is NEGATIVE
💡 Key Tip
✔️ Odd powers keep the sign
✔️ Even powers remove the negative sign
✔️ Multiplying signs decides final result