Simplification

Simplification is a mathematical process where a complex expression is reduced to its simplest form by performing the basic arithmetic operations: addition, subtraction, multiplication, and division. The order of operations is crucial in simplification to arrive at the correct answer. We use the acronym BODMAS or PEMDAS to remember this order.

BODMAS Rule

BODMAS stands for:

• Brackets: Operations inside brackets are performed first. There are different types of brackets: vinculum (bar), parentheses (), braces {}, and square brackets []. The order is typically from innermost to outermost.
• Of: This means multiplication. It is usually applied after brackets but before division.
• Division: Performed next.
• Multiplication: Performed after division.
• Addition: Performed next.
• Subtraction: Performed last.

PEMDAS Rule

PEMDAS is an alternative acronym used in some regions, standing for:

• Parentheses: Similar to Brackets.
• Exponents: Powers and roots.
• Multiplication
• Division
• Addition
• Subtraction

Both BODMAS and PEMDAS represent the same order of operations. For this syllabus, we will primarily focus on BODMAS.

Applying the BODMAS Rule: Step-by-Step

Let's take an example: Simplify 12 + (6 - 2) × 3 ÷ 2.

1. Brackets: First, solve the operation inside the parentheses: 6 - 2 = 4. The expression becomes 12 + 4 × 3 ÷ 2.
2. Of: There are no 'of' operations in this example.
3. Division: Next, perform the division: 3 ÷ 2 = 1.5. The expression is now 12 + 4 × 1.5.
4. Multiplication: Perform the multiplication: 4 × 1.5 = 6. The expression is 12 + 6.
5. Addition: Finally, perform the addition: 12 + 6 = 18.
6. Subtraction: There is no subtraction in this example.

Therefore, the simplified value is 18.

Common Pitfalls in Simplification

• Incorrect order of operations: Forgetting BODMAS can lead to completely wrong answers.
• Mistakes in handling signs: Especially with subtraction and negative numbers.
• Errors in fraction or decimal arithmetic.

Practice Problems

1. Simplify: 20 × [ 5 + ( 10 ÷ 2 ) - 3 ]

2. Evaluate: 18 - 6 ÷ 2 + 4 × 3

Solution to Practice Problem 1:

20 × [ 5 + ( 10 ÷ 2 ) - 3 ]

= 20 × [ 5 + 5 - 3 ] (Innermost bracket first: 10 ÷ 2 = 5)

= 20 × [ 10 - 3 ] (Addition inside bracket: 5 + 5 = 10)

= 20 × [ 7 ] (Subtraction inside bracket: 10 - 3 = 7)

= 140 (Multiplication: 20 × 7 = 140)

Solution to Practice Problem 2:

18 - 6 ÷ 2 + 4 × 3

= 18 - 3 + 12 (Division first: 6 ÷ 2 = 3; Multiplication next: 4 × 3 = 12)

= 15 + 12 (Subtraction from left to right: 18 - 3 = 15)

= 27 (Addition: 15 + 12 = 27)

Percentage

A percentage is a fraction of a whole number, expressed as a part of 100. The word "percent" means "per hundred." The symbol for percentage is "%". For example, 50% means 50 out of 100, which can be written as the fraction 50/100 or the decimal 0.50.

Converting Fractions and Decimals to Percentages

• Fraction to Percentage: Multiply the fraction by 100. Example: To convert 3/4 to a percentage, calculate (3/4) × 100 = 75%.
• Decimal to Percentage: Multiply the decimal by 100 and add the '%' sign. Example: To convert 0.25 to a percentage, calculate 0.25 × 100 = 25%.

Converting Percentages to Fractions and Decimals

• Percentage to Fraction: Divide the percentage by 100 and simplify. Example: To convert 40% to a fraction, calculate 40/100 = 2/5.
• Percentage to Decimal: Divide the percentage by 100. Example: To convert 65% to a decimal, calculate 65/100 = 0.65.

Calculating Percentage of a Number

To find a percentage of a number, convert the percentage to a decimal or fraction and then multiply by the number.

Formula: Percentage of a number = (Percentage / 100) × Number

Example: Find 20% of 150.

Method 1 (using fraction): (20/100) × 150 = (1/5) × 150 = 30.

Method 2 (using decimal): 0.20 × 150 = 30.

Percentage Increase and Decrease

Percentage Increase: Used when a quantity has increased. Percentage Increase = [(New Value - Original Value) / Original Value] × 100

Example: If the price of a book increased from Rs. 100 to Rs. 120, the percentage increase is [(120 - 100) / 100] × 100 = (20/100) × 100 = 20%.

Percentage Decrease: Used when a quantity has decreased. Percentage Decrease = [(Original Value - New Value) / Original Value] × 100

Example: If the number of students in a class decreased from 50 to 40, the percentage decrease is [(50 - 40) / 50] × 100 = (10/50) × 100 = 20%.

Profit and Loss (Related to Percentage)

Cost Price (CP): The price at which an article is bought.

Selling Price (SP): The price at which an article is sold.

Profit: If SP > CP, Profit = SP - CP.

Loss: If CP > SP, Loss = CP - SP.

Profit Percentage: (Profit / CP) × 100

Loss Percentage: (Loss / CP) × 100

Example: A shopkeeper buys a toy for Rs. 200 (CP) and sells it for Rs. 240 (SP).

Profit = 240 - 200 = Rs. 40.

Profit Percentage = (40 / 200) × 100 = 20%.

Simple Interest (Related to Percentage)

Simple Interest (SI) is calculated on the principal amount only.

Formula: SI = (P × R × T) / 100

Where:

• P = Principal amount (the initial sum of money)
• R = Rate of interest per annum (in %)
• T = Time period (in years)

Example: Calculate the simple interest on Rs. 5000 at 8% per annum for 3 years.

SI = (5000 × 8 × 3) / 100 = (50 × 8 × 3) = Rs. 1200.

Amount = Principal + SI = 5000 + 1200 = Rs. 6200.

Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Methods to Find HCF

1. Prime Factorization Method

Step 1: Find the prime factorization of each number.

Step 2: Identify the common prime factors.

Step 3: Multiply these common prime factors together. The result is the HCF.

Example: Find the HCF of 12 and 18.

Prime factorization of 12: 2 × 2 × 3 = 2² × 3

Prime factorization of 18: 2 × 3 × 3 = 2 × 3²

Common prime factors are 2 and 3.

HCF = 2 × 3 = 6.

Example: Find the HCF of 24, 36, and 60.

24 = 2 × 2 × 2 × 3 = 2³ × 3

36 = 2 × 2 × 3 × 3 = 2² × 3²

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Common prime factors with the lowest power are 2² and 3¹.

HCF = 2² × 3 = 4 × 3 = 12.

2. Division Method (Euclidean Algorithm)

This method is efficient for larger numbers.

Step 1: Divide the larger number by the smaller number and find the remainder.

Step 2: Replace the larger number with the smaller number and the smaller number with the remainder.

Step 3: Repeat the division process until the remainder is 0.

Step 4: The last non-zero remainder is the HCF.

Example: Find the HCF of 48 and 180.

180 ÷ 48 = 3 with a remainder of 36. (180 = 3 × 48 + 36)

Now, divide 48 by 36: 48 ÷ 36 = 1 with a remainder of 12. (48 = 1 × 36 + 12)

Now, divide 36 by 12: 36 ÷ 12 = 3 with a remainder of 0. (36 = 3 × 12 + 0)

The last non-zero remainder is 12. So, HCF(48, 180) = 12.

3. Listing Factors Method

This method is suitable for small numbers.

Step 1: List all the factors (divisors) of each number.

Step 2: Identify the common factors.

Step 3: The largest among the common factors is the HCF.

Example: Find the HCF of 8 and 12.

Factors of 8: 1, 2, 4, 8

Factors of 12: 1, 2, 3, 4, 6, 12

Common factors: 1, 2, 4

The largest common factor is 4. So, HCF(8, 12) = 4.

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of each of the numbers.

Methods to Find LCM

1. Prime Factorization Method

Step 1: Find the prime factorization of each number.

Step 2: Identify all the prime factors that appear in any of the factorizations.

Step 3: For each prime factor, take the highest power that appears in any of the factorizations.

Step 4: Multiply these highest powers together. The result is the LCM.

Example: Find the LCM of 12 and 18.

Prime factorization of 12: 2 × 2 × 3 = 2² × 3¹

Prime factorization of 18: 2 × 3 × 3 = 2¹ × 3²

Prime factors involved are 2 and 3.

Highest power of 2 is 2².

Highest power of 3 is 3².

LCM = 2² × 3² = 4 × 9 = 36.

Example: Find the LCM of 24, 36, and 60.

24 = 2³ × 3¹

36 = 2² × 3²

60 = 2² × 3¹ × 5¹

Prime factors involved are 2, 3, and 5.

Highest power of 2 is 2³.

Highest power of 3 is 3².

Highest power of 5 is 5¹.

LCM = 2³ × 3² × 5¹ = 8 × 9 × 5 = 72 × 5 = 360.

2. Division Method

Step 1: Write the numbers in a row.

Step 2: Divide the numbers by a common prime factor that divides at least two of them. Write the quotient below.

Step 3: If a number is not divisible by the prime factor, bring it down as it is.

Step 4: Repeat the process until all numbers in the row are prime to each other (i.e., no common factor other than 1).

Step 5: The LCM is the product of the divisors and the remaining numbers in the last row.

Example: Find the LCM of 12 and 18.

2 | 12, 18 --|------- 3 | 6, 9 --|------- | 2, 3

LCM = 2 × 3 × 2 × 3 = 36.

Example: Find the LCM of 24, 36, and 60.

2 | 24, 36, 60 --|----------- 2 | 12, 18, 30 --|----------- 3 | 6, 9, 15 --|----------- | 2, 3, 5

LCM = 2 × 2 × 3 × 2 × 3 × 5 = 8 × 9 × 5 = 360.

3. Listing Multiples Method

This method is best for small numbers.

Step 1: List the multiples of each number.

Step 2: Identify the common multiples.

Step 3: The smallest among the common multiples is the LCM.

Example: Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Multiples of 6: 6, 12, 18, 24, 30, ...

Common multiples: 12, 24, ...

The smallest common multiple is 12. So, LCM(4, 6) = 12.

Relationship between HCF and LCM

For two numbers, say 'a' and 'b', there is a fundamental relationship:

Product of the numbers = HCF of the numbers × LCM of the numbers

a × b = HCF(a, b) × LCM(a, b)

This formula is very useful. If you know any three of these values, you can find the fourth.

Example: The HCF of two numbers is 12 and their LCM is 72. If one of the numbers is 24, find the other number.

Let the other number be 'x'.

24 × x = 12 × 72

x = (12 × 72) / 24

x = 12 × (72 / 24)

x = 12 × 3

x = 36.

The other number is 36.

Summary of HCF and LCM

Concept	Definition	Method Example (Numbers: 12, 18)	Key Property
HCF (Highest Common Factor)	Largest number that divides both numbers exactly.	Prime Factors: 12 = 22 × 3, 18 = 2 × 32. Common factors: 21, 31. HCF = 2 × 3 = 6.	Product of common prime factors with lowest powers.
LCM (Lowest Common Multiple)	Smallest number that is a multiple of both numbers.	Prime Factors: 12 = 22 × 3, 18 = 2 × 32. All factors: 22, 32. LCM = 22 × 32 = 4 × 9 = 36.	Product of all prime factors with highest powers.