Simplification

Simplification is a fundamental concept in mathematics that involves reducing a mathematical expression to its simplest form. This process often requires applying a set of rules and operations in a specific order to arrive at a single numerical value or a more manageable expression. The most common set of rules used for simplification is BODMAS/PEMDAS.

BODMAS/PEMDAS Rule

The BODMAS rule is an acronym that helps remember the order of operations in an expression:

  • Brackets (or Parentheses)
  • Orders (or Exponents/Indices)
  • Division
  • Multiplication
  • Addition
  • Subtraction

PEMDAS is an alternative acronym used in some regions, where P stands for Parentheses, E for Exponents, M for Multiplication, D for Division, A for Addition, and S for Subtraction. The order of operations is the same.

It's crucial to remember that Division and Multiplication have the same priority, and Addition and Subtraction also have the same priority. When operations of the same priority appear in an expression, they are performed from left to right.

Applying the BODMAS Rule: Step-by-Step

Let's work through an example to understand the application of the BODMAS rule.

Consider the expression: 10 + (6 × 2) - 14 ÷ 7

  1. Brackets: First, we solve the operation inside the brackets.

    6 × 2 = 12

    The expression becomes: 10 + 12 - 14 ÷ 7

  2. Orders: There are no orders (exponents) in this expression.
  3. Division: Next, we perform the division.

    14 ÷ 7 = 2

    The expression becomes: 10 + 12 - 2

  4. Multiplication: There is no multiplication left.
  5. Addition: Now, we perform addition from left to right.

    10 + 12 = 22

    The expression becomes: 22 - 2

  6. Subtraction: Finally, we perform subtraction.

    22 - 2 = 20

Therefore, the simplified value of the expression 10 + (6 × 2) - 14 ÷ 7 is 20.

Common Pitfalls in Simplification

Students often make mistakes by not following the order of operations strictly. For instance, performing addition before division or multiplication can lead to an incorrect answer. Always remember to tackle brackets first, then exponents, followed by division and multiplication (from left to right), and finally addition and subtraction (from left to right).

Fractions and Simplification

Simplification also applies to fractions. To simplify a fraction, we find the Highest Common Factor (HCF) of the numerator and the denominator and divide both by it. This results in an equivalent fraction in its lowest terms.

Example: Simplify the fraction 48/72.

First, find the HCF of 48 and 72.

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The common factors are 1, 2, 3, 4, 6, 8, 12, 24. The HCF is 24.

Now, divide both the numerator and the denominator by the HCF:

48 ÷ 24 = 2

72 ÷ 24 = 3

So, the simplified fraction is 2/3.

Simplification Shortcut: When dealing with multiple operations, always write down BODMAS/PEMDAS to remind yourself of the correct order. For fractions, remember to simplify by dividing both numerator and denominator by their HCF to get the fraction in its lowest terms.

Percentage

A percentage is a way of expressing a number as a fraction of 100. The word 'percent' means 'per hundred'. The symbol used for percentage is '%'. For example, 50% means 50 out of 100, which can be written as 50/100 or 0.5.

Converting Fractions and Decimals to Percentages

To convert a fraction to a percentage, multiply the fraction by 100 and add the '%' sign.

Example: Convert 3/4 to a percentage.

(3/4) × 100% = (300/4)% = 75%

To convert a decimal to a percentage, multiply the decimal by 100 and add the '%' sign.

Example: Convert 0.85 to a percentage.

0.85 × 100% = 85%

Converting Percentages to Fractions and Decimals

To convert a percentage to a fraction, write the percentage value over 100 and simplify the fraction.

Example: Convert 60% to a fraction.

60% = 60/100. Simplifying this fraction by dividing both numerator and denominator by their HCF (20), we get 3/5.

To convert a percentage to a decimal, remove the '%' sign and divide the number by 100 (or move the decimal point two places to the left).

Example: Convert 35% to a decimal.

35% = 35/100 = 0.35

Calculating Percentage of a Number

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

Example: Find 25% of 200.

Using fractions: 25% = 25/100 = 1/4. So, (1/4) × 200 = 50.

Using decimals: 25% = 0.25. So, 0.25 × 200 = 50.

Percentage Increase and Decrease

Percentage Increase: Calculated as ((New Value - Original Value) / Original Value) × 100%.

Percentage Decrease: Calculated as ((Original Value - New Value) / Original Value) × 100%.

Example: A shirt costs ₹500 and is sold for ₹600. Find the percentage increase.

Increase = ₹600 - ₹500 = ₹100

Percentage Increase = (100 / 500) × 100% = (1/5) × 100% = 20%.

Example: A laptop originally priced at ₹40,000 is sold for ₹32,000. Find the percentage decrease.

Decrease = ₹40,000 - ₹32,000 = ₹8,000

Percentage Decrease = (8,000 / 40,000) × 100% = (1/5) × 100% = 20%.

Profit and Loss

These concepts are directly related to percentages. Profit is the gain made when the selling price is greater than the cost price, and loss occurs when the selling price is less than the cost price.

  • Profit % = ((Selling Price - Cost Price) / Cost Price) × 100%
  • Loss % = ((Cost Price - Selling Price) / Cost Price) × 100%

Simple Interest

Simple Interest (SI) is a method of calculating the interest charged on a sum of money at a given rate for a given period. The interest is calculated only on the principal amount.

Formula: SI = (P × R × T) / 100

Where P = Principal amount, R = Rate of interest per annum, T = Time period in years.

Amount = Principal + Simple Interest

Percentage Trick: To quickly calculate X% of Y, you can also calculate Y% of X. For example, 30% of 70 is the same as 70% of 30, which is (70/100) * 30 = 21. This can be faster for certain number combinations. Also, remember that 10% of any number is just the number with the decimal point moved one place to the left.

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. Listing Factors Method

This method involves listing all the factors (divisors) of each number and then identifying the largest factor that is common to all of them.

Example: Find the HCF 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
  • Highest Common Factor: 6

This method is practical for smaller numbers but can be time-consuming for larger ones.

2. Prime Factorization Method

In this method, we find the prime factorization of each number. The HCF is the product of the common prime factors raised to the lowest power they appear in any of the factorizations.

Example: Find the HCF of 24 and 36.

  • Prime factorization of 24: 2 × 2 × 2 × 3 = 23 × 31
  • Prime factorization of 36: 2 × 2 × 3 × 3 = 22 × 32

The common prime factors are 2 and 3.

The lowest power of 2 is 22.

The lowest power of 3 is 31.

HCF = 22 × 31 = 4 × 3 = 12.

3. Division Method (Euclidean Algorithm)

This is a very efficient method, especially for larger numbers. It is based on the principle that the HCF of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until one of the numbers becomes zero, and the other number is the HCF.

A more common form of the Euclidean algorithm involves repeated division.

Example: Find the HCF of 135 and 225.

  1. Divide the larger number (225) by the smaller number (135):

    225 = 1 × 135 + 90

  2. Replace the larger number with the smaller number (135) and the smaller number with the remainder (90). Divide 135 by 90:

    135 = 1 × 90 + 45

  3. Replace 135 with 90 and 90 with 45. Divide 90 by 45:

    90 = 2 × 45 + 0

The last non-zero remainder is the HCF. In this case, the HCF is 45.

HCF of Three or More Numbers

To find the HCF of three or more numbers, you can find the HCF of the first two numbers, and then find the HCF of the result and the third number, and so on.

HCF(a, b, c) = HCF(HCF(a, b), c)

HCF Tip: For numbers ending in 0, remember that 10 is a common factor. For example, HCF(30, 50) = 10 * HCF(3, 5) = 10 * 1 = 10. This simplifies calculations. The Euclidean algorithm is the most efficient for exam purposes.

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. Listing Multiples Method

List the multiples of each number until you find the smallest multiple that is common to all of them.

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, ...
  • Lowest Common Multiple: 12

This method is suitable for small numbers.

2. Prime Factorization Method

Find the prime factorization of each number. The LCM is the product of the highest powers of all prime factors that appear in any of the factorizations.

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

The prime factors involved are 2 and 3.

The highest power of 2 is 22.

The highest power of 3 is 32.

LCM = 22 × 32 = 4 × 9 = 36.

3. Division Method

Arrange the numbers in a row. Divide them by a common prime factor. Write the quotients and any numbers that are not divisible below. Repeat the process until all the numbers in the row are prime to each other. The LCM is the product of the divisors and the remaining numbers.

Example: Find the LCM of 8, 12, and 15.

2 | 8, 12, 15
2 | 4, 6, 15
3 | 2, 3, 15
5 | 2, 1, 5
| 2, 1, 1

LCM = 2 × 2 × 3 × 5 × 2 = 120.

Relationship Between HCF and LCM

For two numbers, say 'a' and 'b', there is a very important relationship between their HCF and LCM:

HCF(a, b) × LCM(a, b) = a × b

This formula is extremely useful for solving problems where you might know three of the four values (HCF, LCM, and the two numbers) and need to 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'.

HCF × LCM = First Number × Second Number

12 × 72 = 24 × x

x = (12 × 72) / 24

x = (1 × 72) / 2

x = 36.

The other number is 36.

LCM of Three or More Numbers

Similar to HCF, you can find the LCM of three or more numbers iteratively.

LCM(a, b, c) = LCM(LCM(a, b), c)

LCM Formula Shortcut: Always remember HCF × LCM = Product of the numbers. This single formula connects HCF, LCM, and the numbers themselves, saving a lot of calculation time in exams. For the division method of LCM, it's best to start dividing by the smallest prime numbers (2, 3, 5, 7...) and continue until no two numbers share a common factor.
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