Simplification and Percentage Calculations

1. Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. It involves applying the order of operations and various mathematical rules to make complex expressions easier to understand and solve. The most common rule used in simplification is BODMAS or PEMDAS.

1.1. The BODMAS Rule

BODMAS is an acronym that helps us remember the order of operations in a mathematical expression. It stands for:

  • Brackets (Parentheses)
  • Orders (Powers and Square Roots, etc.)
  • Division
  • Multiplication
  • Addition
  • Subtraction

When solving an expression, we follow these steps strictly from left to right. If there are multiple operations of the same priority (like division and multiplication, or addition and subtraction), we perform them from left to right as they appear.

1.2. Applying BODMAS with Examples

Let's take an example to understand how BODMAS works:

Example 1: Solve 12 + (6 × 2) - 10 ÷ 5

  1. Brackets: First, solve the expression inside the brackets: (6 × 2) = 12. The expression becomes: 12 + 12 - 10 ÷ 5
  2. Orders: There are no orders (powers, roots) in this expression.
  3. Division: Next, perform the division: 10 ÷ 5 = 2. The expression becomes: 12 + 12 - 2
  4. Multiplication: There is no multiplication in this expression.
  5. Addition: Perform the addition from left to right: 12 + 12 = 24. The expression becomes: 24 - 2
  6. Subtraction: Finally, perform the subtraction: 24 - 2 = 22.

So, 12 + (6 × 2) - 10 ÷ 5 = 22.

Example 2: Simplify 36 ÷ 3 × 4 + (15 - 7)

  1. Brackets: (15 - 7) = 8. Expression: 36 ÷ 3 × 4 + 8
  2. Orders: None.
  3. Division and Multiplication: Perform division and multiplication from left to right. First, 36 ÷ 3 = 12. Expression: 12 × 4 + 8 Next, 12 × 4 = 48. Expression: 48 + 8
  4. Addition: 48 + 8 = 56.
  5. Subtraction: None.

So, 36 ÷ 3 × 4 + (15 - 7) = 56.

1.3. Simplification of Fractions

When simplifying expressions involving fractions, we also follow BODMAS. We often need to find a common denominator for addition and subtraction of fractions.

Example 3: Simplify 1/2 + 1/3 × 1/4

  1. Multiplication: First, multiply 1/3 × 1/4 = 1/12. Expression: 1/2 + 1/12
  2. Addition: To add these fractions, find a common denominator, which is 12. 1/2 = 6/12. Expression: 6/12 + 1/12 = 7/12.

1.4. Simplification of Square Roots and Powers

When dealing with square roots and powers, remember that they fall under 'Orders' in BODMAS. You must simplify these before proceeding with other operations.

Example 4: Simplify (√16 × 2) + 32

  1. Orders: √16 = 4. 32 = 9. Expression: (4 × 2) + 9
  2. Brackets: (4 × 2) = 8. Expression: 8 + 9
  3. Addition: 8 + 9 = 17.
Simplification Shortcut: Always remember BODMAS. If you see nested brackets, solve the innermost bracket first. For multiplication and division, or addition and subtraction, proceed strictly from left to right.

2. Percentage Calculations

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

2.1. Converting Fractions and Decimals to Percentages

To convert a fraction or a decimal to a percentage, multiply it by 100.

Formula: Percentage = (Fraction or Decimal) × 100%

Example 1: Convert 3/4 to a percentage.

Percentage = (3/4) × 100% = 0.75 × 100% = 75%.

Example 2: Convert 0.65 to a percentage.

Percentage = 0.65 × 100% = 65%.

2.2. Converting Percentages to Fractions and Decimals

To convert a percentage to a fraction, divide it by 100 and remove the '%' sign. To convert to a decimal, simply divide by 100.

Formula: Fraction/Decimal = Percentage / 100

Example 3: Convert 40% to a fraction and a decimal.

Fraction = 40/100 = 4/10 = 2/5.

Decimal = 40/100 = 0.40 or 0.4.

2.3. Calculating Percentage of a Number

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

Formula: P% of N = (P/100) × N

Example 4: Find 25% of 200.

25% of 200 = (25/100) × 200 = (1/4) × 200 = 50.

Alternatively, using decimals: 0.25 × 200 = 50.

2.4. Finding the Percentage Increase or Decrease

Percentage Increase: (Increase in Value / Original Value) × 100%

Percentage Decrease: (Decrease in Value / Original Value) × 100%

Example 5: A man's salary increased from ₹10,000 to ₹12,000. Find the percentage increase.

Increase in Value = ₹12,000 - ₹10,000 = ₹2,000.

Percentage Increase = (2000 / 10000) × 100% = (1/5) × 100% = 20%.

Example 6: The price of a book decreased from ₹100 to ₹80. Find the percentage decrease.

Decrease in Value = ₹100 - ₹80 = ₹20.

Percentage Decrease = (20 / 100) × 100% = 20%.

2.5. Finding the Original Value when Percentage Change is Known

If a quantity increases by P%, the new value is (100 + P)% of the original value. If it decreases by P%, the new value is (100 - P)% of the original value.

Example 7: If 20% of a number is 50, find the number.

Let the number be 'x'.

20% of x = 50

(20/100) × x = 50

(1/5) × x = 50

x = 50 × 5 = 250.

Example 8: A number is increased by 10%. If the new number is 220, find the original number.

The new number is (100 + 10)% = 110% of the original number.

Let the original number be 'x'.

110% of x = 220

(110/100) × x = 220

1.1 × x = 220

x = 220 / 1.1 = 2200 / 11 = 200.

2.6. Percentage Related Problems

Many problems in aptitude tests involve successive percentage changes, profit and loss, discounts, and interest, all of which rely on percentage calculations.

Successive Percentage Changes: If a quantity is changed by P1% and then by P2%, the net percentage change is given by:

Net Change % = P1 + P2 + (P1 × P2) / 100

Note: If the percentage change is a decrease, use a negative sign.

Example 9: A man's salary is increased by 10% and then by 20%. Find the total percentage increase.

P1 = 10%, P2 = 20%

Net Change % = 10 + 20 + (10 × 20) / 100

Net Change % = 30 + 200 / 100

Net Change % = 30 + 2 = 32%.

Example 10: A price is first increased by 20% and then decreased by 10%. Find the net percentage change.

P1 = +20%, P2 = -10%

Net Change % = 20 + (-10) + (20 × -10) / 100

Net Change % = 10 + (-200) / 100

Net Change % = 10 - 2 = 8%.

Since the result is positive, it's an 8% increase.

Percentage Shortcut: Remember that P% of N is the same as N% of P. For example, 20% of 50 = 50% of 20 = 20. This can save time in calculations. Also, for successive percentage changes, the formula P1 + P2 + (P1 × P2) / 100 is very useful.

3. Combined Concepts and Exam Application

In competitive exams, simplification and percentage calculations are often combined. You might encounter questions that require you to simplify a complex expression involving percentages, or to use percentage concepts within simplification problems.

3.1. Simplifying Percentage Expressions

Example 1: Simplify: 15% of 200 + 20% of 150 - 10% of 300

  1. Calculate each part:
    • 15% of 200 = (15/100) × 200 = 15 × 2 = 30
    • 20% of 150 = (20/100) × 150 = (1/5) × 150 = 30
    • 10% of 300 = (10/100) × 300 = (1/10) × 300 = 30
  2. Substitute the values back into the expression: 30 + 30 - 30
  3. Perform the addition and subtraction: 60 - 30 = 30

The result is 30.

3.2. Percentage Calculations within Simplification Problems

Example 2: What is 20% of (50% of 400)?

  1. Solve the inner part first: 50% of 400. 50% of 400 = (50/100) × 400 = (1/2) × 400 = 200.
  2. Now find 20% of the result (200): 20% of 200 = (20/100) × 200 = (1/5) × 200 = 40.

The answer is 40.

3.3. Real-World Applications

Understanding simplification and percentages is crucial for everyday financial decisions. For instance:

  • Shopping: Calculating discounts on items (e.g., 30% off a ₹500 shirt means a saving of 30% of ₹500).
  • Taxes: Understanding income tax slabs and GST (Goods and Services Tax).
  • Investments: Calculating returns on investments based on interest rates (percentages).
  • Statistics: Interpreting data presented in percentages in news and reports.
Exam Tip: Practice a variety of problems involving both simplification and percentages. Pay close attention to the wording of the question to correctly identify whether you need to calculate a percentage, a percentage change, or simplify an expression containing percentages. Always double-check your calculations, especially when dealing with decimals and fractions.
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