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Percentage and Ratio

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Learning objective
Calculate percentages and understand ratio concepts

Introduction to Percentage and Ratio

Mathematics is not just about numbers; it is a language that helps us understand the world around us. Two important concepts that often appear in daily life and competitive exams like the Assam Direct Recruitment Examination (ADRE) are percentage and ratio. These concepts help us compare quantities, understand changes, and divide things fairly.

For example, when you see a discount of 20% on a product, or when you share sweets among friends in a certain ratio, you are using these ideas. In this chapter, you will learn how to calculate percentages, understand ratios and proportions, and apply these concepts to solve problems efficiently.

By the end of this section, you will be able to:

  • Calculate the percentage of any quantity
  • Understand and simplify ratios
  • Solve proportion problems
  • Apply these concepts in real-life situations and exam problems

Percentage Calculation

What is Percentage?

The word percentage means "per hundred". It is a way to express a part of a whole as a fraction of 100. For example, 50% means 50 out of 100, or half.

Percentages are useful because they give us a common way to compare different quantities, no matter their size.

Converting Fractions and Decimals to Percentage

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

For example:

  • Fraction: \( \frac{1}{2} = 0.5 \). Multiply by 100: \(0.5 \times 100 = 50\%\)
  • Decimal: 0.25. Multiply by 100: \(0.25 \times 100 = 25\%\)

Converting Percentage to Fraction or Decimal

To convert a percentage to a fraction, write the percentage number over 100 and simplify if possible.

To convert a percentage to a decimal, divide by 100.

Conversion between Fraction, Decimal, and Percentage
Fraction Decimal Percentage
\(\frac{1}{2}\) 0.5 50%
\(\frac{1}{4}\) 0.25 25%
\(\frac{3}{4}\) 0.75 75%
\(\frac{1}{5}\) 0.2 20%
\(\frac{2}{5}\) 0.4 40%

How to Calculate Percentage of a Number?

To find what percentage a part is of a whole, use the formula:

Percentage Formula

\[\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\]

Finds what percent the part is of the whole

Part = portion of the total
Whole = total amount

To find a percentage of a number (for example, 15% of 200 INR), convert the percentage to a decimal and multiply:

\(15\% = \frac{15}{100} = 0.15\)

Then multiply: \(0.15 \times 200 = 30\) INR

Percentage Increase and Decrease

When a quantity grows or shrinks, we often want to know by what percentage it has changed.

Use this formula:

Percentage Increase/Decrease

\[\text{Percentage Change} = \left(\frac{\text{Change}}{\text{Original Value}}\right) \times 100\]

Calculates how much a quantity has increased or decreased

Change = difference between new and original value

Remember, if the new value is more than the original, it is an increase; if less, it is a decrease.

Ratio and Proportion

What is a Ratio?

A ratio compares two quantities by division. It tells us how many times one quantity is compared to another.

For example, if there are 3 apples and 2 oranges, the ratio of apples to oranges is written as 3:2.

Simplifying Ratios

Just like fractions, ratios can be simplified by dividing both terms by their Highest Common Factor (HCF).

For example, the ratio 6:8 can be simplified:

HCF of 6 and 8 is 2

Divide both by 2: \( \frac{6}{2} : \frac{8}{2} = 3:4 \)

What is Proportion?

A proportion states that two ratios are equal.

For example, if \( \frac{a}{b} = \frac{c}{d} \), then these two ratios are in proportion.

This helps us find unknown values when quantities are related.

graph TD    A[Identify given ratios] --> B[Set up proportion equation]    B --> C[Cross multiply]    C --> D[Solve for unknown]

How to Solve Proportion Problems?

Use cross multiplication:

Proportion Equation

\[\frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c\]

Used to solve for unknowns by cross multiplying

a,b,c,d = terms of the two ratios

Real-life Application of Ratios

Ratios are used in cooking (mixing ingredients), dividing money, or comparing speeds. Understanding ratios helps you solve many practical problems.

Worked Examples

Example 1: Finding 15% of 200 INR Easy
Find 15% of 200 INR.

Step 1: Convert 15% to decimal: \( \frac{15}{100} = 0.15 \)

Step 2: Multiply 0.15 by 200: \(0.15 \times 200 = 30\)

Answer: 15% of 200 INR is 30 INR.

Example 2: Dividing 500 INR in Ratio 3:2 Medium
Divide 500 INR between two people in the ratio 3:2.

Step 1: Add the parts of the ratio: \(3 + 2 = 5\)

Step 2: Find one part value: \( \frac{500}{5} = 100 \) INR

Step 3: Calculate each share:

  • First person: \(3 \times 100 = 300\) INR
  • Second person: \(2 \times 100 = 200\) INR

Answer: The shares are 300 INR and 200 INR respectively.

Example 3: Calculating 12% Price Increase on 250 INR Medium
An item costs 250 INR. Its price increases by 12%. Find the new price.

Step 1: Calculate the increase amount:

12% of 250 = \( \frac{12}{100} \times 250 = 30 \) INR

Step 2: Add increase to original price:

New price = \(250 + 30 = 280\) INR

Answer: The new price is 280 INR.

Example 4: Converting Ratio 4:5 to Percentages Easy
Convert the ratio 4:5 into percentages to find each part's share.

Step 1: Add the parts: \(4 + 5 = 9\)

Step 2: Calculate percentage for each part:

  • First part: \( \frac{4}{9} \times 100 \approx 44.44\% \)
  • Second part: \( \frac{5}{9} \times 100 \approx 55.56\% \)

Answer: The shares are approximately 44.44% and 55.56%.

Example 5: Dividing 1200 INR in Ratio 2:3:5 Hard
Divide 1200 INR among three people in the ratio 2:3:5. Find each person's share.

Step 1: Add the parts of the ratio: \(2 + 3 + 5 = 10\)

Step 2: Find the value of one part: \( \frac{1200}{10} = 120 \) INR

Step 3: Calculate each share:

  • First person: \(2 \times 120 = 240\) INR
  • Second person: \(3 \times 120 = 360\) INR
  • Third person: \(5 \times 120 = 600\) INR

Answer: The shares are 240 INR, 360 INR, and 600 INR respectively.

Formula Bank

Percentage Formula
\[ \text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100 \]
where: Part = portion of the total, Whole = total amount
Percentage Increase/Decrease
\[ \text{Percentage Change} = \left(\frac{\text{Change}}{\text{Original Value}}\right) \times 100 \]
where: Change = difference between new and original value
Ratio Simplification
\[ \frac{a}{b} = \frac{a \div \text{HCF}(a,b)}{b \div \text{HCF}(a,b)} \]
where: a, b = terms of the ratio
Proportion Equation
\[ \frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c \]
where: a, b, c, d = terms of the two ratios

Tips & Tricks

Tip: Convert percentages to decimals by dividing by 100 before multiplication.

When to use: When calculating percentage of a number to simplify multiplication.

Tip: Always simplify ratios before solving proportion problems.

When to use: To reduce calculation complexity and avoid errors.

Tip: Use cross multiplication for quick solving of proportions.

When to use: When two ratios are set equal and one term is unknown.

Tip: Remember that percentage increase adds to the original value, decrease subtracts.

When to use: While solving problems related to price changes or population growth.

Tip: Check if the sum of ratio parts matches the total before dividing amounts.

When to use: To avoid incorrect distribution in ratio problems.

Common Mistakes to Avoid

❌ Confusing percentage with percentage points
✓ Understand percentage as a fraction of 100, not just a number
Why: Students often treat percentages as whole numbers leading to calculation errors.
❌ Not simplifying ratios before solving problems
✓ Always find HCF and simplify ratio terms first
Why: Skipping simplification causes unnecessary complexity and mistakes.
❌ Incorrectly applying percentage increase/decrease formulas
✓ Use original value as denominator and add/subtract change accordingly
Why: Misinterpretation of increase/decrease leads to wrong final answers.
❌ Mixing up numerator and denominator in ratio problems
✓ Keep track of which quantity corresponds to which ratio term
Why: Confusion in ratio terms causes wrong allocation or calculation.
❌ Ignoring units (INR, metric) in word problems
✓ Always include and convert units properly before solving
Why: Unit mismatch leads to unrealistic or incorrect answers.
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