Introduction and Learning Objectives

Ratios and percentages form the backbone of quantitative reasoning in everyday life, business, science, and beyond. Understanding how to interpret and manipulate these concepts will sharpen your mathematical skills, helping you analyze data, solve practical problems, and make informed decisions.

This course covers foundational to advanced topics, featuring real-world applications, problem-solving strategies, and actionable practice exercises. Whether you’re a student preparing for exams, an adult learner refreshing math skills, or a professional seeking practical knowledge, you’ll find clear guidance and expert insights here.

Learning Objectives

• Define and correctly represent ratios and percentages in various formats.
• Convert between ratios, fractions, decimals, and percentages fluently.
• Calculate equivalent and simplified ratios for problem-solving.
• Compute percentages of quantities and interpret percentage data effectively.
• Solve word problems employing ratios and percentages across contexts.
• Analyze percentage increase, decrease, error, and distinguish between percentage points and percent change.
• Apply ratio and percentage concepts to advanced scenarios including financial mathematics, mixtures, and data interpretation.

Course Structure

This course is organized into the following topics, each with targeted lessons, quizzes, and assignments to reinforce learning:

• Topic 1: Definition and Notation of Ratios
• Topic 2: Equivalent Ratios and Simplifying Ratios
• Topic 3: Ratio Word Problems
• Topic 4: Introduction to Percentages and their Meaning
• Topic 5: Converting Between Fractions, Decimals, Ratios, and Percentages
• Topic 6: Calculating Percentages of Quantities
• Topic 7: Percentage Increase, Decrease, and Error
• Topic 8: Advanced Applications and Problem-Solving Strategies

Each topic will include lessons, practice quizzes, and assignments designed to build your skills step-by-step.

1. Understanding Ratios

Definition and Notation of Ratios

A ratio is a way to compare two quantities that are related to each other. It tells us how many times one quantity contains or is contained within the other.

Ratios can be expressed in three common forms:

• a to b (spoken form)
• a : b (colon form)
• \(\frac{a}{b}\) (fraction form)

For example, if a recipe uses 2 cups of flour and 3 cups of sugar, the ratio of flour to sugar can be written as 2:3, “2 to 3”, or as the fraction \(\frac{2}{3}\).

Equivalent Ratios and Simplifying Ratios

Equivalent ratios express the same relationship between quantities. For instance, 2:3 is equivalent to 4:6 because both represent the same proportion when simplified.

To simplify a ratio:

1. Find the greatest common divisor (GCD) of the two numbers.
2. Divide both numbers by the GCD.

Example: Simplify the ratio 12:18.

The GCD of 12 and 18 is 6. Dividing both by 6 gives 2:3, the simplified form.

Ratio Word Problems

Ratios often appear in real-life situations such as mixing ingredients, comparing speeds, or splitting quantities.

Since the ratio of red to blue is 5:7, and blue paint amount is 35 liters (7 parts), each part is 35/7 = 5 liters. Therefore, red paint required = 5 parts × 5 liters = 25 liters.

2. Introduction to Percentages

What is a Percentage?

A percentage is a way to express a number as a fraction of 100. The word “percent” means “per hundred.” For example, 45% means 45 out of 100.

Percentages are widely used to describe proportions, growth rates, discounts, taxes, and much more.

Converting Between Fractions, Decimals, Ratios, and Percentages

Conversion skills are critical to moving smoothly between different representations:

• Ratio to Fraction: \(a:b = \frac{a}{b}\)
• Fraction to Decimal: Divide numerator by denominator.
• Decimal to Percentage: Multiply by 100 and add “%.”
• Percentage to Decimal: Divide by 100.
• Percentage to Fraction: Write the percentage over 100 and simplify.

Example: Convert the ratio 3:4 to a percentage.

Ratio as fraction = \(\frac{3}{4} = 0.75\) decimal. Multiply by 100 = 75%. So, 3:4 corresponds to 75%.

3. Calculating Percentages of Quantities

To find a percentage of a quantity:

Percentage of Quantity = (Percentage ÷ 100) × Quantity

Example: What is 20% of 150?

20% of 150 = (20 ÷ 100) × 150 = 0.20 × 150 = 30.

Percentage Increase and Decrease

Percentage change shows how much a quantity has increased or decreased relative to its original value.

• Percentage Increase: \(\frac{\text{New} – \text{Original}}{\text{Original}} \times 100\%\)
• Percentage Decrease: \(\frac{\text{Original} – \text{New}}{\text{Original}} \times 100\%\)

Example: A jacket originally priced at $80 is now $60. What is the percentage decrease?

Decrease = 80 – 60 = 20

Percentage decrease = \(\frac{20}{80} \times 100 = 25\%\)

Percentage Error and Percentage Points

Percentage error measures the accuracy of an estimate compared to the true value:

Percentage error = \(\frac{|\text{Measured} – \text{True}|}{\text{True}} \times 100\%\)

Percentage points differ from percent change. For example, if the interest rate rises from 5% to 7%, the increase is 2 percentage points, which is a 40% increase relative to the original rate.

4. Advanced Concepts and Applications

Compound Percentages and Successive Percentage Changes

Sometimes, values change by several percentages in sequence. To find the overall effect, multiply the successive change factors.

Formula: \((1 + p_1)(1 + p_2)(1 + p_3) \cdots – 1\), where each \(p_i\) is expressed as a decimal (increase positive, decrease negative).

Example: A product price increases by 10% one year, then decreases by 5% the next year. What is the overall percentage change?

Overall change factor = \(1.10 \times 0.95 = 1.045\)

Overall change = (1.045 – 1) × 100 = 4.5% increase.

Ratio Applications in Mixtures and Solutions

Ratios are critical in mixing substances to achieve desired concentrations. For example, if a solution requires a 3:5 ratio of acid to water, adjusting volumes while maintaining the ratio is essential.

Using Ratios and Percentages in Financial Mathematics

Financial concepts such as interest rates, discounts, profit and loss all rely heavily on percentage calculations.

Discount = 15% of 120 = 0.15 × 120 = $18.

Sale price = 120 – 18 = $102.

Inverse Ratios and Their Applications

Inverse ratios describe scenarios where one quantity increases while the other decreases to keep a product constant.

Example: Speed and travel time are inversely proportional. If speed doubles, travel time halves.

Ratio Scales and Maps

Ratios are used to represent scale in maps and models, e.g., a 1:50,000 map means 1 unit on the map represents 50,000 units in reality.

Using Algebra to Solve Complex Ratio and Percentage Problems

Algebraic methods help handle unknowns within ratio and percentage problems. For example, setting variables to unknown quantities and forming equations based on ratio relationships.

Let number of boys = 3x, girls = 4x.

After 5 boys join, boys = 3x + 5, girls = 4x.

New ratio: \(\frac{3x + 5}{4x} = \frac{4}{5}\)

Cross multiply: \(5(3x + 5) = 4 \times 4x \Rightarrow 15x + 25 = 16x \Rightarrow 25 = x\)

Original boys = 3 × 25 = 75; girls = 4 × 25 = 100.

5. Real-World Examples and Case Studies

Increase = 60 – 50 = 10 million.

Percentage increase = \(\frac{10}{50} \times 100 = 20\%\).

This reveals a 20% increase over 10 years, helpful for planners and policymakers.

Sugar amount = \(1.5 \times \frac{1}{2} = 0.75\) cups.

Using ratios ensures ingredients remain proportional for consistent taste.

Profit = 250,000 – 200,000 = 50,000.

Profit margin = \(\frac{50,000}{250,000} \times 100 = 20\%\).

This percentage helps assess financial health and compare with competitors.

Practical Exercises and Assignments

Quiz Questions