Lesson 5.2: Ratios, Proportions, and Percentages

Introduction

In this lesson, students, we will delve into the concepts of ratios, proportions, and percentages—key mathematical skills that are essential for solving a variety of problems in the ACT Mathematics section. Understanding these topics will not only help you on the test but will also enrich your general problem-solving skills.

Learning Objectives

Setting up and solving ratio and proportion problems.
Understanding percent change, percent of, and combined percentage scenarios.
Solving multistep ratio, proportion, and percentage problems.
Translating word problems into proportional or percentage equations.
Explaining the main ideas and terminology behind ratios, proportions, and percentages.

Ratios

A ratio is a comparison of two quantities that shows how many times one value contains or is contained within the other. Ratios can be expressed in several ways:

As a fraction (e.g., $\frac{a}{b}$)
With a colon (e.g., $a:b$)
In words (e.g., "a to b")

Setting Up Ratios

To set up a ratio, identify the two quantities you are comparing. For example, if a recipe calls for 2 cups of flour and 3 cups of sugar, the ratio of flour to sugar can be expressed as:

$$\text{Flour to Sugar} = \frac{2}{3}$$

Example 1: Calculating a Ratio

Consider a classroom with 20 boys and 15 girls. To find the ratio of boys to girls, we set it up as:

$$\text{Boys to Girls} = \frac{20}{15}$$

We can simplify this ratio by dividing both terms by their greatest common divisor, which is 5:

$$\frac{20 \div 5}{15 \div 5} = \frac{4}{3}$$

Thus, the ratio of boys to girls is 4:3.

Proportions

A proportion states that two ratios are equal. It can be represented as:

$$\frac{a}{b} = \frac{c}{d}$$

Setting Up Proportions

To set up a proportion, use the cross-multiplication method to find an unknown quantity. For example, if you know that 4 out of 10 students passed a math test, and you are trying to find how many out of 25 students are expected to pass, set up the proportion:

$$\frac{4}{10} = \frac{x}{25}$$

Example 2: Solving a Proportion

To solve for $x$, apply cross-multiplication:

$$4 \cdot 25 = 10 \cdot x$$

This simplifies to:

$$100 = 10x$$

Now divide both sides by 10:

$$x = \frac{100}{10} = 10$$

So, we expect 10 out of 25 students to pass the test.

Common Misconception: Simplifying Ratios and Proportions

A common mistake is to confuse simplifying ratios and solving proportions. Remember, simplifying a ratio may involve finding a common factor, while solving directly involves establishing equality between two ratios and finding an unknown value.

Percentages

Percentages are a way to express a number as a fraction of 100. The symbol for percentage is “%.” To convert a fraction to a percentage, multiply by 100:

$$\text{Percentage} = \frac{a}{b} \times 100$$

Example 3: Calculating Percentages

If a student scores 18 out of 25 on a test, to find the percentage, we calculate:

$$\text{Percentage} = \frac{18}{25} \times 100 = 72\%$$

Percent Change

Percent change represents the degree of change over time and can be calculated using the formula:

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$$

Example 4: Calculating Percent Change

If a stock price increases from $50 to $65, the percent change is:

$$\text{Percent Change} = \frac{65 - 50}{50} \times 100 = \frac{15}{50} \times 100 = 30\%$$

Combined Percentage Scenarios

In many scenarios, you may have to deal with combined percentages. This often occurs in problems that involve successive percentage changes.

Example 5: Solving Combined Percentage Problems

For instance, if a jacket originally priced at $80 is discounted by 25%, then an additional 10% is applied to the discounted price, how much is the jacket after both discounts?

Calculate the first discount:

$$\text{First Discount} = 80 \times 0.25 = 20$$

The new price after the first discount is:

$$80 - 20 = 60$$

Calculate the second discount:

$$\text{Second Discount} = 60 \times 0.10 = 6$$

The final price after both discounts is:

$$60 - 6 = 54$$

Thus, the jacket now costs $54.

Conclusion

In this lesson, students, we covered the foundational concepts of ratios, proportions, and percentages. We explored how to set up and solve different types of problems, encompassing the fundamental principles that are necessary for success on the ACT. Understanding these skills will not only aid you in passing the exam but will also enhance your everyday mathematical reasoning.

Study Notes

A ratio compares two quantities and can be expressed as a fraction or in words.
A proportion equates two ratios and can be solved using cross-multiplication.
Percentages are a way to express numbers out of 100 and are useful in a variety of contexts.
Percent change quantifies the difference between an original and a new value.
Be mindful of combined percentages and calculate each change step-by-step to avoid errors.