Lesson 2.4: Ratios and Proportions

Introduction

In this lesson, we will delve into the concepts of ratios and proportions, crucial elements of arithmetic that form the basis of quantitative reasoning. Understanding these concepts will not only enhance your problem-solving skills but will also aid you in efficiently tackling GMAT questions.

Learning Objectives

• Set up and simplify ratios and proportions.
• Understand part-to-part versus part-to-whole reasoning.
• Scale quantities and combine ratios.
• Translate ratio language into solvable relationships.
• Solve multi-part ratio and proportion problems.

What is a Ratio?

A ratio is a way to compare two or more quantities. It expresses the relationship between them and is often written in the form of $a:b$, where $a$ and $b$ are two quantities. For example, if there are 2 apples and 3 oranges, the ratio of apples to oranges can be expressed as $2:3.

Working Example 1: Understanding Ratios

Problem: There are 4 red balls and 5 blue balls in a jar. What is the ratio of red balls to total balls?

Solution:

1. Calculate the total number of balls:

Total balls = Red balls + Blue balls = 4 + 5 = 9

1. Then, express the ratio:

Ratio of red balls to total balls = Red balls: Total balls = $4:9.

Common Misconceptions

• Misconception: Ratios can only compare two groups.

Clarification: Ratios can compare more than two groups. For instance, the ratio of red, blue, and green balls could be represented as $4:5:3 if there are 4 red, 5 blue, and 3 green balls.

What is a Proportion?

A proportion is an equation that states that two ratios are equivalent. The general form of a proportion is $ \frac{a}{b} = \frac{c}{d} $, which means the ratio of $a$ to $b$ is the same as the ratio of $c$ to $d$. For example, if the ratio of red balls to blue balls is $2:3 and the ratio of green balls to blue balls is $4:6, these ratios can be set up in a proportion as:

$$ \frac{2}{3} = \frac{4}{6} $$

Working Example 2: Setting Up Proportions

Problem: If $\frac{2}{3} = \frac{x}{12}$, find the value of $x$.

Solution:

1. Set up the proportion:

$ \frac{2}{3} = \frac{x}{12} $

1. Cross-multiply:

$ 2 \cdot 12 = 3 \cdot x $

1. Solve for $x$:

$ 24 = 3x \rightarrow x = \frac{24}{3} = 8 $

Thus, $x = 8$.

Part-to-Part versus Part-to-Whole Reasoning

Understanding the difference between part-to-part and part-to-whole ratios is essential.

• Part-to-Part Ratio: Compares one part of a whole to another part. E.g., in a class of 10 boys and 15 girls, the ratio of boys to girls is $10:15.
• Part-to-Whole Ratio: Compares one part of a whole to the entire whole. Using the previous example, the ratio of boys to the total number of students is $10:(10 + 15) = 10:25$ or simplified to $2:5.

Working Example 3: Part-to-Part and Part-to-Whole

Problem: In a survey of 50 students, 30 like Math and 20 like Science. What is the part-to-part ratio of students who like Math to those who like Science? What is the part-to-whole ratio of students who like Math?

Solution:

1. Part-to-Part Ratio (Math to Science):

Ratio = $30:20 = 3:2$.

1. Part-to-Whole Ratio (Math to Total Students):

Ratio = $30:50 = 3:5$.

Scaling Quantities and Combining Ratios

When you want to scale up a ratio, you can multiply each part of the ratio by the same factor. This is useful in real-world applications, such as cooking or mixing substances.

Working Example 4: Scaling Ratios

Problem: If the ratio of sugar to flour in a recipe is $1:4, how much sugar and flour do you need for 12 cups of flour?

Solution:

1. Identify parts of the ratio:

Sugar = 1 part, Flour = 4 parts. Total parts = 1 + 4 = 5 parts.

1. Determine the scale factor:

Each part represents:

$ \frac{12\text{ cups}}{4\text{ parts}} = 3\text{ cups per part} $

1. Calculate amounts:

Sugar = $1 \times 3 = 3$ cups

Flour = $4 \times 3 = 12$ cups. So, you need 3 cups of sugar and 12 cups of flour.

Translating Ratio Language into Solvable Relationships

Ratios are often presented in word problems. Learning to translate this language into mathematical expressions is vital.

Working Example 5: Word Problem

Problem: If the ratio of cats to dogs is $5:3 and there are 20 cats, how many dogs are there?

Solution:

1. From the ratio, if there are $5 + 3 = 8$ parts total.
2. Set up the relationship:

$ \frac{cats}{dogs} = \frac{5}{3} $

1. With 20 cats:

$ \frac{20}{dogs} = \frac{5}{3} $

1. Cross-multiply to find dogs:

$ 20 \cdot 3 = 5 \cdot dogs $

$ 60 = 5 \cdot dogs \rightarrow dogs = \frac{60}{5} = 12 $

Thus, there are 12 dogs.

Multi-Part Ratio and Proportion Problems

These problems involve more than two parts and require careful setup.

Working Example 6: Multi-Part Problems

Problem: In a class, the ratio of boys to girls is $4:5. If there are 36 boys, how many girls are there?

Solution:

1. Set up the ratio:

Boys:Girls = $4:5.

1. Write this as a proportion:

$ \frac{4}{5} = \frac{36}{girls} $

1. Cross-multiply:

$ 4 \cdot girls = 5 \cdot 36 $

1. Solve for girls:

$ 4 \cdot girls = 180 \rightarrow girls = \frac{180}{4} = 45 $

Thus, there are 45 girls in the class.

Conclusion

In this lesson, we explored the foundations of ratios and proportions. We examined their definitions, built intuition through definitions, examples, and addressed common misconceptions, all of which are crucial for quantitative reasoning. Understanding and applying the concepts of ratios and proportions will significantly enhance your ability to solve GMAT problems efficiently.

Study Notes

• Ratios compare two or more quantities.
• Proportions express equality between two ratios.
• Part-to-part ratios compare segments of a whole.
• Part-to-whole ratios compare a segment to the entire whole.
• To scale a ratio, multiply each part by the same factor.
• Translate word problems into mathematical ratios or proportions.
• Multi-part problems require careful setup and solving with proportions.