Ratios and Proportions

Introduction to Ratios and Proportions

Ratios and proportions are fundamental concepts in algebra that allow us to compare quantities and solve real-world problems involving relative sizes and scaling. Understanding how to write, interpret, and solve ratios and proportions is essential for success in College Algebra and its applications.

Writing Ratios

Definition of a Ratio

A ratio is a comparison of two quantities using division. It expresses how many times one value contains or is contained within the other. Ratios can be written in several forms:

• a to b

• a : b

• \( \frac{a}{b} \)

When writing ratios, it is important that both quantities are in the same units. If not, convert one so that both are comparable.

Examples of Writing Ratios

• Example 1: The ratio of 7 yards to 4 yards is \( 7:4 \) or \( \frac{7}{4} \).

• Example 2: To find the ratio of 8 feet to 6 yards, first convert 6 yards to feet: \( 6 \times 3 = 18 \) feet. The ratio is \( 8:18 \) or \( \frac{8}{18} = \frac{4}{9} \) after simplification.

• Example 3: The ratio of 9 hours to 2 hours is \( 9:2 \) or \( \frac{9}{2} \).

• Example 4: To find the ratio of 10 minutes to 5 seconds, convert 10 minutes to seconds: \( 10 \times 60 = 600 \) seconds. The ratio is \( 600:5 \) or \( 120:1 \).

Visual Examples

Ratios can also be represented visually. For example, if there are 6 flowers and 4 bees, the ratio of flowers to bees is 6:4 or 3:2 after simplification.

Similarly, if there are 7 footballs and 8 basketballs, the ratio of footballs to basketballs is 7:8.

Solving Proportions

Definition of a Proportion

A proportion is an equation stating that two ratios are equal. It is written as:

where b and d are not zero. We read this as "a is to b as c is to d." In a proportion, the cross products are equal:

Solving Proportions

• Set up the proportion so that the ratios are equal.

• Cross-multiply to form an equation.

• Solve for the unknown variable.

Example: Solve \( \frac{2}{3} = \frac{x}{51} \)

Cross-multiply:

Divide both sides by 3:

Solving Proportions with Variables

When variables are present in the numerators or denominators, use the same cross-multiplication method.

Example: Solve \( \frac{w-4}{3} = \frac{w+1}{6} \)

Cross-multiply:

Expand:

Subtract from both sides:

Add 24 to both sides:

Divide by 3:

Solving Applied Problems Using Proportions

Unit Price and Best Buy Problems

Proportions are useful for comparing unit prices to determine the best buy. To find the unit price, divide the total price by the number of units (e.g., ounces).

The lowest unit price indicates the best buy. In this example, the 40-oz jar is the best buy.

Solving Real-World Proportion Problems

• Example 1: If 2 liters of sea water contain 70 grams of salt, how much salt is in 32 liters?

Set up the proportion:

Cross-multiply:

grams

• Example 2: On a scaled drawing, a building measures 4.5 cm tall. If the scale is 25 meters per 2 cm, how tall is the actual building?

Set up the proportion:

Cross-multiply:

meters

• Example 3: A factory can make 20 toasters in 0.5 hours. How many toasters can it make in 6.5 hours?

Set up the proportion:

Cross-multiply:

toasters

• Example 4: A farmer gets 340 bushels of corn from 2 acres. How many bushels from 15 acres?

Set up the proportion:

Cross-multiply:

bushels

Summary Table: Key Concepts