Learn on PengiBig Ideas Math, Course 2Chapter 5: Ratios and Proportions

Lesson 4: Solving Proportions

In this Grade 7 lesson from Big Ideas Math Course 2, Chapter 5, students learn to solve proportions using two key methods: the Multiplication Property of Equality and the Cross Products Property. Through hands-on activities and worked examples, students practice finding unknown values in proportions involving fractions, decimals, and real-world contexts such as science solutions and turnpike tolls. This lesson builds on ratio table concepts and aligns with Florida Standards MAFS.7.RP.1.2b and MAFS.7.RP.1.2c.

Section 1

Choosing and Applying Solution Methods for Proportions

Property

Three methods for solving proportions: Mental Math (when numbers have obvious relationships), Multiplication Property of Equality (multiply both sides by the LCD), and Cross Products Property: If $\frac{a}{b} = \frac{c}{d}$ , then $a \cdot d = b \cdot c$ .

Examples

Section 2

Mental math with scale factors

Property

To solve problems involving proportional variables, we can use a build-up strategy. This involves finding a scale factor that relates a known quantity to a desired quantity. If we multiply one variable by this scale factor, we must multiply the other variable by the same scale factor to maintain the proportional relationship. This process can be organized in a ratio table.

Examples

A recipe for soup requires 3 cups of broth to serve 4 people. To serve 12 people, you use a scale factor of 3 (since $4 \times 3 = 12$ ). Therefore, you need $3 \times 3 = 9$ cups of broth.
A car travels 180 miles in 3 hours. To find how far it travels in 40 minutes, we use a scale factor of $\frac{2}{9}$ (since 40 minutes is $\frac{2}{3}$ of an hour, and we are starting from 3 hours, so $\frac{2/3}{3} = \frac{2}{9}$ ). The distance is $\frac{2}{9} \times 180 = 40$ miles.
If 5 comic books cost 22 dollars, how much do 20 comic books cost? The number of books is multiplied by a scale factor of 4, so we multiply the cost by 4: $22 \times 4 = 88$ dollars.

Explanation

This is like resizing a photo. To keep the picture from looking stretched or squished, you have to scale the height and width by the same percentage. With proportions, you multiply both variables by the same scale factor to get the right answer.

Section 3

Method 2: Multiplication Property of Equality

Property

A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$ , where $b \neq 0$ , $d \neq 0$ . The proportion is read " $a$ is to $b$ as $c$ is to $d$ ." Since a proportion is an equation with fractions, we can solve it by multiplying both sides of the equation by the LCD (least common denominator) to clear the fractions. This method uses the multiplication property of equality to eliminate denominators.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Choosing and Applying Solution Methods for Proportions

Property

Examples

Section 2

Mental math with scale factors

Property

Examples

A recipe for soup requires 3 cups of broth to serve 4 people. To serve 12 people, you use a scale factor of 3 (since $4 \times 3 = 12$ ). Therefore, you need $3 \times 3 = 9$ cups of broth.
A car travels 180 miles in 3 hours. To find how far it travels in 40 minutes, we use a scale factor of $\frac{2}{9}$ (since 40 minutes is $\frac{2}{3}$ of an hour, and we are starting from 3 hours, so $\frac{2/3}{3} = \frac{2}{9}$ ). The distance is $\frac{2}{9} \times 180 = 40$ miles.
If 5 comic books cost 22 dollars, how much do 20 comic books cost? The number of books is multiplied by a scale factor of 4, so we multiply the cost by 4: $22 \times 4 = 88$ dollars.

Explanation

Section 3

Method 2: Multiplication Property of Equality

Property

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Choosing and Applying Solution Methods for Proportions

Property

Examples

Section 2

Mental math with scale factors

Property

Examples

A recipe for soup requires 3 cups of broth to serve 4 people. To serve 12 people, you use a scale factor of 3 (since $4 \times 3 = 12$ ). Therefore, you need $3 \times 3 = 9$ cups of broth.
A car travels 180 miles in 3 hours. To find how far it travels in 40 minutes, we use a scale factor of $\frac{2}{9}$ (since 40 minutes is $\frac{2}{3}$ of an hour, and we are starting from 3 hours, so $\frac{2/3}{3} = \frac{2}{9}$ ). The distance is $\frac{2}{9} \times 180 = 40$ miles.
If 5 comic books cost 22 dollars, how much do 20 comic books cost? The number of books is multiplied by a scale factor of 4, so we multiply the cost by 4: $22 \times 4 = 88$ dollars.