Learn on PengiBig Ideas Math, Course 1Chapter 5: Ratios and Rates

Lesson 1: Ratios

In this Grade 6 lesson from Big Ideas Math Course 1, Chapter 5, students learn how to define and write ratios as comparisons of two quantities, including part-to-part, part-to-whole, and whole-to-part relationships. Students practice expressing ratios using multiple formats such as "a to b" and a:b notation through real-world contexts like counting coins and mixing paint colors. Tape diagrams are also introduced as a visual tool for representing the relationship between two quantities.

Section 1

Representing Ratios

Property

A ratio is commonly described as a pair of positive numbers, written $a : b$ and read as “ $a$ to $b$ .” When describing a ratio, the order of the quantities must be the same as the order of the numbers.

Examples

In a garden, there are 5 rose bushes for every 2 lilac bushes. The ratio of roses to lilacs is 5:2.
A pancake recipe requires 2 cups of flour for every 1 cup of milk. The ratio of flour to milk is 2:1.
For every 3 games the team wins, they lose 1. The ratio of wins to losses is 3:1.

Explanation

A ratio is a recipe for comparing two amounts. It tells you, "for every this, you have that." The order is critical—a ratio of 2:3 is not the same as 3:2, just like putting on shoes then socks doesn't work!

Section 2

Introduction: Part-to-Part and Part-to-Whole Ratios

Property

A part-to-part ratio compares the sizes of different parts of a given population. For example, the ratio of right-handed people to left-handed people.
A part-to-whole ratio compares one part of a population to the entire population. For example, saying “one in ten people is left-handed” is a part-to-whole ratio.

Examples

In a fruit basket with 5 apples and 8 bananas, the part-to-part ratio of apples to bananas is $5:8$ . The part-to-whole ratio of apples to all fruit is $5:13$ .
A classroom has 12 girls and 15 boys. The part-to-part ratio of girls to boys is $12:15$ , which simplifies to $4:5$ . The part-to-whole ratio of boys to students is $15:27$ , which simplifies to $5:9$ .
A bag contains 10 red marbles and 6 blue marbles. The part-to-part ratio of blue to red is $6:10$ . The part-to-whole ratio of blue to all marbles is $6:16$ .

Explanation

Think of it like a pizza! A part-to-part ratio compares pepperoni slices to mushroom slices. A part-to-whole ratio compares pepperoni slices to the total number of slices in the whole pizza.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Representing Ratios

Property

Examples

In a garden, there are 5 rose bushes for every 2 lilac bushes. The ratio of roses to lilacs is 5:2.
A pancake recipe requires 2 cups of flour for every 1 cup of milk. The ratio of flour to milk is 2:1.
For every 3 games the team wins, they lose 1. The ratio of wins to losses is 3:1.

Explanation

Section 2

Introduction: Part-to-Part and Part-to-Whole Ratios

Property

Examples

In a fruit basket with 5 apples and 8 bananas, the part-to-part ratio of apples to bananas is $5:8$ . The part-to-whole ratio of apples to all fruit is $5:13$ .
A classroom has 12 girls and 15 boys. The part-to-part ratio of girls to boys is $12:15$ , which simplifies to $4:5$ . The part-to-whole ratio of boys to students is $15:27$ , which simplifies to $5:9$ .
A bag contains 10 red marbles and 6 blue marbles. The part-to-part ratio of blue to red is $6:10$ . The part-to-whole ratio of blue to all marbles is $6:16$ .

Explanation

Think of it like a pizza! A part-to-part ratio compares pepperoni slices to mushroom slices. A part-to-whole ratio compares pepperoni slices to the total number of slices in the whole pizza.

Overview

Lesson overview

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

Overview

Section 1

Representing Ratios

Property

Examples

In a garden, there are 5 rose bushes for every 2 lilac bushes. The ratio of roses to lilacs is 5:2.
A pancake recipe requires 2 cups of flour for every 1 cup of milk. The ratio of flour to milk is 2:1.
For every 3 games the team wins, they lose 1. The ratio of wins to losses is 3:1.

Explanation

Section 2

Introduction: Part-to-Part and Part-to-Whole Ratios

Property

Examples

In a fruit basket with 5 apples and 8 bananas, the part-to-part ratio of apples to bananas is $5:8$ . The part-to-whole ratio of apples to all fruit is $5:13$ .
A classroom has 12 girls and 15 boys. The part-to-part ratio of girls to boys is $12:15$ , which simplifies to $4:5$ . The part-to-whole ratio of boys to students is $15:27$ , which simplifies to $5:9$ .
A bag contains 10 red marbles and 6 blue marbles. The part-to-part ratio of blue to red is $6:10$ . The part-to-whole ratio of blue to all marbles is $6:16$ .

Explanation

Think of it like a pizza! A part-to-part ratio compares pepperoni slices to mushroom slices. A part-to-whole ratio compares pepperoni slices to the total number of slices in the whole pizza.