Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 6: Ratios and Percents

Lesson 1: Basic Ratio Problems

In this Grade 4 lesson from AoPS: Introduction to Algebra, students learn how to interpret and apply basic ratios, including writing ratios as fractions or in colon notation and converting part-to-part ratios into part-to-whole ratios. Using problems aligned with AMC 8 and AMC 10 competition math, students practice setting up equations and solving for unknown quantities when given ratio and total information. This lesson builds foundational skills in proportional reasoning that are essential for more advanced algebra and competition problem-solving.

Section 1

Writing Ratios in Three Notations

Property

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$ , $\frac{a}{b}$ , or $a:b$ .

Examples

The ratio 20 to 36 can be written as $20\ \text{to}\ 36$ , $20:36$ , and $\frac{20}{36}$ , which simplifies to $\frac{5}{9}$ .

The ratio 45 to 18 can be written as $45\ \text{to}\ 18$ , $45:18$ , and $\frac{45}{18}$ , which simplifies to $\frac{5}{2}$ .

Section 2

Calculating the Whole from the Parts of a Ratio

Property

Part-to-part ratios compare individual parts: $a:b$

Part-to-whole ratios compare one part to the total: $a:(a+b)$ or $b:(a+b)$

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Writing Ratios in Three Notations

Property

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$ , $\frac{a}{b}$ , or $a:b$ .

Examples

The ratio 20 to 36 can be written as $20\ \text{to}\ 36$ , $20:36$ , and $\frac{20}{36}$ , which simplifies to $\frac{5}{9}$ .

The ratio 45 to 18 can be written as $45\ \text{to}\ 18$ , $45:18$ , and $\frac{45}{18}$ , which simplifies to $\frac{5}{2}$ .

Section 2

Calculating the Whole from the Parts of a Ratio

Property

Part-to-part ratios compare individual parts: $a:b$

Part-to-whole ratios compare one part to the total: $a:(a+b)$ or $b:(a+b)$

Overview

Lesson overview

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

Overview

Section 1

Writing Ratios in Three Notations

Property

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$ , $\frac{a}{b}$ , or $a:b$ .

Examples

The ratio 20 to 36 can be written as $20\ \text{to}\ 36$ , $20:36$ , and $\frac{20}{36}$ , which simplifies to $\frac{5}{9}$ .

The ratio 45 to 18 can be written as $45\ \text{to}\ 18$ , $45:18$ , and $\frac{45}{18}$ , which simplifies to $\frac{5}{2}$ .

Section 2

Calculating the Whole from the Parts of a Ratio

Property

Part-to-part ratios compare individual parts: $a:b$

Part-to-whole ratios compare one part to the total: $a:(a+b)$ or $b:(a+b)$