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Ratios

Covers the definition of ratios, methods for simplifying and comparing ratios, and applications such as scaling and unit-rate calculations.

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Ratios

Study Guide

Key Definition

A ratio is a comparison of two numbers. It can be written as $1 : 5$, $1 \text{ to } 5$, or $\frac{1}{5}$.

Important Notes

Ratios can be simplified like fractions: $\frac{6}{4}$ simplifies to $\frac{3}{2}$
The reciprocal of a ratio $3 : 2$ is $2 : 3$
Ratios are used to compare quantities
Order matters in ratios; $1 : 2$ is different from $2 : 1$
Ratios can be expressed in different forms but represent the same comparison

Mathematical Notation

$\frac{a}{b}$ means division of a by b

$:$ represents a ratio

$\times$ represents multiplication

$\div$ represents division

Remember to use proper notation when solving problems

Why It Works

Ratios express the relative size of two quantities by division. Simplifying a ratio by dividing both terms by their greatest common divisor preserves the proportional relationship.

Remember

Always simplify ratios by dividing both terms by their greatest common divisor, and note that the order of terms matters: a : b is not the same as b : a.

Quick Reference

Reducing Ratios:$\frac{a}{b} = \frac{c}{d}$ if $a \times d = b \times c$

Part-to-Whole Conversion:For part a and total a+b, the ratio of part to whole is $a : (a + b)$

Scaling Ratios:Multiplying both terms by k yields $a : b = (k a) : (k b)$

Unit Rate:Converting a ratio $a : b$ to a unit rate gives $(a/b) : 1$

Understanding Ratios

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Video explanation of this concept

Beginner

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Beginner Explanation

A ratio compares two quantities by division. For example, if there are 4 apples and 2 oranges, the ratio of apples to oranges is $4 : 2$, which simplifies to $2 : 1$ by dividing both numbers by 2.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the simplified ratio of $8 : 4$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If you have 10 apples and 15 oranges, what is the ratio of apples to oranges in simplest form?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a car travels 60 miles in 2 hours, what is the ratio of miles to hours?

Challenge Quiz

Single Choice Quiz

Advanced

If $a : b = 3 : 4$ and $b : c = 5 : 6$, what is $a : c$?

Recap

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Review key concepts and takeaways