Introduction

We can find equivalent ratios by multiplying or dividing all numbers in a ratio by the same value.

To demonstrate, let's consider the following ratio:

2:6

If we multiply both numbers by 5, for example, we get the ratio 10:30.

\begin{array}{llllll} \,\,{\color{blue}2}& &: &\,\,\,&\,\, {\color{red}6}& \\ \,\,\downarrow\times\, 5&&&& \,\,\downarrow \times\, 5& \\ \,10& &:&& \,30& \end{array}

Therefore, the ratios 2:6 and 10:30 are equivalent.

Similarly, if we take the ratio 2:6 and divide both numbers by 2, we get 1:3.

\begin{array}{llllll} \,\,{\color{blue}2}& &: &\,\,\,&\,\, {\color{red}6}& \\ \,\,\downarrow\div\, 2&&&& \,\,\downarrow \div\, 2& \\ \,\,1& &:&& \,\,3& \end{array}

Therefore, the ratios 2:6 and 1:3 are also equivalent.

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Finding Missing Values in Equivalent Ratios

Suppose that the following two ratios are equivalent:

3 : 5, \qquad 12 : \fbox{[math]\phantom{0}[/math]}

What is the missing number?

To find a ratio equivalent to 3 : 5, we multiply or divide both numbers by the same value.

In this case, we need to get 12 from {\color{blue}{3}}, so we multiply {\color{blue}{3}} by 4.

\begin{array}{llllll} \,\,{\color{blue}3}& &: &\,\,\,&\,\, {\color{red}5}& \\ \,\,\downarrow\times\, 4&&&& \,\,\downarrow \times\, 4& \\ 12& &:&& \,\fbox{[math]\phantom{0}[/math]}& \end{array}

So, to get \,\fbox{[math]\phantom{0}[/math]}\, from {\color{red}{5}}, we multiply {\color{red}{5}} by 4 as well:

{\color{red}5} \times 4 = \bbox[3px,lightgray]{\color{red}20}

Therefore, the missing number is 20.

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Example: Identifying Equivalent Ratios

Which of the following ratios is equivalent to 12 : 9?

  1. 2 : 3
  2. 4 : 3
  3. 5 : 3
EXPLANATION

All of the answer choices have the second number equal to 3. So our equivalent ratio is of the form \,\fbox{[math]\phantom{0}[/math]} : 3.

To find a ratio that's equivalent to 12 : 9, we divide both numbers by the same value. In particular, we need to get 3 from {\color{red}{9}}, so we divide {\color{red}{9}} by 3.

\begin{array}{llllll} {\color{blue}12}& &:&\,\,\,&\,\,\, {\color{red}9}& \\ \,\,\downarrow\div\,\, 3&&&& \,\,\,\downarrow \div\,\, 3& \\ \,\fbox{[math]\phantom{0}[/math]}& &:&&\, \,\,3& \end{array}

So to get \,\fbox{[math]\phantom{0}[/math]} from {\color{blue}{12}}, we divide 12 by 3 as well:

{\color{blue}12} \div 3 =4

Therefore, 12:9 is equivalent to 4:3.

So, the correct answer is "II only."

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Practice: Identifying Equivalent Ratios

Determine the missing number that makes the ratio $8 : \fbox{$\phantom{0}$}\,$ equivalent to $2 : 3.$

a
 $12$
b
 $14$
c
 $10$
d
 $15$
e
 $16$
Practice: Identifying Equivalent Ratios

The ratio $15 : 35$ is equivalent to the ratio $3:$

a
b
c
d
e
Expressing Ratios in Their Simplest Form

We can simplify ratios by dividing both numbers in the ratio by a common factor.

To demonstrate, let's simplify the following ratio:

18:9

The numbers 18 and 9 share a common factor of 9, so we can divide both numbers by 9 as follows:

\begin{array}{llllll} \,\,{\color{blue}18}& &:&\,\,\,\,&\,\,\, {\color{red}9}& \\ \,\,\downarrow\div\,\, 9&&&& \,\,\,\downarrow \div\,\, 9& \\ \,\,2& &:&&\, \,\,1& \end{array}

Therefore, 18:9 is equivalent to 2:1.

Since 2 and 1 have no common factors, 2:1 is the simplest form of this ratio. We often prefer to work with ratios in their simplest form, where possible.

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Example: Simplifying Ratios

What ratio describes the number of triangles to the total number of shapes? Give your answer in its simplest form.

EXPLANATION

First, we count the number of triangles. We see that there are 4 triangles.

Then, we count the total number of shapes. We see that there are 6 shapes in total.

So the ratio of triangles to total shapes is 4:6.

We can simplify this ratio by dividing both numbers in the ratio by 2\mathbin{:}

\begin{array}{llllll} \,\,{\color{blue}4}& &:&\,\,\,\,&\,\,\, {\color{red}6}& \\ \,\,\downarrow\div\,\, 2&&&& \,\,\,\downarrow \div\,\, 2& \\ \,\,2& &:&&\, \,\,3& \end{array}

Therefore, the ratio of triangles to the total shapes is 2:3.

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Practice: Simplifying Ratios

Express the number of circles to the number of pentagons as a ratio in its simplest form.

a
b
c
d
e
Practice: Simplifying Ratios

What ratio describes the number of rhombuses to the total number of shapes?

a
 $1:4$
b
 $2:3$
c
 $1:1$
d
 $1:2$
e
 $3:2$
Example: Completing a Statement

What number is missing from the boxes in the statement below?

The ratio of squares to circles is 1 :\fbox{[math]\phantom{0}[/math]}\, because there is 1 square for every \fbox{[math]\phantom{0}[/math]}\, circles.

EXPLANATION

First, we count the number of squares. We see that there are 2 squares.

Then, we count the number of circles. We see that there are 2 circles.

So the ratio of circles to squares is 2:2.

We can simplify this ratio by dividing both numbers by 2\mathbin{:}

\begin{array}{llllll} \,\,{\color{blue}2}& &:&\,\,\,\,&\,\,\, {\color{red}2}& \\ \,\,\downarrow\div\,\, 2&&&& \,\,\,\downarrow \div\,\,2 & \\ \,\,1& &:&&\, \,\,1& \end{array}

This means that 2:2 is equivalent to 1:1.

So, the complete statement is "The ratio of squares to circles is 1:\bbox[3px,lightgray]{1}\, because there is 1 square for every \bbox[3px,lightgray]{1}\, circle."

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Practice: Completing a Statement

What number is missing from the boxes in the statement below?

The ratio of triangles to hexagons is $1 :\fbox{$\phantom{1}$}\,$ because there is $1$ triangle for every $\fbox{$\phantom{1}$}$ hexagon.

a
 $3$
b
 $4$
c
 $1$
d
 $6$
e
 $2$
Practice: Completing a Statement

There are $18$ parrots and $15$ ducks in a zoo. What number is missing from the boxes in the statement below?

The ratio of parrots to ducks is $6 :\fbox{$\phantom{0}$}\,$ because there are $6$ parrots for every $\fbox{$\phantom{0}$}\,$ ducks.

a
 $5$
b
 $1$
c
 $13$
d
 $7$
e
 $11$
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