Ratios and proportions | Lesson (article)

What are ratios and proportions?

A ratio is a comparison of two quantities. The ratio of $a$ ‍ to $b$ ‍ can also be expressed as $a : b$ ‍ or $\frac{a}{b}$ ‍.

Pepper has $3$ ‍ hats and $2$ ‍ scarves.

The ratio of hats to scarves is $3$ ‍ to $2$ ‍, $3 : 2$ ‍, or $\frac{3}{2}$ ‍.

The ratio of scarves to hats is $2$ ‍ to $3$ ‍, $2 : 3$ ‍, or $\frac{2}{3}$ ‍.

A proportion is an equality of two ratios. We write proportions to help us establish equivalent ratios and solve for unknown quantities.

A recipe calls for $3$ ‍ cups of flour for every $1$ ‍ cup of water. If Shaun follows the recipe and uses $6$ ‍ cups of flour, how many cups of water does he need?

In the recipe, the ratio of flour to water is $3$ ‍ to $1$ ‍. To follow the recipe, Shaun's flour-to-water ratio must also be $3$ ‍ to $1$ ‍.

Using $x$ ‍ to represent the number of cups of water Shaun needs:

$\frac{3 cups of flour}{1 cup of water} = \frac{6 cups of flour}{x cups of water}$ ‍

We can solve the equation for $x$ ‍ to find the number of cups of water Shaun needs.

What skills are tested?

Identifying and writing equivalent ratios
Solving word problems involving ratios
Solving word problems using proportions

How do we write ratios?

Two common types of ratios we'll see are part to part and part to whole. For example, when we make lemonade:

The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.
The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.

$part : whole = part : sum of all parts$ ‍

To write a ratio:

Determine whether the ratio is part to part or part to whole.
Calculate the parts and the whole if needed.
Plug values into the ratio.
Simplify the ratio if needed. Integer-to-integer ratios are preferred.

There are $20$ ‍ seniors and $5$ ‍ lower-level students on the varsity soccer team. What is the ratio of lower-level students to total number of students on the varsity soccer team?

The ratio is that of $lower-level students$ ‍ to $total number of students$ ‍ on the varsity soccer team. It is a part to whole ratio.

There are $5$ ‍ members of the varsity soccer team who are $lower-level students$ ‍. The $total number of students$ ‍ includes both seniors and lower-level students: $20 + 5 = 25$ ‍.

The ratio of $lower-level students$ ‍ to $total number of students$ ‍ is $5$ ‍ to $25$ ‍, which can also be written as $5 : 25$ ‍ or $\frac{5}{25}$ ‍.

$\frac{5}{25} = \frac{1 \times 5}{5 \times 5} = \frac{1}{5}$ ‍

The ratio of lower-level students to total number of students on the varsity soccer team is $1$ ‍ to $5$ ‍. This is equivalent to:

How do we use proportions?

If we know a ratio and want to apply it to a different quantity (for example, doubling a cookie recipe), we can use proportional relationships, or equations of equivalent ratios, to calculate any unknown quantities.

$\begin{aligned} a : b & = c : d \\ \frac{a}{b} & = \frac{c}{d} \end{aligned}$ ‍

To use a proportional relationship to find an unknown quantity:

Write an equation using equivalent ratios.
Plug in known values and use a variable to represent the unknown quantity.
If the numeric part of one ratio is a multiple of the corresponding part of the other ratio, we can calculate the unknown quantity by multiplying the other part of the given ratio by the same number.
If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.

There are $340$ ‍ students at Armin Academy. If the student-to-teacher ratio is $17 : 2$ ‍, how many teachers are there?

We can set up a proportional relationship using equivalent student-to-teacher ratios. Using $x$ ‍ to represent the number of teachers:

$\begin{aligned} \frac{students}{teachers} & = \frac{students}{teachers} \\ \frac{340 students}{x teachers} & = \frac{17 students}{2 teachers} \\ \frac{340}{x} & = \frac{17}{2} \end{aligned}$ ‍

Some of us might notice that $340$ ‍ is $20$ ‍ times $17$ ‍. As such, $x$ ‍ must also be $20$ ‍ times $2$ ‍ for the two ratios to be equivalent: $20 \times 2 = 40$ ‍ teachers.

Alternatively, we can solve the equation for $x$ ‍:

$\begin{aligned} \frac{340}{x} & = \frac{17}{2} \\ \frac{340}{x} \times x & = \frac{17}{2} \times x \\ 340 & = \frac{17}{2} x \\ 340 \times \frac{2}{17} & = \frac{17}{2} x \times \frac{2}{17} \\ 40 & = x \end{aligned}$ ‍

There are $40$ ‍ teachers.

Your turn!

TRY: WRITING A RATIO

A pancake recipe uses $\frac{1}{4}$ ‍ cup of all-purpose flour and $\frac{1}{4}$ ‍ cup of rice flour. What is the ratio of all-purpose flour to rice flour in the recipe?

Choose 1 answer:

(Choice A)
$1 : 4$ ‍
(Choice B)
$1 : 2$ ‍
(Choice C)
$1 : 1$ ‍
(Choice D)
$2 : 1$ ‍
(Choice E)
$4 : 1$ ‍

TRY: WRITING A RATIO

Pippin owns $2$ ‍ cats, $3$ ‍ dogs, and a lizard as pets. What is the ratio of the number of cats to the total number of pets Pippin owns?

Choose 1 answer:

(Choice A)
$\frac{1}{6}$ ‍
(Choice B)
$\frac{1}{3}$ ‍
(Choice C)
$\frac{2}{5}$ ‍
(Choice D)
$\frac{1}{2}$ ‍
(Choice E)
$\frac{2}{3}$ ‍

TRY: SOLVING USING A PROPORTIONAL RELATIONSHIP

Nicholas drinks $8$ ‍ ounces of milk for every $5$ ‍ cookies he eats. If he eats $20$ ‍ cookies, how many ounces of milk does he drink?

ounces

TRY: SOLVING USING A PROPORTIONAL RELATIONSHIP

The ratio of fiction books to non-fiction books in Roxane's library is $7$ ‍ to $4$ ‍. If Roxane owns $182$ ‍ fiction books, how many non-fiction books does she own?

Things to remember

A ratio is a comparison of two quantities.

A proportion is an equality of two ratios.