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# Ratios and proportions | Lesson

## 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, colon, b or start fraction, a, divided by, b, end fraction.
A proportion is an equality of two ratios. We write proportions to help us establish equivalent ratios and solve for unknown quantities.

### 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.
start text, p, a, r, t, end text, colon, start text, w, h, o, l, e, end text, equals, start text, p, a, r, t, end text, colon, start text, s, u, m, space, o, f, space, a, l, l, space, p, a, r, t, s, end text
To write a ratio:
1. Determine whether the ratio is part to part or part to whole.
2. Calculate the parts and the whole if needed.
3. Plug values into the ratio.
4. Simplify the ratio if needed. Integer-to-integer ratios are preferred.
Equivalent ratios are ratios that have the same value. Given a ratio, we can generate equivalent ratios by multiplying both parts of the ratio by the same value.

## 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 \\\\ \dfrac{a}{b}&=\dfrac{c}{d} \end{aligned}
To use a proportional relationship to find an unknown quantity:
1. Write an equation using equivalent ratios.
2. Plug in known values and use a variable to represent the unknown quantity.
3. 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.
4. If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.

TRY: WRITING A RATIO
A pancake recipe uses start fraction, 1, divided by, 4, end fraction cup of all-purpose flour and start fraction, 1, divided by, 4, end fraction cup of rice flour. What is the ratio of all-purpose flour to rice flour in the recipe?

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?

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.
To write a ratio:
1. Determine whether the ratio is part to part or part to whole.
2. Calculate the parts and the whole if needed.
3. Plug values into the ratio.
4. Simplify the ratio if needed. Integer-to-integer ratios are preferred.
Equivalent ratios are ratios that have the same value.
\begin{aligned} a:b &= c:d \\\\ \dfrac{a}{b}&=\dfrac{c}{d} \end{aligned}
To use a proportional relationship to find an unknown quantity:
1. Write an equation using equivalent ratios.
2. Plug in known values and use a variable to represent the unknown quantity.
3. 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.
4. If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.