Ratios and Proportions

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Ratios and proportional relationships

Ratios and proportions are similar figures and concepts that are as easily confused as toads and frogs (all toads are frogs, but not all frogs are toads). Ratios compare values, while proportions compare ratios.

What are ratios?

Ratios compare values. You can compare the number of brown-haired boys to the number of blond-haired boys, or to the number of pencils in the classroom, or to the number of brown-haired girls, or … well, you get the idea. Ratios compare values of the same things or things that are different.

Say you have 10 brown-haired girls in a class, and 6 blonde-haired girls in the same class. You can set up six different ratios:

1. $\frac{10}{16}$: Brown-haired girls to all girls

2. $\frac{6}{10}$: Blonde-haired girls to brown-haired girls

3. $\frac{6}{16}$: Blonde-haired girls to all girls

4. $\frac{10}{6}$: Brown-haired girls to blonde-haired girls

5. $\frac{16}{10}$: All girls to brown-haired girls

6. $\frac{16}{6}$: All girls to blonde-haired girls

Three of those ratios are improper fractions; that is okay! Ratios can be written as proper or improper fractions. They can also be written with a semicolon, like this:

1. 10:16

2. 6:10

3. 6:16

4. 10:6

5. 16:10

6. 16:6

What are proportions?

When you compare two ratios, you use proportions. You are asking if the first ratio is the same, less than, or more than the second ratio. Compare the ratios of brown-to-all girls and blonde-to-all girls:

You can see these two ratios are not equal, so they are not proportional:

How to solve ratios and proportions

What would proportional fractions look like? Let's add eight class pets to the classroom: 5 hamsters and 3 frogs. The ratios you can create are:

• 5:3 (hamsters to frogs)

• 3:5 (frogs to hamsters)

• 5:8 (hamsters to all pets)

• 3:8 (frogs to all pets)

• 8:5 (all pets to hamsters)

• 8:3 (all pets to frogs)

Proportions can tell us if two ratios are equal or not. Compare the ratio of hamsters to all pets and the ratio of brown-haired girls to all girls:

You can check these fractions in a few ways, such as simplifying $\frac{10}{16}$ to $\frac{10}{16}$, or by cross-multiplying and dividing: $\frac{5\times 16}{10}=8$

These two ratios are proportional to each other. The ratio of hamsters to all class pets is the same as the ratio of brown-haired girls to all girls in the class.

Ratios and proportions word problems

Cooking, comparing prices, driving, engineering, construction and finance are just some areas where ratios and proportions work every day.

Here is a recipe for hamster food to feed one hamster:

• 20g of five-cereal blend

• 10g small seed blend

• 10g rolled oats

• 10g dried vegetables

• 5g nuts

• 5g dried fruit

One hamster gets 60 grams of hamster chow. How much should you mix for five hamsters?

Whatever you multiply 1 times to get 5, multiply 60 times the same number. You need 300 grams.

How much of each ingredient should you mix?

For every 60 grams of hamster chow for one hamster, 20 grams is five-cereal blend, a ratio of 20:60 or 1:3.

If you want to feed five hamsters, you have to mix more of everything in the right proportions. How many grams of five-cereal blend will you need?

Say you did your calculations and mixed the five-cereal blend at a ratio of 50:300. Is that correct? Check: Is $\frac{50}{300}$ proportional to $\frac{20}{60}$ or $\frac{20}{60}$?

You can cross-multiply and divide to check: $\frac{50\times 3}{300}$.

You see that $\frac{150}{300}=\frac{1}{2}$, not 1. So your mix is not in the right proportion because 50 is not one third of 300.

You needed 100 g of five-cereal blend to maintain the right proportions.

Ratios and proportions examples

Perhaps you have a part-time job in a grocery store, assembling gift baskets of fruit. Your manager tells you to maintain a ratio of 2:3 of pears to apples for every size of basket. A small basket gets 2 pears and 3 apples. An extra-large basket must have the same ratio, 2:3, but be five times larger.

The ratio of pears:apples is 2:3, so multiply both parts of the ratio times 5 to get the new ratio: 10:15.

Your extra-large gift basket needs 10 pears and 15 apples.

Ratios and proportions practice

The class of 10 brown-haired and 6 blonde-haired girls also has boys in it. Of the 12 boys in the class, 4 have blond hair and 8 have brown hair.

Write three ratios using this new information.

Many ratios can be written from the information. See if you can figure out what these ratios describe:

• 4:12

• 8:12

• 4:28

• 8:10

• 28:8

Did you get these answers?

• 4:12 (Blond-haired boys to all boys)

• 8:12 (Brown-haired boys to all boys)

• 4:28 (Blond-haired boys to all students)

• 8:10 (Brown-haired boys to brown-haired girls)

• 28:8 (All students to brown-haired boys)

Lesson summary

You have learned that ratios compare values, while proportions compare ratios. Proportions are most often used to ensure ratios are equal when they increase or decrease. You can write ratios as either a fraction or with a colon between them, like this: $\frac{10}{16}$ or $10:16$. Ratios can compare like and unlike things. Both ratios and proportions are useful in many aspects of everyday life.