# Ratios and Proportions

### Written by tutor Jeff S.

## Ratios

We use Ratios to describe the relationship between two similar items. Here’s an example of how they work:

A middle school Science teacher decides to use M & M’s candy to help her class understand ratios. Students count 250 M & M’s (total) in a large bag of the candies. Next, students count the number of red M & M’s and find out there are 50 red M & M’s in the bag. Their teacher tells them that the ratio of red M & M’s to total M &M’s is 50 to 250.

We can write this ratio three different ways:

- 50 to 250
- 50:250
- 50/250

One thing to remember about *ratios* is that we *cannot compare different types of things using
ratios*. For example, we cannot use ratios to compare things like the number of M & M’s in a
bag and the ounces of soda we drink in a day. This is because an ounce refers to a weight, which
isn’t the same as a simple count of the number of M & M’s in a bag. Another way to think of
this is that you can’t compare the ounces of M & M’s in a large bag of the candies to ounces of
soda because soda is a liquid and M & M’s small solid candies.

The first step to using ratios is to make sure the items you’re comparing really are similar. Next, decide exactly what you want to compare. For example, do you want to know how many red M & M’s there are compared to green, or do you want to know how many red M & M’s there are compared to the total number in the bag?) Then, make the calculations needed to find the numbers for your ratio. Finally, choose one of the three methods above for writing ratios and write your numbers in ratio format.

**Exercise 1: Read the following paragraphs. First, decide whether ratios can be used to compare the
items. Then, type the ratio in the answer box below each item.**

**A.** Student council decides to compare the number of students who want to have a pizza party
at the end of the semester with students who want to watch a movie and have popcorn instead.
They ask all students to vote during homeroom to find out which they prefer. After counting the
ballots, they find out that 156 students voted to have a pizza party and 210 students would rather
watch a movie and have popcorn.

Yes because we’re comparing two things that are alike: groups of students.

We could write this answer as 156:210, 156 to 210, or 156/210. All three mean the same thing!

**B.** Joey and his brother are arguing about who has had a worse week at school. Joey says that
his week was the worst because he forgot to turn in his homework twice and his Math teacher
gave him four (4) pages of extra homework today. His brother says that his week was worse
because he was late to three (3) different classes and he’ll get detention if he is late to any of
them the rest of the semester.

Yes, because Joey’s extra homework his brother’s tardies are both consequences they earned for breaking school rules. Otherwise, you wouldn’t be able to compare homework to tardies because doing homework is nothing like being late to class.

We could write this answer as 4:3, 4/3, or 4 to 3. Another way to say it is that Joey earned 4 consequences to his brother’s three.

## Proportions

Proportions are comparisons of two ratios. There are three ways to write proportions:

- 50:250 = 1:5
- 50/250 = 1/5
- =

We would say the above proportion this way: “Fifty is to two-hundred and fifty as one is to five”. This explains the relationship between the numbers in the two ratios we are comparing.

We can also use an equation to find the fourth number in a *proportion* if we know the
other three numbers. Let’s use the equation to find one of the numbers in the *proportion*
above and see if it works.

Remember, in proportions, the following is true: a:b = c:d. This is another way to explain mathematically that the ratios in a proportion are equal and constant. The formula can be written this way:

a = (b x c)/d and c = (a x d)/b

b = (a x c)/d and d = (b x c)/a

We can remember this equation with this sentence, “The product of the means is equal to the product of the extremes.” Here’s how it works:

We cross multiply and get (a x d) = (b x c)

*“a” and “d” are the “extremes” and “b” and “d” are the “means”.* “Extreme” means
that these two numbers are the first and last numbers in the proportion. Written another
way, it looks like this: a:b = c:d. (The “extremes” are in red.) “Mean” means that the
numbers are in the middle of the proportion if written in the same format: a:**b** = **c**:d.
(The “means” are in **red**.)

It might help you to remember this sentence by remembering that a mean is a type of average in a set of numbers, or number “in the middle” of a set of numbers.

Let’s use our original proportion to see if this works. Here it is again:

Let’s pretend we didn’t know one of the terms in our proportion. Here’s our problem:

Here, we used “a” to represent the number we don’t know.

First, we set up our equation:

250 x 1 = 50 x a

Then, we multiply both sides and solve:

250 = 50a *Divide both sides by 50 to isolate the variable.*

5 = a *Our answer is 5 equals a.*

Our answer is 5 (which we already knew was correct)! Here are some practice problems. Use the formula, plug in the numbers, and solve the equation to find the unknown number.

What does a equal?

1 x a = 10 x 6 *Multiply both sides.*

a = 10 x 6 *In this case we can drop the unnecessary 1 and just multiply the right side.*

a = 60 *Our answer is a equals 60, which makes sense in our proportion.*

What does a equal?

3 x 24 = 4 x b *Multiply both sides.*

72 = 4b *Divide both sides by 4 to isolate the variable. *

18 = b *Our answer is b equals 18, which makes sense in our proportion.*