- Learn the definition of a ratio in math
- Learn how to write ratios using: a colon, a fraction, or "to"
- Learn how to simplify ratios
- Learn how to find the unit rate
- Learn the definition of a proportion in math
- Learn how to determine if two ratios are a proportion

## Ratios in Math

We often get word problems that involve ratios. Let's take a look at a quick example.

Example 1:

A bag of marbles has 3 green marbles for every 5 red marbles and every 7 yellow marbles. If there are a total of 330 marbles in the bag, how many of each type of marble is present (how many green marbles, red marbles, yellow marbles)?

To solve this problem, let's first think about the fact that we will have 3 green marbles, 5 red marbles, and 7 yellow marbles in each group of (3 + 5 + 7) 15 marbles. So now, it becomes pretty clear that we need to find how many groups of 15 can be made out of the number 330. (330 ÷ 15 = 22), which tells us we would have 22 such groups.

green marbles » 22 groups with 3 marbles in each group (22 x 3 = 66)

red marbles » 22 groups with 5 marbles in each group (22 x 5 = 110)

yellow marbles » 22 groups with 7 marbles in each group (22 x 7 = 154)

We can conclude that the bag of marbles contains 66 green marbles, 110 red marbles, and 154 yellow marbles. This is consistent with our information given since: $$66:110:154 \hspace{.2em}»\hspace{.2em}3:5:7$$ The simplified ratio is the same. $$66 + 110 + 154 = 330$$ The individual quantities sum to the total given.

## Unit Rate in Math

When we think about a rate in math, we are also thinking about a ratio. A rate is just a bit different and involves a comparison in which the units are not the same. Normally, we will be concerned with how much of something there is per a single amount of another. This is known as a unit rate. We see this pretty much everywhere, even if we don't realize it: mpg (miles per gallon), cost per pound, dollars earned per hour, so on and so forth... To set up a unit rate, we set up a fraction and divide the quantity in the numerator by the quantity in the denominator. This will give us the amount of the numerator per single unit of the denominator. To further clarify, let's look at an example.

Example 2

A legal printer can print 1287 pages in 13 minutes. What is the unit rate for pages printed per minute? To solve this problem, let's set up a fraction with the total number of pages being printed in the numerator and the number of minutes it takes in the denominator: $$\frac{1287 \hspace{.1em}\text{pages}}{13 \hspace{.1em}\text{minutes}}$$ To find the unit rate, just divide the number in the numerator by the number in the denominator. This will give us the pages "per" minute. $$\frac{1287 \hspace{.1em}\text{pages}}{13 \hspace{.1em}\text{minutes}}=\frac{99 \hspace{.1em}\text{pages}}{1 \hspace{.1em}\text{minute}}$$ Our printer has an expected output of 99 pages per minute.

## Proportions in Math

A proportion in math states that two ratios are equal. In order to determine if two ratios are equal, we check to see if the cross products of the number parts are equal. Let's look at an example.Example 3

On Monday, a certain pet park had 4 cats and 7 dogs. The following day, the park had 16 cats and 28 dogs. Does the ratio of cats to dogs for the two separate days represent a proportion?

On Monday, the first day, we know there were 4 cats and 7 dogs. So the ratio of cats to dogs was 4:7. The next day, the pet park had 16 cats and 28 dogs, so the ratio of cats to dogs was 16:28. We can just set these ratios up as fractions and check to see if the cross products are equal. $$\frac{4 \hspace{.1em}\text{cats}}{7 \hspace{.1em}\text{dogs}}? \frac{16 \hspace{.1em}\text{cats}}{28 \hspace{.1em}\text{dogs}}$$ Check the cross products: $$7 \cdot 16=112$$ $$28 \cdot 4=112$$ Since the cross products are equal, we have a proportion.

#### Skills Check:

Example #1

There are 4 turtles, 6 chickens, and 8 pigs on a farm. What is the ratio of turtles to chickens to pigs?

Please choose the best answer.

Example #2

At Malcolm High, the ratio of passing students to failing students is 7:1. If the school has 880 total students, how many are failing?

Please choose the best answer.

Example #3

A wholesaler sells 3125 gallons of milk for $5,843.75. Find the price per gallon.

Please choose the best answer.

Example #4

To make sugar cookies, Lamont Bakery runs small batches and uses 2 cups of flour and 1 cup of sugar. A competing baker, Jerome Bakery makes sugar cookies by running large batches. They use 29 cups of flour and 15 cups of sugar. Consider the two baker's ratio of flour to sugar and determine if the ratios represent a proportion.

Please choose the best answer.

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