Ratios, Rates, & Proportions Lesson

What exactly is a ratio in math? A ratio tells us how much of one thing there is when compared to how much of some other thing there is. As a simple example, suppose we have some fruit: 3 oranges, 6 pineapples, and 9 apples. A ratio is a perfect tool to make comparisons about how much of one type of fruit we have versus how much of some other type of fruit that we have. Let’s start by writing a ratio to compare the number of oranges to the number of pineapples. This can be done with a colon, a fraction, or the word "to". Since we have 3 oranges and 6 pineapples present, we can write our ratio of oranges to pineapples as: $$3:6$$ $$\frac{3}{6}$$ $$3\hspace{.2em}\text{to}\hspace{.2em}6$$ In each case, the number of oranges is listed in the first position. This is because the order matters with a ratio; if we reversed the numbers, it would no longer be true. If we ask for oranges to pineapples, we want the number of oranges listed first, followed by the number of pineapples listed second. With one exception, we can simplify a ratio in the same way we did with fractions (the exception is covered below). Since each part of the ratio is divisible by 3, we can write our ratio as: $$1:2$$ $$\frac{1}{2}$$ $$1\hspace{.2em}\text{to}\hspace{.2em}2$$ If we rearrange the picture a bit, it becomes completely clear that for each orange, there are two pineapples. So what if we reversed the order of the words in our ratio? We would want to reverse the numbers involved. What is the ratio of pineapples to oranges? $$\frac{6}{3}=\frac{2}{1}$$ Now, here is where our exception mentioned before comes into play. In this case, we need to be careful. We know when we work with a fraction and we get something like 2/1, this can just be written as the number 2. When we work with a ratio, we need to keep the 1 since it is showing the other part of our ratio. If we see 2:1 or 2/1, we know it's 2 pineapples for every 1 orange. In some cases, we may see a ratio that compares three or more quantities. We can proceed with this type of situation using the same logic. Again, referencing our picture above, what is the ratio of oranges to pineapples to apples? Here, we would use the colon notation and put each of our numbers in the order of the words given. $$3:6:9$$ Again each part is divisible by 3, so we can really say the ratio of oranges to pineapples to apples is: $$1:2:3$$ If we look at each row of our sorted picture, we can see this is the case. For each orange, we have two pineapples and three apples.
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

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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.

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