Lesson Video: Ratios Mathematics • 6th Grade

In this video, we will learn how to use a ratio to describe the relationship between two quantities and use this to solve real-world problems.

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Video Transcript

In this video, we will learn how to use a ratio to describe the relationship between two quantities. We’ll also look at how to write ratios in their simplest form. So first things first. What is a ratio? Very simply, a ratio is a comparison of one value to another. If we think about a recipe, we can find ratios of one ingredient to another. For example, we could find the ratio of sugar to milk. In this ratio, we’re going to compare the amount of sugar to the amount of milk. When working with ratios, order is one of the most important things.

Since we’re asked for the ratio of sugar to milk, we need the sugar value to come first and the milk value to come second. In this recipe, there is one cup of sugar and two cups of milk. We can write this ratio with the word to between. We could say one cup to two cups. But since the units are both cups, we would write the ratio of sugar to milk is one to two, one part sugar to two parts milk. This is one way to write a ratio. A second way to write it is with a colon in the middle. This second line still says 1 to 2. When there’s a colon between the values, you read it as the word to. And the third option for writing this ratio is writing it as a fraction, one over two.

Using the same recipe, let’s consider another ratio, the ratio of flour to milk. Again, the order is very important. The first value will be the flour, and the second value will be the milk. There are three cups of flour in this recipe and again two cups of milk. First, we can write this with words as a ratio of three to two and then, with the colon, 3:2. And finally, three over two in fraction form. Note that when we’re working with fractions, the first value will be the numerator, and the second value will be the denominator.

Now that we’ve seen some basic ratios, we’re going to consider finding the simplest form of ratios. To consider the simplest form of ratios, let’s imagine at a school there’s a running club and a basketball club. The running club has 150 members, and the basketball club has 75 members. And we want to know what the ratio of runners to basketball players are at the school. Again, order is very important here, so we have runners first and basketball players second. And so we write 150 to 75. But to find its simplest form, we want to know if 150 and 75 share any common factors. 150 and 75 are both divisible by 25. 150 divided by 25 equals six. And 75 divided by 25 equals three.

Our first row said that the number of runners to basketball players was 150 to 75. Another way to say that is for every 150 runners, there are 75 basketball players. But a simpler way to say that would be for every six runners, there are three basketball players. However, we should notice that three and six have a common factor of three. Six divided by three is two, and three divided by three is one. The ratio of runners to basketball players is two to one.

We could see this another way. Remember, we can write ratios as fractions, with the first value as the numerator and the second value as the denominator. And when we’re working with ratios just like when we’re working with fractions, if we divide one of the values by something, we must divide the other value by that same amount.

So 150 over 75 simplifies to six over three, which can be simplified one more time to two over one. Now we could have recognized that 75 is a factor of 150. And that means we could have initially divided both sides by 75, which would also give us two to one. Before we look at some examples, there’s one more thing we should note about ratios. And that is equivalent ratios.

Equivalent ratios express the same relationship between values. What we’ve shown here — 150 to 75, six to three, and two to one — are equivalent ratios. They’re all showing the relationship between runners and basketball players at the school clubs. We can find equivalent ratios as we’ve done here by dividing both sides of the ratio by the same amount. You could also find an equivalent ratio by multiplying both sides of your ratio by the same amount. This ratio 20 to 10 is an additional equivalent ratio of two to one. 20 to 10 is a multiple of two to one.

With this information, we’re ready to consider our first example.

For every three male fish in a fishbowl, there are seven females. What is the ratio of male to female fish in the bowl?

We know that a ratio is a comparison of two different values. In this case, it will be the male fish to the female fish. Order here is important to maintain. The ratio of male to female means the value that comes first will be the number of male fish. There are three male fish for every seven female fish. And that means we have a ratio of three to seven. We can write it with a colon in the middle, 3:7, or in its fraction form, three over seven. Both of these values correctly label the ratio.

But before we move on, we should note that that does not mean that they’re only three male fish in the bowl and seven female fish in the bowl. This only tells us the ratio of male to female. For example, in this fish bowl, there are six males and 14 females. If we group three males together and seven females, we can do this twice. In this case, we have six males and 14 females, which simplifies to three over seven. And so although we can say with certainty the ratio is three to seven, we can’t say for sure exactly how many fish are in the bowl.

Our next example is asking us to find a ratio in its simplest form.

Find the ratio of the number of squares to the number of triangles in its simplest form.

When we work with ratios, we know that order is very important. In this case, we’re looking for the ratio of squares to the number of triangles. However, we notice that we do wanna find this in simplest form. The first thing we need to do is find out how many squares and how many triangles are in our image. In total, we have 14 squares and eight triangles. Since the ratio is squares to triangles, the number of squares come first and triangles second. And that means we have a ratio of 14 to 8. And when written as a fraction, 14 is the numerator and eight is the denominator.

To find this value in simplest form, we need to determine if eight and 14 have any common factors. Both 14 and eight are even numbers and therefore divisible by two. 14 divided by two is seven. Eight divided by two is four. So we have an equivalent ratio of seven to four. We know that seven is a prime number, and that means seven and four will not share any additional factors apart from one. And that means seven to four is the simplest form of the ratio of squares to triangles here.

If we want to visualize this, we could say for every seven squares, there are four triangles. We take a second group of seven squares and we get a second group of four triangles, which confirms a square-to-triangle ratio of seven to four.

In our next example, we’re given a ratio, and we need to interpret what that ratio means.

Complete the following: At an ice cream shop, the ratio of small ice cream cones sold to large ice cream cones sold was five to nine. For every blank large ice cream cones sold, there were blank small ice cream cones sold.

First, let’s think about what we know. We have this ratio of five to nine. In order for us to understand this five to nine, we need to know what relationship it represents. It is the ratio of small ice cream cones sold to large ice cream cones sold. And that means our first value of five represents the number of small ice cream cones sold, while the nine represents the value of the large ice cream cones sold in this ratio. Five to nine is then small to large.

If we read the sentence we need to solve carefully, we’ll see that we want to know for every blank large ice creams cones sold, there were blank small ice cream cones sold. And that means for us, the nine value must go with the large, and the five value must go with the small. For every nine large ice cream cones sold, there were five small ice cream cones sold because the ratio of small to large was five to nine. The first blank should be nine and the second blank should be five.

Let’s consider another ratio where we’ll need to find a simplest form.

Given that there are 50 boys and 20 girls in a class, calculate the ratio of the number of girls to boys in its simplest form.

We’re looking for a ratio. That’s a comparison of two different values. Our comparison is going to be of the number of girls to boys in a class. When we’re working with ratios, order is extremely important. This ratio should be the number of girls to the number of boys. But when we read our problem, we notice that they’ve given our information in the opposite order. They’ve given 50 boys and 20 girls. This isn’t a problem. We just have to read carefully and notice that there are 20 girls and 50 boys.

We’ll plug in the value of 20 for the girls and 50 for the boys so that we have a ratio of girls to boys as 20 to 50. We’re not finished here, however, because we’re looking for the simplest form. A ratio is in its simplest form when the two values do not share any common factors apart from one. Currently, we see that 20 and 50 are both divisible by 10. To keep this ratio equivalent, we divide both sides by 10. 20 divided by 10 equals two. 50 divided by 10 equals five. And we have an equivalent simplified ratio of two to five. Two and five are both prime numbers, which means they don’t share any more common factors and are, therefore, in simplest form. The ratio of the number of girls to boys in this class in simplest form would be two to five.

The key to solving this problem was to know which order you were looking for. If you had written five to two as the ratio, that would be the ratio of boys to girls and was not what we were looking for and would therefore be incorrect. When solving these problems, careful reading is required.

For our next example, we’ll have to take an additional step before we can find a ratio.

For every seven bags, Isabella has five pairs of shoes. What is the ratio of the bags to the total number of bags and shoes?

Here, we’re looking for a ratio and we know that a ratio is a comparison of two different quantities. When solving ratio problems, we always need to carefully identify what quantities we are comparing. Our ratio will be of the bags to the total number of bags and shoes. Our first value is the number of bags, but our second value will be the number of bags and shoes together. We know that for every seven bags, Isabella has five pairs of shoes. We can write seven in place of the number of bags. But for the total, we’ll need seven and then five. Seven plus five is 12. This ratio is then seven to 12.

Again, the key to solving this problem is correctly identifying what each piece of the ratio is. We knew that the first piece represented the number of bags Isabella had. But we had to know that the second piece, the second quantity we were comparing, were bags and shoes, a total value, which required an additional step of adding the shoes and the bags. When we look at seven and 12, they do not have any common factors apart from one. And therefore, seven to 12 is in its simplest form.

In our final example, we’ll look at the ratio between lengths of two line segments.

What is the ratio between lengths 𝐴𝐶 and 𝐴𝐸 in its simplest form?

Our ratio here will be a comparison of lengths. The first segment we’re interested in is 𝐴𝐶, which is this distance. And the second distance we’re interested in is 𝐴𝐸, which is this distance. Before we do anything else, we should note that our ratio is going to be 𝐴𝐶 to 𝐴𝐸. This is because when our ratio is listed, 𝐴𝐶 came first and 𝐴𝐸 came second. And that determines the order of our ratio. But when we look closer at the line segment, we realize that we’re not given any exact distances.

But we are told that each of these segments are equal in length. And 𝐴𝐶 contains two of those equal segments, while 𝐴𝐸 contains four of those equal segments. And that means we could list a ratio as two to four. If you’re still not sure that this is true because we don’t have a measurement, imagine that the length of segment 𝐴𝐵 was five inches. That would mean each of these segments was five inches. That would make 𝐴𝐶 10 inches and 𝐴𝐸 20 inches. That would be an equivalent ratio. The ratio of the side lengths will be two to four no matter how we’re measuring the distance.

However, we’ve been told that we want the simplest form. And that means we need to consider if two and four share any common factors. Both of these values are divisible by two. Two divided by two is one. Four divided by two is two. And that means, in simplest form, the ratio of these side lengths would be one to two. The ratio of 𝐴𝐶 to 𝐴𝐸 is one to two. If you want one more way to visualize this, for every one segment on 𝐴𝐶, there are two segments on 𝐴𝐸. Again, we can find one segment on 𝐴𝐶 and two segments on 𝐴𝐸, a ratio of one to two.

Now we’re ready to briefly summarize what we’ve learned. A ratio is a comparison of one quantity to another. Here are the three notations we generally use when writing ratios ⁠— apples to bananas, apples colon bananas, which we read as apples to bananas, or apples over bananas. And when dealing with ratios, we would still read the apples over bananas as apples to bananas. And finally, a ratio is in its simplest form when the two quantities have no common factors.

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