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.
