Hi!  Welcome to Math Antics.
In this lesson, we’re gonna learn what proportions are and how we can use them to find an unknown value.
The good news is, if you know about equivalent fractions, then you already know a lot about proportions.
To see what I mean, let’s start with the simple fraction 1 over 2 (or one-half).
Now let’s look at a pair of equivalent fractions, 1 over 2 and 5 over 10.
These fractions are equivalent because,
even though they have different top and bottom numbers, they have the same value.
1 is half of 2, and 5 is half of 10, so they represent the same amount.
Okay, but to understand what a proportion is, we need to start with a ratio instead.
A ratio is basically just a fraction that’s used in a certain way.
If you don’t remember what a ratio is, you can watch our video about them.
So let’s imagine that a student, who’s a really good reader, can read 1 book in 2 days.
We could say that the ratio of books to days is 1 over 2 (one book per two days).
Alright, but what if our student reads books at that same rate for 10 days.
How many books would they read?
Well, if they finish 1 book every 2 days, then in 10 days they’ll have read 5 books,
so that ratio would be 5 books per 10 days.
Ah, do you see what we have here?
These are equivalent ratios!
Just like the equivalent fractions, they represent the same amount so we can put an equals sign between them.
When we do that, we have a proportion.
A proportion is just two ratios that are equivalent (or equal).
Oh, and one thing that’s really important to remember…
In order for two ratios to be equivalent, they not only have to have the same value,
they also have to have the same units.
That is, they have to be representing the same thing on the top and bottom.
Let me show you what I mean.
This is a proportion because the top numbers both refer to ‘books’
and the bottom numbers both refer to ‘days’.
But what if we change the top unit of the second ratio to be ‘pizzas’ instead of ‘books’?
5 pizzas in 10 days is NOT equivalent to 1 book in 2 days.
So even though the numbers are still the same, this is no longer a proportion.
Or, what if we keep the same units and just switch them in the second ratio
so that the ‘days’ are on top and ‘books’ on the bottom.
Are they still equivalent?
Nope! This is not a proportion anymore either.
5 days per 10 books is NOT equivalent to 1 book per 2 days.
So the units have to be exactly the same for both ratios to form a proportion.
Alright then… So a proportion is a pair of equivalent ratios.
But why do we care?  What are proportions good for?
Well, it turns out that proportions are really good for figuring out something you DON’T know from something you DO know,
and that makes them very useful!
For example, let’s suppose that our student (who’s a good reader),
has a BIG stack of books that they want to read (23 books to be precise),
and they want to know how many days it will take them to finish.
How do we figure that out?
Well, let’s start with what we DO know.
We know that they can read 1 book in 2 days.
So let’s take that ratio and set up an equivalent ratio for 23 books.
They key in setting up that equivalent ratio is to make sure that the units are the same as the first ratio;
books on the top and days on the bottom.
We know that the number of books that they want to read is 23, so that goes on top.
But the number of days it will take is unknown, so instead of putting a number there,
we’re going to put the letter ‘n’ there temporarily to stand for the number that we don’t know.
This is how you’ll usually see and use proportions in math.
Three of the proportion’s numbers will be known and one will be unknown.
Fortunately, if you know three of the numbers,
you can find the missing number easily using a procedure called cross-multiplying.
Cross-multiplying is just a short-cut way of doing some basic algebra
to re-arrange our proportion so we can find the unknown number.
To do it, we first start by writing down a new equal sign because cross-multiplying will give us another equation.
Next, imagine that a criss-cross shape (like an ‘x’) is overlaid on the proportion.
This cross shape tells you which numbers to multiply together on each side of the new equals sign.
1 and ‘n’ will be multiplied together on this side of the equation,
and 2 and 23 will be multiplied together on the other side of the equation.
Oh, and as long as you follow the criss-cross guides, it doesn’t matter which pair goes on which side.
Okay, so our proportion has been re-arranged… now what?
Well, on one side of the new equation we have two numbers being multiplied together.
The next step is to go ahead and simplify by doing that multiplication.  2 × 23 = 46.
But what about the other side of the equation.
That has a number being multiplied by our unknown letter ‘n’.
How can we multiply when one of the numbers is unknown?
Actually, we can’t!
Fortunately, we don’t need to because we’re just trying to figure our what our unknown number is.
What does it equal?
In other words, we need to keep re-arranging our equation
until the unknown value is all by itself on one side of the equals sign
and all the known values have all been combined on the other side of the equals sign.
Then we’ll have found the unknown.
In this problem, getting the ‘n’ by itself is easy because it’s just being multiplied by the number 1.
And what happens to a number when we multiply it by 1?
Yep - absolutely nothing.
1 times ‘n’ is exactly the same thing as just plain ‘n’, so we can just ignore or get rid of the 1.
And look!  Our equation is now n = 46.
That means that we know what ’n’ equals.
We’ve figured out what the missing number of our proportion is.
If our student can read 1 book in 2 days, then they can read 23 books in 46 days.
We’ve used a proportion to solve for an unknown.
Alright, let’s see another example of using a proportion to find an unknown.
This one involves a map.
Have you ever noticed that maps are a lot smaller than the real-life places that they show?
A map is a good example of something called a “scaled drawing”,
which is just a drawing that's either larger or smaller than the real thing it depicts,
but it’s still ‘in proportion’ to that thing.
For example, this map of Hawaii is a lot smaller than the actual Hawaii.
But even though the map is smaller, it’s still proportional to the real island
and there’s even a scale on it to show the relationship between the two sizes.
It says that 5 centimeter on the map is equal to 9 miles on the real island.
Okay, suppose that we want to know how many miles it is from the Hawaiian volcano, Mona Loa, to the city called Hilo.
We can set up a proportion to figure that out.
The ratio that we already know is 9 miles per 5 cm.
Now we just need to set that equal to an equivalent ratio that has the unknown distance in it.
Because we have the map, we can use a ruler to measure how many centimeters it is from Mona Loa to Hilo.
It looks like about 20 cm, so the bottom number of the equivalent ratio is 20 cm
and the top number is the number of miles, which is unknown.
Again, we’ll just use a letter ‘n’ to stand for that missing number.
To solve this proportion for the unknown number, we use our cross-multiplying procedure.
First, we write a new equal sign and then we imagine the criss-cross to show us what we multiply together on each side.
On the first side, we have 9 × 20. And on the other side, we have 5 × n.
On the side that has two numbers, we can go ahead and simplify.  9 × 20 = 180.
On the other side, we have 5 multiplied by our unknown value ‘n’.
We can’t multiply that, but we don’t need to.
Instead, we want to get the ‘n’ all by itself.  How do we do that?
Well, we can’t just ignore the 5 like we ignored the 1 in the last problem.
Instead, to get the ‘n’ by itself, all we have to do
is divide both sides of the equation by the number that ’n’ is being multiplied by.
In this case, that’s 5.  So on the first side, 180 divided by 5 equals 36.
And on the other side, 5 times ‘n’ divided by 5 is just ‘n’ since the fives cancel out.
There, now we know what the unknown value in our proportion is:
36 = n, which is the same as n = 36.
That’s the number of miles it is from the volcano Mona Loa to Hilo.
Alright, so in this video, we learned that a proportion is a pair of equivalent ratios,
and we learned how we can set up a proportion that has an unknown number
and then find out what that number is by cross multiplying.
Proportions are really important!
If you understand how they work, you can use them to solve all sorts of real-world math problems.
And the best way to understand them is to practice what you’ve learned in this video by working some problems on your own.
Thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com
