It’s Professor Dave, I wanna tell you about fractions.
We’ve learned about division, and any time we perform division there must be a dividend,
a divisor, and a quotient.
At first, we looked at examples where the dividend is larger than the divisor.
This can sometimes result in a quotient that is a whole number, like fifteen divided by
five, which gives us three.
But sometimes the divisor will be larger than the dividend, in which case we will get a
fraction that has a value somewhere between zero and one.
It’s a piece of one, or a fraction of one, hence the name: fraction.
To see how these work, let’s look at a pizza.
You’ll typically see a pizza cut up into eight equal slices.
To represent this, we can take the number one, to represent the pizza, and divide by
the number eight, to show that it has been divided into eight sections.
That means that each slice is one eighth of the pizza.
If you eat two slices, you have eaten two eighths.
You can have three, or four, all the way up to eight eighths if you are very hungry, in
which case you have eaten the entire pizza.
Let’s say the pizza was instead cut into fourths.
Then each slice would be one fourth of the pizza.
We can see that eating one fourth of the pizza is the same as eating two eighths, and that’s
because these fractions are equivalent.
Two divided by eight is the same as one divided by four, and two eighths can therefore reduce
to one fourth.
When we have a fraction like this, we can manipulate it any way we want by simply doing
the same thing to both the top, or numerator, and the bottom, or denominator.
If we divide both the two and the eight by two, we get one fourth, which again means
that these fractions must be equivalent.
If two fractions have the same denominator, it is easy to see which one is greater.
If the denominators are different, we can do a number of things.
Let’s say we are comparing three eights and three tenths.
Which is greater?
Well let’s draw two circles; one can be divided into eight pieces, and the other into
ten.
Then, let’s shade three pieces in each circle.
We can clearly see that three eights covers a larger area, so it’s the larger fraction.
This makes sense, because the numerator, three, is being divided fewer times than with three
tenths.
We could also do things like convert to decimal notation, or find the least common denominator,
both of which we will learn later.
Of course, not all fractions are less than one; fractions can also be greater than one.
This will be the case if the dividend is greater than the divisor but the quotient is not a
whole number.
Take for example four divided by three, or four thirds.
We know that three thirds would equal one, and therefore six thirds would equal two,
so four thirds is somewhere in between one and two.
These are called improper fractions, and we can either leave these as they are, or convert
them to mixed numbers, like one and one third.
This can be useful in a certain context, like ordering pizzas.
Let’s say you’re having a pizza party, so you want to order some pizzas from your
favorite restaurant.
They cut their pizzas into eight slices, and your ten friends each want three slices.
Doing the multiplication, three slices each times ten friends means thirty slices.
Since each slice is one eighth of a pizza, we could say that we need thirty eighths of
pizza.
But when we call to order, we wouldn’t say it that way.
We need to order a specific number of whole pizzas.
So how many do we need?
Well, eight eighths comprise a whole, so eight eights is one pizza, sixteen is two pizzas,
and twenty-four gives us three whole pizzas.
Then, we still need six more slices.
So, instead of thirty eighths, we could say that we need three whole pizzas plus six more
slices, or three and six eighths pizza, which reduces to three and three fourths.
That means we need to order four pizzas, and we will have two slices left over.
We could also go the other way.
Say that we ate four and three eighths pizzas, so how many slices did we eat?
Well each pizza has eight slices, so we multiply the four by eight to get thirty-two.
Then we add on the remaining three, and we get thirty-five slices.
We should be able to convert between improper fractions and mixed numbers in this fashion,
so before we move on, let’s check comprehension.