Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures, we saw that we can change a fraction’s form,
without changing its value,
if we multiply its numerator and denominator by the same number.
For example, the fraction one-third can be written as two-sixths,
if we multiply the top and bottom by two.
Or by multiplying the top and bottom by two again, it can be written as four-twelfths.
Of course doing so, will create an equivalent fraction with a larger numerator and denominator.
But it is also possible to do this process in reverse,
by dividing the numerator and denominator by the same number.
This creates an equivalent fraction with a smaller numerator and denominator,
so this is called "reducing" or "simplifying" the fraction.
In order for a fraction to be reduced,
it must have a numerator and denominator which can both be divided by the same number.
In other words they must both have a "common divisor".
But how do we know when this is the case?
Sometimes it is obvious.
If we look at the fraction four eights,
it is easy to recognize that the numerator and denominator are both divisible by four,
and so this fraction can be written as one-half.
And most people would recognize that the numerator and denominator of twenty thirtieths
are both divisible by ten.
But then, sometimes it's not so obvious.
The surest way to determine whether or not a fraction can be reduced,
is to "factor" the numerator and denominator.
For example, say we wish to determine if the fraction fifteen twenty-firsts can be reduced.
Factoring the numerator and denominator,
we can see that they both have a factor of three in common,
so we can divide the top and bottom of this fraction by three.
Of course, this has the same effect
as simply removing a factor of three from the top and bottom.
In other words we "cancel out" a factor of three in the numerator and denominator.
This reduces the fraction to five sevenths.
As another example, let’s simplify the fraction thirty forty-seconds.
Factoring the top and bottom,
we see that they both have a factor of two and a factor of three in common.
After canceling out the factors of two and three, we are left with five sevenths.
Let’s do a couple more examples.
Let’s reduce the fraction thirty seventy-seconds.
We first factor the top and bottom.
This time the bottom has three factors of two, and two factors of three.
However, since the top has only one two and one three,
we can only cancel out a single factor of two and a single factor of three,
which leaves us with five twelfths.
As our last example, let’s reduce seven sixty-thirds.
The seven in the top is a prime number, but we can factor the sixty-three in the bottom.
When we cancel out the factors of seven in the top and bottom,
we will be left with a nine in the bottom.
But what about the top?
Since seven is the same as seven times one,
when we cancel out the sevens, we are left with one ninth.
We have seen that the process of reducing a fraction
involves determining which factors the numerator and denominator have in common.
In this example, the fraction has several common factors.
These common factors include the prime numbers two, and three.
But we could also name other factors common to both the numerator and denominator.
For example, the composite numbers four, six, and twelve.
The product of all the common prime factors, is the "greatest common factor".
In this example, the greatest common factor is twelve.
And once we eliminate the greatest common factor,
the fraction is reduced to its simplest form, which in this example is five sixths.
The fraction five sixths is called a "proper fraction",
since it represents a value less than one.
However, sometimes arithmetic operations result in fractions greater than one,
called "improper fractions".
In the next lecture, we will learn how to convert these improper fractions
to a different form, involving an integer plus a proper fraction.