Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture, we saw how fractions are multiplied.
When multiplying fractions,
the numerators are multiplied to create the numerator of the result,
and the denominators are multiplied to create the denominator of the result.
But how do we divide fractions?
Let’s start by talking about different ways to write a division operation.
You have probably seen long division problems written like this.
The number to be divided is called the "dividend",
the number which divides the dividend is called the "divisor",
and the result is called the "quotient".
Division can also be written using a division sign.
Once again, the dividend is divided by the divisor to create the quotient.
Yet another way to write a division operation is to write it as a fraction,
placing the dividend over the divisor.
When doing Algebra problems, the most common ways to indicate division
are either by writing the dividend and divisor as a fraction,
or by using a division sign.
Both of these notations may also be used when dividing fractions.
The easiest way to divide with fractions is to turn the division into a multiplication.
Instead of dividing fraction A by fraction B,
we can multiply fraction A by the "reciprocal" of fraction B.
So what is a reciprocal?
Every number except zero has a reciprocal.
A number's reciprocal, is the number that you must multiply it by to get one.
For this reason, a number’s reciprocal is also called its "multiplicative inverse".
The number zero does not have a reciprocal
since there is no number which we can multiply it by to get one.
And the number one is its own reciprocal.
We can visualize the concept of reciprocals
by using a square with a width and height of one.
Since the area of a square or rectangle is equal to its width times its height,
the area will be one.
Now if we slice the square into two equal pieces,
and lay them end to end to create a rectangle,
the area of the rectangle will be the same as the square, one.
However, its width will be two and its height will be one-half.
So we can see that two times one-half is one.
Therefore, two and one-half are reciprocals.
Likewise, if we slice the square into three equal pieces,
the area of the rectangle will still be one.
However, its width will be three, and its height will be one-third.
Therefore three times one-third is one, so three and one-third are reciprocals.
From this process, we can see that any natural number n, and the unit fraction one over n,
are reciprocals of each other.
Since dividing by a number gives the same result as multiplying by its reciprocal,
reciprocals are useful for turning division problems with fractions
into multiplication problems.
For example, if we want to divide six by one-third,
we can just multiply six by the reciprocal of one-third,
in other words six times three,
which is equal to eighteen.
So six divided by one-third is eighteen.
Now let’s try dividing one-third by one-third.
Instead of dividing by one-third, we can multiply by its reciprocal, three.
One-third times three is three-thirds, which is equal to one.
Or course, it makes sense that one-third divided by one-third is equal to one,
since anything divided by itself is one.
Let’s try one more example, dividing one-third by three.
Once again, this is the same as multiplying one-third times the reciprocal of three,
or one-third times one-third.
Multiplying the numerators and denominators, we get one-ninth.
So one-third divided by three is one-ninth.
In our examples so far, we have divided by unit fractions,
but we can also divide by a common fraction if we know it’s reciprocal.
So how do we find the reciprocal of a common fraction?
Recall that a number's reciprocal is the number that you must multiply it by to get one.
So we need to know what to multiply a common fraction by to get one.
If we multiply our fraction
times the same fraction with the numerator and denominator swapped,
the numerator of the result will be n times m,
and the denominator of the result will be m times n.
Using the commutative property to swap the n and m in the numerator,
we can see that the numerator and denominator are the same, so the result is one.
So a common fraction, times the same fraction with the numerator and denominator swapped,
is one.
Therefore, these two fractions are reciprocals.
The reciprocal of any fraction is created by swapping the numerator and denominator.
For example, the reciprocal of three-fifths is five-thirds.
If we multiply three-fifths times five-thirds,
the product’s numerator and denominator are both fifteen, and the result is one.
We can use this rule for unit fractions as well.
Swapping the numerator and denominator of a unit fraction, one over m,
gives us m over one, which is the same as the integer m.
Now that we know how to take the reciprocal of any fraction,
let’s try dividing two common fractions, two-thirds and five-sevenths.
We multiply two-thirds by the reciprocal of five-sevenths, which is seven-fifths.
Multiplying the numerators and the denominators, we get fourteen-fifteenths.
We have shown that if we wish to divide by a fraction,
we simply multiply by the reciprocal of that fraction.
But will this always work for any fraction?
Maybe we were just lucky, picking numbers in our examples which happened to work.
We can prove this will always work if we can show that dividing by a fraction
is always equivalent to multiplying by the reciprocal of that fraction.
Let’s take the most general case we can come up with.
Instead of using specific numbers, we will use letters which can represent any number.
We will do an example, dividing a fraction n over m by another fraction p over q.
Now n, m, p, and q can be any integer,
except that since a fraction’s denominator can never be zero, m and q can’t be zero.
Also p can’t be zero, since this would make the fraction p over q equal to zero,
and we can’t divide the first fraction, n over m, by zero.
As we saw earlier, another way to write a division operation is to write it as a fraction,
placing the dividend over the divisor.
So we can write this division operation between two fractions as one big fraction.
Since one is the multiplicative identity,
we can always multiply anything by one without changing its value.
Let’s replace the number one by another number which is equal to one.
Any fraction where the numerator and denominator are the same number is equal to one,
so we will replace one, by a fraction
where the numerator and denominator are both q over p.
Now we multiply these two big fractions by multiplying their numerators and denominators.
Notice that the denominator of the big fraction is now the fraction p over q,
times the reciprocal of that fraction, q over p.
Anything multiplied by its reciprocal is equal to one,
so we can replace this denominator with one.
Since the denominator of the big fraction is one,
the value of that fraction is just equal to its numerator.
So we have shown that dividing by any fraction, p over q,
is equivalent to multiplying by the reciprocal of that fraction, q over p.
Since n, m, p, and q can be any reasonable integer, this will always work.
Let’s repeat this one more time with specific numbers.
We first write the division operation as a fraction, placing the dividend over the divisor.
Then we multiply by one.
Since one is the multiplicative identity, the value is not changed.
Then we replace the one, by a fraction equal to one.
Any fraction where the numerator and denominator are the same number is equal to one,
so we replace one by a fraction where the numerator and denominator are both seven-fifths.
Now we multiply these two big fractions by multiplying their numerators and denominators.
The denominator of the big fraction is now five-sevenths times its reciprocal,
so it is equal to one.
Since the denominator of the big fraction is one,
the value of the fraction is just equal to its numerator.
So dividing two-thirds by five-sevenths, is equivalent to multiplying two-thirds by seven-fifths.
So we have seen that when working with fractions,
division can be converted to multiplication by the divisor’s reciprocal.
We also saw in the previous chapter, how to add and subtract fractions,
as long as the fractions have the same denominators.
In the next lecture, we will see how to add and subtract fractions with different denominators
by finding a common denominator.