Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures, we saw how to multiply and divide fractions.
We also saw that adding and subtracting fractions is easy, as long as they have the same denominators.
In that case, it’s very much like adding and subtracting integers.
For example, on the number line, adding the fractions
two-thirds, minus six-thirds, minus two-thirds, plus four-thirds
looks a lot like adding the integers two, minus six, minus two, plus four,
except that each division represents one-third unit instead of one whole unit.
This is why we simply add the numerators of each fraction,
since the numerators specify the number of divisions in each fraction.
Since the size of the divisions is not changed,
the denominators are not affected.
But how can we add or subtract fractions who’s denominators are not the same?
For example, let’s say that we want to add the fractions one-third and one-half.
We can use the number line to visualize this.
Since the fraction one-third has a denominator of three,
each unit length on the number line is divided into three equal divisions,
and the fraction one-third is represented by one of these divisions.
On the other hand, the fraction one-half has a denominator of two,
so each unit length on the number line is divided into two equal divisions,
and the fraction one-half is represented by one of these divisions.
Unfortunately, when we add these two fractions, the sum won’t fit into either set of divisions.
This is true whether we use three divisions per unit or two divisions per unit.
Since each fraction is expressed using different sized divisions,
it’s like trying to add apples and watermelons.
The solution is to create new divisions which will work for both fractions.
We need to divide the unit length in such a way that each fraction
will fit perfectly into these new divisions.
If we do this, then both fractions can be represented by some number of these new divisions.
Since both fractions will then use the same-sized divisions, they will have the same denominator.
In other words, they will have a common denominator.
We can then simply add their numerators.
For instance, in this example we can split each division on the top number line in half.
Each unit length on the top line will then be divided into six equal pieces,
and the fraction one-third will be represented by two of these divisions, or two-sixths.
Now if we split each unit on the bottom number line into thirds,
each unit on the bottom line will also be divided into six equal pieces.
The fraction one-half will be represented by three of these divisions, or three-sixths.
Both fractions now represent some number of the same-size divisions,
so now we can add their numerators,
and we see that these two fractions add up to five-sixths.
So when adding or subtracting fractions with different denominators,
we must find a single denominator which will work for all of the fractions.
In other words, we must find a "common denominator".
So how do we find this common denominator?
Finding a common denominator is easy.
We simply multiply each fraction’s denominator by the denominator of the other fraction.
This way, both fractions end up with a denominator equal to the product of the two denominators.
But when we change a fraction’s denominator we must also change its numerator
so that the fraction still represents the same quantity as it did before.
Let’s see how this works on the number line.
In the previous example, we started with the fraction one-third.
We then split each division on the number line into two pieces.
There were then twice as many divisions per unit length,
so we had to multiply the denominator by two.
The fraction one-third then represented twice as many of these new smaller divisions,
so we also had to multiply the numerator by two.
So the new equivalent fraction was created
by multiplying both the numerator and the denominator by two.
Likewise, we took the fraction one-half,
and split each division on the number line into three pieces.
There were then three times as many divisions per unit length,
so we had to multiply the denominator by three.
The fraction one-half then represented three times as many of these new divisions,
so we also had to multiply the numerator by three.
So this new equivalent fraction was created
by multiplying both the numerator and the denominator by three.
Multiplying the numerator and denominator of a fraction by the same number
doesn’t change its value,
since it is the same as multiplying the entire fraction by one.
And it is this trick which will allow us to change any fraction
into an equivalent fraction with a different denominator.
When adding or subtracting two fractions,
the simplest way to create a common denominator
is by multiplying the numerator and denominator of each fraction
by the other fraction’s denominator.
In our example, both the numerator and the denominator of the fraction, one-third,
were multiplied by two,
which was the denominator of the one-half.
and the numerator and denominator of the one-half were both multiplied by three,
the denominator of the one-third.
So we multiplied the top and bottom of each fraction by the other fraction’s denominator.
This gave us two-sixths plus three sixths,
and two-sixths plus three sixths is five-sixths.
So we found that one-third plus one-half is equal to five-sixths.
Let’s try another example, adding the fractions two-fifths and three-sevenths.
To create the common denominator,
we multiply the numerator and denominator of the fraction two-fifths,
by the denominator of three-sevenths.
And we multiply the numerator and denominator of three-sevenths,
by the denominator of two-fifths.
This gives us fourteen thirty-fifths plus fifteen thirty-fifths.
And fourteen thirty-fifths plus fifteen-thirty-fifths is twenty-nine thirty-fifths.
So two-fifths plus three-sevenths is twenty-nine thirty-fifths.
One last example, subtracting the fractions three-halves minus two-thirds.
Once again, to create a common denominator,
we multiply the top and bottom of the fraction three-halves, by the denominator of two-thirds.
And we multiply the top and bottom of the fraction two-thirds,
by the denominator of three-halves.
This gives us nine-sixths minus four-sixths, which is equal to five-sixths.
So three-halves minus two-thirds is five-sixths.
By finding a common denominator, we can add or subtract fractions with different denominators.
However, sometimes when we multiply the denominators of fractions together to find a common denominator
we create an unnecessarily large common denominator.
We will see in the next lecture how to find the smallest possible common denominator,
otherwise known as the "Least Common Denominator".