In the last video we learned how to find what I like to call the ‘easiest common denominator’.
And personally, I prefer to use that method of getting a common denominator whenever I can
because it’s quick and easy to remember.
But, I can think of at least two cases
where it would be better to use a different method to convert ‘unlike’ fractions into ‘like’ fractions.
The first case is when one of the fraction’s bottom numbers is a multiple of the other fraction’s bottom number.
And the second case is when your teacher says that you HAVE to use this other method!
This new method is called finding the ‘least common denominator’,
because it involves using the smallest common denominator we can find
instead of just using the product of the bottom numbers like we did in the first method.
To find the smallest (or least) common denominator of two fractions,
we need to figure out what the smallest (or least) common multiple of the two bottom numbers is.
Now in case you’ve forgotten, a ‘multiple’ of a number is just the result of multiplying it by another whole number like 1, 2 or 3.
So here’s what we’re gonna do to find the Least Common Denominator (or LCD for short).
First we’ll take the two different bottom numbers and start making multiples of each of them.
We start with 1 times the numbers… and then 2 times, and then 3 times, and 4 times, and so on…
It’s helps to arrange these multiples in a small table, almost like a scoreboard, so that you can keep things organized and easy to find.
We’ll stop making multiples as soon as we find an answer that’s the same for both numbers.
That answer is called the ‘least common multiple’ and it will become our new common denominator.
Once we know what the least common denominator is,
we have to figure out which ‘whole fractions’ we will need to multiply our unlike fractions by
to get equivalent fractions with that common denominator.
The solution is to use the same numbers that resulted in the common multiple.
For example if you multiplied by 4 to get the common multiple for the first unlike fraction,
then you’ll use 4 over 4 as your ‘whole fraction’.
And if you multiplied by 3 to get the common multiple for the second unlike fraction,
then you’ll use 3 over 3 as your ‘whole fraction’ for it.
Have I lost you yet? It should make a lot more sense after you’ve seen a few examples.
Let’s start with this problem: 3 over 8 plus 5 over 24
Step one is to take our two bottom numbers and make multiples of them to see if we can find a common multiple.
First let’s multiply them both by 1. That’s easy! …we have 8 and 24.
Next we multiply them both by 2. That gives us 16 and 48.
I still don’t see anything in common, so let’s multiply them both by 3.
3 times 8 is 24 and 3 times 24 is 72.
But look! We have something in common now. We have a ’24’ in each column.
We have found the least common multiple of the numbers 8 and 24, and it happens to be 24.
That makes sense if you remember your multiplication tables that 3 × 8 = 24.
So now we know we’re gonna use 24 as our common denominator, but what ‘whole fractions’ do we need to get it?
The answer lies in our multiples chart.
To get our common multiple we had to multiply our 8 by 3, so we’re gonna use the ‘whole fraction’ 3 over 3 for our first fraction.
And our common multiple for the 24 was just itself. We multiplied by 1.
So we could use the ‘whole fraction’ 1 over 1, but we really don’t need to since multiplying by 1 won’t change anything.
We already have a denominator of 24 on that side, se we don’t need to change it.
Okay, now we multiply.
On top… 3 times 3 gives us 9,
and on the bottom, 8 times 3 gives us 24, just like we wanted.
Now we have ‘like’ fractions and we can use our simple procedure to add them.
We add the top numbers: 9 + 5 = 14.
And we keep the same bottom number: 24
Okay …ready for one more example?
Let’s find the LCD for these fractions: 2 over 9 and 7 over 12.
Again, let’s start by making a list of multiples for our two bottom numbers to look for a common multiple.
9 times 1 is 9, and 12 times 1 is 12… of course!
9 times 2 is 18, and 12 times 2 is 24
9 times 3 is 27, and 12 times 3 is 36
9 times 4 is 36… Ah ha! We found it!
36 is the least common multiple of 9 and 12, so we’ll use that as our common denominator.
Now let’s figure out which ‘whole fractions’ we need to use to make our fractions have that denominator.
We’ll use 4 over 4 for our first unlike fraction since 4 times 9 was 36,
and we’ll use 3 over 3 for our second unlike fraction because 3 times 12 is 36.
There, now when we multiply, we get two new (but equivalent) fraction that have a common denominator.
Now we can add them with our simple procedure.
On the top: 8 + 21 = 29.
And we keep the same bottom number: 36
So, that’s how you use the least common denominator method. And it’s really not that hard once you get the hang of it.
So don’t forget to do the exercises for this video, because the way to really learn math is to do it.
Good luck and I’ll see you next time.
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