>> Welcome to the Cypress College Math Review
On multiplying and dividing fractions.
Jumping into objective number one, we need to do some multiplying of fractions.
Our first example tells us to use the diagram to answer the following:
Using the diagram, we want to address Part A first.
What fraction is shaded?
Looks like I've got 1, 2, 3, 4, 5, 6, 7, 8 of these sections that are shaded.
Of course, that means that we've got the remaining sections that are not shaded.
I'm going to label those as 9, 10, 11, and 12.
Meaning, all together we've got a total of 12 sections in this diagram.
In order to address what fraction is shaded then,
we simply count up the shaded sections, which we have done already.
We had 8 shaded sections and a total of 12 sections.
Therefore, the fraction that is shaded would be 8 over 12.
Of course, mathematically we can reduce this to become a 2 over 3.
That takes us over to Part B of the question then,
where they are asking us: what is 1/2 of 2/3?
Now, the key part here is to recognize that 2/3 was represented by the shaded region.
So this question is asking us to take 1/2 of the shaded region, but we still have to account
for the total number of sections.
As we look at the shaded region, 1/2 of that would be represented by the four sections
that we can take -- whether it's top or bottom or however we organize it doesn't matter.
But we had 8 sections that are shaded.
Half of those then would mean we've got four shaded sections.
So of the shaded sections, we've got 4 shaded sections,
representing 1/2 of the shaded sections.
When the question asks, what is 1/2 of 2/3, though,
we have to remember it's out of the total diagram.
Therefore, we end up with 12 total sections.
One-half of our 2/3 then would be 4 over 12.
And of course, we can reduce that to simply be a 1/3.
That takes us to our next example with multiplying fractions.
First, we want to note that when we multiply by a proper fraction, the answer is less
than the original number to keep context on what we've done.
When we multiply fractions, our first step is to cross cancel
and our second step will be to multiply across.
Let's go ahead and jump into the example then.
In our example, they are asking us to multiply 3 over 15 times 20 over 21.
I'm just going to rewrite the problem in order to create some spacing.
So I've got 3 over 15 being multiplied by my 20 over 21.
And again, our first step is to cross cancel.
So what I'm looking to do is cancel with common factors.
The first thing I see is that I've got a common factor of 5 between 15 and 20.
Since I have a common factor of 5, I'm going to cancel out the 15 and 20
and reduce those by the factor of 5.
Reducing 15 by the factor of 5 means I'm left with a 3.
Reducing 20 by a factor of 5 means I am left with 4.
Now, looking at 3 and 21, I can cross cancel with another factor of 3.
When I cancel 3 and 21, 3 reduces to just 1 since 3 divided by 3 is 1, and 21, of course,
reduces to 7 since 21 divided by 3 is 7.
It appears that I can't cancel anymore with the remaining factors,
so I'm going to gather my terms together.
I now have 1 over 3 times 4 over 7.
It doesn't appear I can cancel out anymore since I have no shared factors, which means,
like you see in step two, I multiple across.
Multiplying across, I'm left with 1 times 4, which is just a 4,
and 3 times the 7, giving me a 21.
Which means our answer is 4 over 21.
And of course we cannot reduce that fraction.
Let's go ahead and jump into our next example now, where we are being asked
to multiply four fractions, which means organization is becoming more important.
The example asks us to multiply 2 over 9 times 3 over 10 times 4 over 11 times 5 over 12.
I'm going to rewrite the problem again, just to create some spacing.
Because, remember, our first step will be to cross cancel.
Anywhere we've got some common factors.
Now, the issue here is to remember that we don't have to cross cancel only fractions
that are right next to each other.
Meaning, our options are not only to look for cross cancelling amongst consecutive fractions.
We can cross cancel anywhere available from numerator to denominator.
To demonstrate that, I'm going to cross cancel 2 and 12.
Again, just to be clear, we could cross cancel 2 and 10, since they have a common factor of 2.
Instead, just to show that this is an option, I'm going to cross cancel 2 and 12.
Sharing the common factor of 2, 2 reduces to be just 1,
and of course 12 divided by 2 leaves us with 6.
Looking for other options, I'm going to cross cancel 3 and 9 here with the common factor
of 3 -- 9 reduces to 3, 3 reduces to 1.
Continuing our search, I'm going to cross cancel 5 and 10, since they have a common factor
of 5 -- 10 reduces to be 2, 5 reduces to be 1.
Now at this point we've got a lot of numbers being manipulated here.
So I'm going to regather my fractions to see where I'm at to ensure I don't make a mistake.
After all this cross cancelling, I now have 1 over 3 times 1
over 2 times 4 over 11 times 1 over 6.
We have an option to cross cancel using the numerator of 4.
You can either cross cancel 4 with 2 or we can cross cancel 4 with 6.
I'm going to cross cancel 4 with 2.
That leaves me with 1 and 2 in the numerator.
Recognize that we can also cancel 2 with 6 since they have a common factor of 2.
Two, of course, reduces to 1 and 6 reduces to 3.
Gathering my fractions again, now that we've done more cross cancelling, it appears I have 1
over 3 times 1 over 1 times 1 over 11 times 1 over 3.
Which means at this point, double check to make sure we can't cross cancel.
We have one's in the numerator all the way across the board, so we can move on to step two
to multiply straight across: 1 times 1 times 1 times 1 is of course just 1.
And in the denominator, we have 3 times 1 times 11 times 3 to get us our 99.
Since we cannot reduce that fraction, our answer is the 1 over 99.
Pause the video and try these problems
>> That takes us to our objective two, which is dividing fractions.
First, we want to discuss reciprocals for a fraction.
We need to interchange the numerator and the denominator
when we are trying to find the reciprocal.
For example, the reciprocal of 3 over 10 is 10 over 3.
Jumping into the first example, then,
they ask us to use the following diagram to answer the following.
As we did previously, Part A asks, what fraction is shaded?
Looking at our diagram like we did previously, it looks like I've got one section,
two sections that are shaded, while this third and fourth section are not shaded.
Which means, to address what fraction is shaded,
we've got two shaded sections out of four total sections.
Which, of course, we can reduce to be 1/2.
That takes us over to Part B of the question.
What is 1/2 of 1/2?
Remember, 1/2 represented the shaded section, and they are asking us to take 1/2 of 1/2.
So instead of considering both shaded sections, we are now only taking half
of the two or one of those sections.
Which means I have 1 shaded section, but of course, this is out of the total 4 sections.
Therefore, 1/2 of 1/2 is 1/4.
This brings us to an important note that dividing by 2 is the same as multiplying by 1/2.
What we're trying to accomplish here is to make the connection
between multiplication and division of fractions.
Now, let's go ahead and jump into our last example of dividing fractions.
Note first to multiply the first fraction by the reciprocal of the second fraction.
That is, we need to take the reciprocal of the divisor in order to divide these fractions.
Our example asks us to divide 2 over 15 divided by 1 over 5.
Now, as previously stated, we want to take the first fraction here, 2 over 15,
but we want to multiply it by the reciprocal of 1 over 5, which would be 5 over 1.
Now that we have converted this into multiplication, I can go into the steps
of multiplication where step number one is to cross cancel.
Cross cancelling 15 and 5 with a common factor of 5 means that 15 reduces to 3, 5 reduces to 1.
We can't cross cancel 2 and 1 at all.
Therefore, I can multiply straight across: 2 times 1 gives me 2, 3 times 1 gives me 3.
We can't reduce 2/3.
Therefore, my answer is 2/3.
Pause the video and try these problems
Thank you again for joining us with the Cypress College Math Review.