| Math Mojo - Making Math Meaningful |
This was the question:
How do you multiply and divide fractions?
Professor Homunculus' answer:
Great question! Everyone should be
able to multiply and divide fractions easily. Let's see if we can get you to
do that, too.
Let's take a simple set of fractions, like
2/3 * 4/5.
The number on the top of the fraction
sign is called the dividend. The one on the bottom is the divisor.
That is because a fraction is pretty much simply a division problem that hasn't
been worked out yet. What I mean is that 2/3 is a fraction, but it is also
the division problem "what is two divided by three?" So in this case,
2 is the dividend, and 3 is the divisor.
I only explain that here to make explaining operations with fractions easier
in the rest of this lesson.
In order to multiply the above problem, (2/3 * 4/5) you simply multiply the
dividends by each other (2 * 4 = 8) and put that over a new fraction bar.
Now multiply the divisors by each other (3 * 5 =15) and put it below the
bar. So the answer would be 8/15. (That's a little more than half of a
whole).
But why does this work?
(Warning - There are no pictures in this explanation. There shouldn't be, either. Make sure you make your own pictures in your mind's eye for each step of the way. If you have to, make your own illustrations on paper for each step, in ways that make sense to you and truly represent the following explanation. Once you have done that for yourself, you won't need the crutch of having actual pictures, and you will have begun to learn what it is like to think math problems out in your mind. This sounds like a pain in the beginning, but it is only a minor pain. One main difference between people who are good at math and others who "don't get it" is the ability to use their mind's eye. This is one of the most important things that are overlooked in "standard curricula" and public schools, and is entirely ignored in testing. Also, this is not just meant for adults - very young children should be taught how to use their mind's eye. It should be reinforced at every opportunity, so they won't develop the crutch of needing other people to make pictures for them. Getting rid of that crutch has a very scientific name - it's called "learning to think.")
Let's imagine you had a pie (why do people always use pies for examples like this?) and someone had eaten a third of it. So you'd have two-thirds of a pie left. That's 2/3. Now we want to take four-fifths of that. That's 4/5.
(By the way, when you are multiplying, you can say "of" or "groups of" instead of "times" - that may make more sense to you. 2/3 * 4/5 is the same as "two-thirds of four-fifths" or "four-fifths of two-thirds.")
To take four-fifths of two-thirds of the pie, you'd basically chop each of those two-thirds into fifths, and then, from each of the two-thirds, take four of those fifths. Now you've got eight pieces.
But what do you call those pieces? We can't keep calling them fifths, because "fifths" really means "fifths of a whole." And they aren't fifths of whole, are they? They are fifths of a third.
If you chop a third into fifths, you'd have five pieces altogether, right? But there are three thirds in a whole. So to make a whole, you'd need three groups of those five pieces, giving you a total of fifteen.
Therefore, when you chop the thirds up into fifths, although each of those pieces is a fifth of a third, they are each a fifteenth of a whole.
Let that sink in before you go on.
Remember, from above, that we took four of these pieces from each of the two thirds that we had, so we have eight of these pieces. Now we also know what to call the pieces - they are fifteenths. Eight-fifteenths is 8/15, which is a bit more than half.
In a nutshell:
If you are a person who likes rules, here's the rule for multiplying fractions in a nutshell:
Reducing
At
this point you normally have to do what is called "reducing the fraction
to simplest terms." That doesn't apply here, because the fraction is
already in simplest form. I say "normally," because that is what
teachers and tests want you to do. It is not really necessary in real life.
It just makes things simpler, so we usually do it.
But what does simplest form mean?
Let's try and example that would show us:
2/3 * 3/4.
That would be (2*3 =) 6 over (3*4=) 12.
That is 6 over 12, or 6/12.
It turns out that there is a number that can divide evenly into both the dividend
and divisor. If such is the case, you can divide that number into them, and
that will reduce your fraction into simplest terms.
So what number goes into both 6 or 12 evenly? Well, you could use 2, 3 or 6.
Use the largest one, if you know it. This is called the greatest common factor,
or GCF.
In our example, when you divide 6 into 6, you get 1, and when you divide it
into 12, you get 2. So 6/12 = 1/2.
If you look at both fractions you will see that that is true.
What if you had 288/32?
You might not know that the GCF for them is 32. (If you did, you would immediately know that this fraction works out to be 9/1, or simply 9.)
But if you didn't know that, you could surely see that both numbers have 2 as a common factor. So divide both by two and get 144/16.
Then you could see that both of those numbers have 2 as a common factor. So divide both by two again and get 72/8.
Then you could see that both of those numbers have 2 as a common factor. So divide both by two again and get 36/4.
At that point you might notice that 4 is a factor of 36, so the GCF is 4. Divide both by 4 and get 9/1, which is 9.
If you hadn't noticed that 4 is a factor of 36, you could still reduce. By now you should be able to figure out how.
This has been a long, drawn out explanation. But the operation is simple in
practice.
Try 5/7 * 3/5 and reduce it (if it is reducable).
Things to think about:
And now for something (not so) completely different...
For now, the
simplest explanation for dividing fractions, is that you take the second
fraction, and turn it upside down (that's called getting the "reciprocal"
of the fraction). For instance, 4/9 becomes 9/4.
So 2/3 * 5/6 becomes 2/3 * 6/5, and you do it like a multiplication problem.
The answer will be correct. Weird, but true. If
you have considered the above things to think about, that may help you figure
out why that is so.
I have to tell you that math is a
lot cooler than just understanding this. Being able to do fractions is a "must"
for your general education. But you won't get the real "juice" out
of doing them unless you understand WHY they work like they do. That is something
they don't usually teach or care about in school. Make your teacher explain
WHY fractions work like that, until you get it. Bug him/her to do it. If he/she
is a good teacher, they will love the challenge. If they don't love the challenge,
realize that soon you will get a better teacher than that jerk when you are
in another grade. You can survive bad teachers. It just takes patience.
Have fun,
Professor Homunculus
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