multiplication / division of fractions

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They told me that it would be easier than the adding and subtracting of. They were wrong. I find them very confusing.

It IS very easy. If you can multiply 2 numbers together, you can multiply and divide fractions. Let's see if I can show this here:

1 3 (1 x 3)
-- X -- is done just like this: ------- just multiplying straight across, the top and bottom numbers never have to meet each other,
2 4 (2 x 4)

so your answer is 3 Now if you want to DIVIDE those same fractions, you just flip the numbers of the second fraction then multiply.
--
8



So 1/2 divided by 3/4 is just 1/2 X 4/3, and you do that just the same way I did the first one up there. I hope I typed all that out before somebody else did.

Ah jeez, that didn't work out at all. It looked a LOT clearer before it posted like that. OK, somebody else try typing that up so it looks right.

I think what he is saying is using "fact families" to help you memorize both multiplication and division.

So the three numbers 2,6,12 are related.

2*6=12

6*2=12

12/6=2

12/2=6

You can memorize it that way. If you need to learn WHY the answer is the way it is:

Multiplication is just repeated addition.

2+2+2 = 2 * 3

2+2+2+2= 2 * 4


Division is taking a number, and splitting it up among groups to see how many are in each group. This works well to have manipulatives. So 12 / 4 is saying you have 12 items and 4 groups, you split the 12 up evenly into each group and see how many there are in each, which will be 3.


Does that help at all, or no?

They are probably skipping steps and "cross cancelling". This used to confuse me in elementary school and I had no idea what they were doing until college.

I will explain. It's a simple concept but it can take a while to realize what is being communicated to you so bear with me.

When working with or simplifying equations, there is something called "The Order of Operations". It tells you which part of the equation or expression you work on first.

It goes.
1. Parenthesis

And then from left to right.
2. Multiple
3. Divide
4. Add
5. Subtract

*Later in your education you will see that sometimes steps 4 and 5 can be switched.

I'll give you an example. It's long so you can skip it if you want. After I will tell you how this applies to fractions.



The fractions

Say I wanted to multiply (1/2)(5/3)

Multiplying the tops together, and then the bottoms together separately I get.

(1*5)/(2*3)= 5/6

A lot of times they want you to multiply and then get the fraction in the simplest form. But 5/6 doesn't reduce so this is it's simplified form.

What about (2/5)(25/3) ?

We could multiply it out the same way we did above like this.
(2*25)/(5*3)= 50/15

This answer is correct but it's not in it's simplest form because the top numer, 50, and the bottom number, 15, have a common factor. That is, there is a number that fits nicely into both of them.

The number is 5. 5 goes into 15, 3 times because 3*5=15. When we factor the 5 out of 15 (the opposite of multiplying so we also say divide out depending on how you want to look at it), we are left with the 3. So in the bottom we will have a 3.

Factoring the 5 out of the 50 in the top we are left with 10, using the same logic as above.

So we are left with (5*10)/(5*3)

The above step is usually omitted and people usually just write 10/3, but I wanted to show you what is actually happening. Do you see how we took the 50 and 15 apart by breaking each into a common factor (the 5) and another number?

This is the same as writing (5/5)(10/3) Can you see where the 5 goes? Any number divided by itself is 1 so 5/5 can also be written as 1. We say "the 5 cancels"

1*(10/3) is just 10/3 since any number.

There's an easier way to do this. Going back to our original fraction multiplication...

(2/5)(25/3)

(2*25)/(5*3)

Before we actually perform the multiplication, look at the expression above.

In in the top, we have a 2 and a 25 that we want to multiply together. In the bottom we have a 5 and a 3 that we want to multiply together.

There is a number in the top that shares a common factor with a number in the bottom. The 25 shares a common factor with the 5. We can rearrange the numbers in the top as long as we keep them in the top, and the numbers in the bottom as long as we keep them in the bottom, when doing multiplication and division so we can write.

(2*25)/(5*3)

as

(25/5)(2/3) if it makes it easier for you to see what we are about to do.

We ask our selves, how many times does the 5 go into the 25? Or equivalently, what is 25/5?

25/5=5

So we have

5*(2/3)

Which is

(5*2)/3

10/3


By finding the common factor early on, when it was more visible, we avoided having to find it after it became combined with larger numbers 50 and 15 and the extra step of simplifying at the end.

In the case of 50/15 it might be apparent to many people that the common factor is 5, but sometimes you will have large numbers that the common factor can be difficult to find in.

Dividing fractions.

(1/2)/(3/7)

You can divide fraction by taking one of the fractions, flipping it, and multiplying it with the other.

For example, flipping 3/7 we get 7/3

Multiplying

(1/2)(7/3)
(1*7)/(2*3) = 7/6

What about

(3/5)/(9/8)

Flipping the 9/8 we get 8/9

And multiplying

(3/5)(8/9)
(3*8)/(5*9)

But wait, we have a number in the top and bottom that share a common factor. The 3 and the 9 both have a factor of 3.

We can ask ourselves how many times the 3 goes into each.....once for the 3 in the top and 3 times for the 9 in the bottom, and arrive at (1*8)/(5*3)= 8/15, but so you can see what's happening again we will write it out the long way.

(3*1*8)/(5*3*3)

Rearranging the numbers in the top and the numbers in the bottom.
(3/3)(1*8)/(3*5)

3/3=1 so we have

(1*8)/(3*5)

8/15

The fast way.

There's a faster way to divide fractions but it takes more visualization and a concept of what's going on.

(5/4)/(2/3)

The fast way involved multiplying the bottom in the top term, and the bottom in the bottom term, by both the top and bottom term.

Here, the bottom in the top term is 4.

4(5/4) = (4*5)/4 = (4/4)(5) So the 4 cancels out and we have only the 5 in the top term

5/(2/3)

Multiplying the 4 with the bottom term....

4(2/3)=(4*2)/3 = 8/3

So we have

5/(8/3)

Now we multiply the top and bottom terms by the bottom in the bottom term, which is 3.

(3*5)=15

(3*8)/3 = (3/3)*8 The 3 cancels out so we are left with 8

Thus we have...

15/8

If I didn't make any mistakes.

15/8?

That was a lot of work, by the way. Thank you.

But 15/8 is 1 7/8

( a/b ) X ( c/d ) =( aXc ) / ( bXd) both b and d must be different from 0

( a/b ) / ( c/d ) = ( aXd )/ ( bXc ) all b, c, d must be different from 0

15/8 is right.

1 + 7/8 is also right.

Writing 1 next to 7/8 is ambiguous, since if you don't leave enough space it's 17/8 and if you leave too much space it looks like 1 and like 7/8 written separately. Also, it could be confused with 1*(7/8). I know people get told in elementary school that we shouldn't write "improper fractions" and that "mixed numbers" are better for a reason that they aren't going to bother telling you. That's silly, though, and they really should stop telling people that.

I think the clearest way to think about it is to think about it in terms of inverses. a*(1/a) = 1 for all a (except zero, where it doesn't make sense). So if you have (a/b)*(c/d) that's really a*(1/b)*c*(1/d), so you can see that if a = b that a*(1/b) is just one, so you're left with 1*c*(1/d) = c/d. Since you can rearrange things that are multiplied together in any order you want, that's also true if a = d, except you're left with c/b instead.

It also makes sense of the "invert and multiply" rule. (a/b) / (c/d) = a * (1/b) * (1/(c*(1/d)) = a * (1/b) * (1/c) * (1/(1/d)) = a * (1/b) * (1/c) * d which is just (a*d) / (b*c). Trying to memorize a slew of a's, b's, c's, and d's and keep track of which goes where is a pain, understanding how inverses work is easy.

Adding and subtracting fractions can be more difficult since you have to find a common multiple in the denominators first. Take 2/5 + 1/3. To add these, you have to convert them to 15ths, since 15 is the least common multiple of 5 and 3:

6/15 + 5/15

and then you get 11/15. Furthermore, you're also dividing your LCM by each original denominator to come up with the new numerators.

If I'm multiplying 2/5 by 1/3, I just say (2*1)/(5*3) or 2/15.

I have just realized I should have used numbers that the website does not interpret emotionally.

I suppose it really likes math....as it should being a computer and all.