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Intuitively, I mean. I know how to do operations with fractions, but unlike with addition and subtraction, fraction multiplication to me amounts to just following a set of instructions, not really getting the concept behind it. I dont understand why the operation works the way it does, or why it even works at all. How did we come up with the way we currently multiply/divide fractions? Any tips on how to get better at it? Thanks for reading, any help is much appreciated :)
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Ok, a way always think and explain multiplication by fractions is by multiplication by unit fractions (I hope this is the correct english expression). These are all the fractions with 1 in the numerator.

So let's say you have 3/10. Intuitively, this is something split in ten parts from which you take 3. So far so good.

Now if you multiply it by something like 1/4 i.e. 3/10\*1/4, well you are taking a fourth of what you had previously taken OR even taking 3 pieces out of the thing split in 4\*10, so 40 parts.

Clear up to this point?

Well, going from 3/10\*1/4 to 3/10\*3/4 is basically the same as (3/10\*1/4)\*3.

We already know that 3/10\*1/4 equates to 3 pieces out of a 40 piece thing, now multiplying by 3 is simply taking 3 of those blocks.

I hope this makes it a bit more intuitive :D
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One (maybe not the most intuitive) way to see why fraction multiplication has to be defined this way is to derive it from basic properties of multiplication and division.

Let's say you have two fractions, written x = a/b and y = c/d, that you wish to multiply.   
Doing this directly would need fraction multiplication, so let's assume we don't yet know how to do it. Instead, let's just try to do everything by multiplication.

We can restructure the definitions of x and y so that there are no fractions involved: x\*b = a and y\*d = c. This is like saying 7 = 14/2 is equivalent to 7\*2 = 14, agree so far?

Now we can just multiply the left hand sides and right hand sides to get (x\*b)\*(y\*d) = a\*c.

By the standard rules of multiplication we can try to get out x\*y, that is, the fractions that we wanted to multiply:

(x\*y)\*(b\*d) = a\*c

divide both sides by b\*d:

x\*y = (a\*c)/(b\*d)

that looks familiar, doesn't it?

This looks like x\*y can be written as a fraction, with a\*c as numerator and b\*d as denominator.
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A fraction is just a multiplication of the numerator and the denominator to the negative 1 power. So when you multiply two fractions, you multiply (numerator 1) and (numerator 2) and (denominator 1)^(-1) and (denominator 2)^(-1) all together
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Imagine that you slice a pizza into four equal slices. Then each slice is 1/4 of the pizza.

Now imagine that you cut each of those four slices in half so each turns into two equal slices. You now have eight equal slices each of which is 1/8 of the original pizza.

If you wanted to mathematically calculate the number of slices that you end up with, you could start with the four and double it: 4•2 = 8.

But what if you wanted to know the size of each of the final slices? You could take the reciprocal of the number: 8 = 8/1 so flipping it gives 1/8.

Or you could start with the size of the four slices and and cut them in half: 1/4 • 1/2 = 1/8.

The two methods are finding the same thing, the size of the eight slices, so they need to produce the same result.

And the way to do that, as you can see above, is to multiply the numerators of the fractions and also multiply the denominators of the fractions.
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3/4 x 5/7 is 3/4 of 5/7. “Times” means “of”. So, how do you find 3/4 of 5/7? Well, you can find an equivalent fraction of 5/7 which has a numerator divisible by 4. That would be 20/28. So, 3/4 of 5/7 is the same as 3/4 of 20/28. What’s 3/4 of 20 twenty-eighths? Well, 1/4 of 20 is 5. So 3/4 of 20 is 15. So, 3/4 of 20 twenty-eighths is 15 twenty-eighths. That is, 3/4 x 5/7 equals 15/28. As if by magic, that’s 3x5 on the top and 4x7 on the bottom. To see why it’s actually not magic, try this exercise again with a/b x c/d, and see what always ends up happening.

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