Why does "invert and multiply" work when dividing fractions?

Answer 1

Yes.

(a) / (x/y) = (a) * (y/x)

Answer 2

For almost any number you give me, I can give you another number that when you multiply your number by my number, you get 1. The numbers are called "reciprocals" of each other by laymen. By mathematicians, they're called "multiplicative inverses." Numbers, called "fractions" are really what mathematicians call "rational numbers."

When dealing with integers, you will find that the reciprocal of a number is simply a fraction with the numerator 1 and the denominator being the number for which you wish the reciprocal. So some people say the "reciprocal of a number" is "one over the number." In a sense they are correct, but it's an additional mathematical step that is unnecessary.

So, if you give me 5, I'll give you 1/5. If you give me -17/23, I'll give you -23/17. If you give me (3x-5)/(y+2), I'll give you (y+2)/(3x-5).

If you look at the rules governing the operations on real numbers, i.e., the associative, commutative, and distributive properties, there is no mention of division. There's a reason for that. What most people call "division" is really multiplying by the divisor's reciprocal. So, when somebody says "divide 10 by 5," either they don't know enough about mathematics, or think you don't, to say "multiple 10 by the reciprocal of 5."

So, if somebody tells you to "divide 7/12 by 21/4" they mean multiply 7/12 by the reciprocal of 21/4. The reciprocal of 21/4 is that number that you multiply 21/4 by to get 1. Well, (21/4)(4/21)=1 so you multiply 7/12 by 4/21 to get the answer.

WARNING!!! I said for ALMOST any number you give me I can give you a reciprocal. The number 0 (zero) has the unique property that whatever you multiply it by you get 0. There is, therefore, no number I can give you that you can multiply it by to get 1. Consequently, it has no reciprocal. Now remember, people who don't know math, or people who think you don't know math are inclined to call multiplying by a number's reciprocal "division." Since 0 has no reciprocal, they say, "you can't divide by 0." And now you know why.

Answer 3

Example: (2/3)/(5/6)

Here we have a big fraction with little fractions as numerator and denominator

If I multiply 5/6 by its reciprocal 6/5 I get 1

If I multiply the denominator by something I must also multiply the numerator by the same to thing to retain the mathematical value of the fraction

If i multiply the numerator and denominator by 6/5, I get

(2/3)(6/5)/(5/6)(6/5) = (2/3)(6/5)/1 = (2/3)(6/5) = 4/5

Answer 4

Hi,
Well, let's say that we want two divide fractions as follows:
(The dashed lines are fraction bars.)
1/3
-------
1/4
Then we can multiply both fractions by 4/1 to reduce the denominator to 1.
(4/1) (1/3)
--------------
(4/1)(1/4)

(4/1)(1/3)/1
=(4/1)(1/3)
Which is the same as inverting the divisor and multiplying.

Hope this helps.
FE

Answer 5

Good question. Let's say you're dividing two fractions, a/b and c/d. So you have:
(a/b) / (c/d)

You can think of this as being one big fraction (dividing two numbers is always the same thing as writing one over the other as a fraction). Therefore you can multiply the top and bottom by the same number. Multiply by d/d and you get:

(a/b)d / (c/d)d
(a/b)d / c

In algebra, the associative law says we can always write (x*y) / z as x*(y/z), so this gives us
(a/b) * (d/c).

So no matter what a, b, c, and d are, you always end up with a/b * d/c when you divide a/b by c/d.

Answer 6

essentially you're doing the same thing with whole numbers... lets take the fraction (3/1) and divde it by (4/1) we know the answer should be (3/4) right??

ok ...

multiplying fractions easy as pie; flip the 2nd number and multiply.

(3/1) x (1/4) is (3/4)!!

its just a property of numbers that seems to work 100% of the time

Answer 7

It works the same way with simple fractions.

If you have 3/5, it is the same as 3 * (1/5) which is the inverse of the denominator. We use it all the time the other way but dont think about it being a two way street.

Answer 8

It works because multiplication and division are inverse operations of each other