Why do you have to invert fractions?

Question

In math, when you divide fractions, why do you always have to invert the divisor? I know because it is a rule and if you don't your problem will turn out wrong: but why?
If you know that's really great because I'm really curious. Thank You!

Answer 1

OK - I'll try to explain.

Let's say you have the following fraction : 5 / (1/2)
So 5 is in the numerator and 1/2 is in the denominator. If you changed the fraction to a decimal it would be .5 so you would have 5/.5. Now if youwant to make the denominator 1 you could mutiply the top and the bottom by 2 and you would have 10/1 =10. With me so far?

Ok - let's go back to 5 / (1/2). Now if we were to mutiply both the top and the bottom by 2 you would end up with 10/1 = 10.

Well guess what? 2 is the invert of 1/2. So there is a logic.

Hope this helps!

Answer 2

For convenience sake. In a division operation you invert the divisor and proceed as in multiplication and the result will be the same as when you divide the dividend. Example:

30 divided by 3
30 is equal to 30/1 as 3 is equal to 3/1
To divide 3/1 becomes 1/3 for convenience as follows:
= (30/1) / (3/1)
= 30/1 * 1/3 inverting the divisor and you arrive at:
= 30/3 or 10/1 (reduced to lowest terms) or just 10 as 10 divided 1 is 10.

To prove simply multiply the quotient with the divisor:
10 (quotient) multiplied by 3 or
10/1 * 3/1 and you arrive at the dividend as before:
30/1 or simply 30.

Answer 3

Because you are balancing the equation and that is a rule just like addition equals the combination while subtraction equals less than the original.

The proper way to switch the fraction across the equal sign is to invert it. There is a mathematical proof that I am sure both you and I wouldn't understand, so I am not even going to try and show it to you. You just have to take it as one of the fundamentals of math like 2+2=4.

Answer 4

As a thought experiment, what is 5 divided by 1/2 ?
It's the same as asking how many halves are in 5? Clearly there are 10 halves in 5, precisely because 5x2 = 10.

We can prove it algebraically for the general case. Let's say we want to divide one fraction, a/b, by another, c/d.

We can write this as one fraction over the other:
a/b
----
c/d

We can multiply top and bottom by d without changing the answer:
ad/b
------
c

Then we can divide top and bottom by c without changing the answer:
ad/bc
------
1
= ad/bc

So a/b divided by c/d is ad/bc
But ad/bc = a/b * d/c
So a/b divided by c/d = a/b times d/c

Answer 5

you invert and then multiply, as i'm sure you know. inverting isn't the ultimate unchanging rule about fractions. technically you could do long division, it's just easier the other way. i like to think of it as a double negative: division is the opposite of multiplication, and a fraction is the opposite of it's reciprocal. 2 negatives make a positive; inverting two elements of a division problem gives you the same answer as the original problem.

Answer 6

if you have 1 and you divide it by "halves" you get 2 pieces which are halves:
1 / (1/2) = 2

The names of some of the easy fractions ("one third", "one quarter", "one seven") tell you how many times they would fit into 1 ... they would 'fit in' 1 them by what ever their bottom number is .. ( 1/2 fits in 1 by "2"). and as all numbers can be re-writen as that number multiplied by one (22 is the same as 22*1) then "dividing by a fraction" is the same as "multiplying by how many times it goes into one ... and we knwo for the easy fractions (1/x .. where x is anything you want).. will always be x, then it shows "divide by 1/x" is the same as "multiply by x/1"
.. and we all know anything "over one" means that number ... (EG 7/1 = 7)

Answer 7

I assume you are talking about fractions and the properties of multiplication and division.

Because multiplication and division are "inverse operations" (similar to addition and subtraction) it turns out that dividing one number by another is exactly the same as multiplying by its multiplicative inverse.

So, for example,

a/b = a x 1/b

etc.

Answer 8

Because when M=math and S=vomit, M=S

Answer 9

I think because it just works like that