Why EXACTLY do you take the reciprical of a fraction to multiply if original dividing?

Question

1/2 divided by 3/4.

it would end up being 1/2 multiplied by 4/3. WHY?

Answer 1

We wish to discover what happens when you divide some number n by a fraction p/q. Set the result equal to x. Then:

x=n/(p/q)

Multiply both sides by p/q:

x*p/q = n

Multiply both sides by q:

x*p = n*q

Divide both sides by p:

x = n*q/p

But x=n/(p/q). Therefore n/(p/q) = n*(q/p) and we see that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Q.E.D.

Answer 2

This is just an illusion.
remember a/b x b/a = 1 where a and b not equal to zero.
then what we get when we divide 1 by any of the two fractions?
just reversal of the letters. That is the truth of multiplication identity. thus when we are dividing 1/2 by 3/4 , we have first done the operation of division of 1 by 3/4 which results in 4/3. And then we are multiplying two fractions. 1/2 and 4/3. That's all.

You can also give another explanation.
what we are trying to do is ascertaining the ratio of one fraction of the other. But the two having two different bases, we convert one of the two into other's base. thus we can make 1/2 as 2/4 or we can make 3/4 as (3/2)/ 2. For fractions we know, when the bases are equal, the fractions are in the same ratio as their numerators. i.e 1: 3/2 or 2:3 and converted to fraction again, it would be 2/3.

Answer 3

Because just as subtraction is an addition of a negative, division is the multiplication of a reciprocal. All numbers are fractions, as every number is itself divided by 1. For example, you could write 1 as 1/1, or 2 as 2/1, or 15 as 15/1, etc... Now you can also do this with numbers that are already fractions, such as 1/2 = (1/2)/1= (1/1)/(2/1).
Now, to divide 1 into 2 parts, you would write the value:
1/2
this really means (1/1)/(2/1) as shown above, but it also means
(1/1) * (1/2) = 1/2
now, notice that by separtating out the denominator from the numerator our 2/1 is really 1/2! So this shows that division is a multiplication of reciprocals.

So to look at the problem you've presented: (1/2)/(3/4)
We can look at it like (1/2)/1 * 1/(3/4) by just separating out the denominator from the numerator.
Now take a closer look at the second term: 1/(3/4)
Really we can show it is 1/(0.75)
thus it can be written as 1/1 * 1/0.75 = 1/1 *4/3 = 4/3
Thus we are really looking at the equation:
(1/2)/(3/4) = 1/2 * 4/3 = 2/3 = 0.66...

Hope this helps

Answer 4

Multiplying by a fraction is the same as dividing by a whole or mixed number. (2*1/2=1) So in that same line of reasoning, dividing by a fraction is the same as multiplying by a whole or mixed number (2/ 1/2=4). So all you do is flip the second fraction and there you have the equivelant number that you multiply by.