Why does the "Multiplying by the Reciprocal" work?

Question

My teacher asked us in class today and nobody knew injcluding her.

Answer 1

Multiplying by the reciprocal works as follows.
Suppose you have 3 divided by 1/2, than that's equal to 3 time 2/1, according to the rule of multiplying by the reciprocal.
3 / (1/2 ) = 3 * (2/1) = 6

Now let's start easy, 3 divided by 1, you know that's

3 / 1 = 3

But now your dividing by something that's two times smaller, your dividing 3 by 1/2. You know if you have ever divided cake, if you divide a cake into half the number pieces you get twice as much!
So your answer is
3 /1 * 2

Now I rearrange this into
3/1*2 = 3 * 2/1 so indeed 3/ (1/2) = 3* (2/1). This works for all numbers [ Try that yourself!].

Answer 2

Multiplying By The Reciprocal

Answer 3

Multiplying by the Reciprocal is the same as Multiplicative Inverse.

1/7 is 7 since 7(1/7) = 1

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Answer 4

Multiplying by the reciprocal is how you perform fraction division - I assume this is what you meant.

It works because the two are algebraically equivalent.

a/b ÷ c/d = (a/b)/(c/d)
Multiply the top and bottom by b*d:
(a/b*b*d)/(c/d*b*d)
Cancel b's on top and d's on bottom, and you get:
(a*d)/(b*c)
Which, because of association, is the same as...
(a/b)*(d/c)

Answer 5

multiplying by 1/x is the same as dividing by x.

29 NOV 06, 2016 hrs, GMT.

Answer 6

take

x/(1/y)

this is the same as

x/y^-1

which is the same as

x(y^-1)^-1 = x(y)

Why are those steps correct? because of the properties of the exponent.

2^-1 = 1/(2^1) = 1/2
(2^2)^2 = 2^(2*2) = 2^4 = 16