Please explain division with fractions for me.

Factual Questions
1

Why does multiplication by a reciprocal give you the answer for division for fractions?

2

There are 4 quarter inches in one inch. If you divide one inch into quarter inches you get 4 of them. 1/(1/4) = 4.

3

It’s by definition. The result of the division is the number such that, when multiplied by the second number will produce the first number. I.e. a divided by b equals c means that b * c = a. Similarly, the reciprocal of a given number is the value which when multiplied by that number gives a result of 1.

So, call d = 1/c

b * c = a
(b * c) * d = a * d
b * (c * d) = a * d
b * 1 = a * d
b = a * d
b = a * 1/c

4

Because all division is, is multiplying the dividend by the divisor’s multiplicative inverse. For instance, when dealing with whole numbers, you calculate 8/2 by multiplying 8 by 2’s multiplicative inverse(in this example, 1/2).

It may be easier to look at it this way:


             1     X
X ÷ Y = X * --- = ---
             Y     Y

That X ÷ Y == X/Y should be intuitively obvious. And since we’re using variables, you can see that it works for any number, whole or fraction. The multiplicative inverse of a fraction is simply its recripocal, so dividing something by a fraction is, obviously, the same as multiplying by its recripocal.

QED

5

(a/b)/(c/d) is that (unique) number which when multiplied by c/d gives a/b.

(ad)/(bc) * c/d = (acd)/(bcd) = a/b

Since ad/bc satisfies the property and such a number is unique, ad/bc is it.

6

Quite simply, because a fraction is a representation of a division in and of itself. 1/4 is not merely a different way to write 0.25 but also the answer to the question, if you take unity and divide it into fourths, what product will result?

Since multiplicative inverses are equivalent to division by the base number (X times 0.25 is equivalent to X divided by four), dividing by a fraction is equivalent to multiplying by the inverse of the fraction, i.e., the numbers of the fraction inverted so that numerator becomes denominator and vice versa. Dividing by 17/23 is equivalent to multiplying by 23/17, for example.

7

Taking David Simmons’ example a step further:
If we want to divide 3 inches into quarter inches, how many quarter inches does that equal?

3 inches ÷ ¼ = 12 quarter inches.

Here’s a tutorial (and calculator) for fraction division and multiplication:
http://www.1728.com/frctons2.htm

8

You obviously know what you’ re talking about, Mathochist, but I don’t think I’ve ever been able to follow one of your explanations.

9

I say what I mean, or in other words I mean what I say.

In general, given numbers p and q, d=p/q is defined as a number satisfying qd=p. Further, there is only one such number, if any (for our purposes, there always is such a number unless q=0).

Now, consider dividing a/b by c/d. That is, find the number x such that x*(c/d) = a/b (following the above criterion). We can easily check that (ad)/(bc), which is the product of a/b and the reciprocal of c/d satisfies this equation.

10

Darn you Mathochist, that’s exactly what* I* was going to say !!! :rolleyes: