# Product of Simple Fractions

The applet below displays three fractions and their graphical representation with rows or columns of rectangular areas. In two black fractions, the digits of both the numerator and the denominator can be modified by clicking a little of their vertical midline. Clicking to the right of a digit will cause it to increase by 1; clicking to the left will decrease the digit by 1.

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In the middle rectangle, the representations of the two fractions overlay and the middle red fraction is just the product of the other two. The product of two fractions is defined as follows

 p q
×
 r s
=
 pr qs

The applet illustrates why the product is defined this way. A simple fraction p/q which is read "p qth," signifying p parts out of q. So a rectangle on the left is split into q equal parts (horizontal rectangles) of which p are colored green. The fraction on the right, if thought of as r/s, means taking r parts out of s, the latter are represented by vertical blue rectangles. Each of these rectangles can be split into q smaller ones: r/s = rq / sq. The two fractions are equivalent, meaning that the operation of taking r parts out of s is the same as taking rq parts out of sq. To multiply this fraction by p/q means taking p parts out of q or taking pr parts out of rq. Combining the two operations we are left with pr parts out of sq (or qs, which is the same.)

Here are additional pages related to the definitions, properties of and operations on, fractions:

### Fractions Copyright © 1996-2018 Alexander Bogomolny

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