## Fractions

We can view the expression:

\[ \frac{X}{Y} \]

as asking, and answering, the question:

How many \(Y\)s are there in each \(X\)?

So for example if \(X = 1 \) and \(Y = 2,\)

then the fraction:

\[ \frac{X}{Y} = \frac{1}{2} ,\]

is asking:

How many twos are there in one?

And the answer is one-half.

Given two fraction, say, \(\frac{1}{3},\) and \(\frac{4}{5},\) we can multiply them together by multiplying the two numerators (the numbers upstairs) and the two denominators (the numbers downstairs), separately.

So for example:

\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}.\]

Or in general:

\[ \frac{P}{M} \times \frac{ Q}{N} = \frac{ P \times Q }{ M \times N}.\]

When we multiply fractions we are really taking fractions of fractions. Imagine that we have divided a pie into five equal parts. One slice falls on the floor and so we have four slices left; i.e., we have \(\frac{4}{5}\) of the original pie. Now even though we had cut out pie into five pieces, because we had expected five people to show up for our party, only three people actually show up.

We want to divide what is left of the pie up evenly among our three guests, and so each guest gets

\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}\]

of a pie.

In practice, this means that we would take each of the pieces we originally cut, and cut them all into three parts. Then we would give each guest four of these smaller pieces.

Because we had originally cut the pie into five pieces, and then cut each one of these pieces (except the one that fell on the floor) into three pieces, each little piece is equal to one fifteenth of a pie. Then we give each guest for each of these smaller pieces.

\[ \frac{ 4}{15}.\]

Suppose we have two pies.

One pie is cut into three equal pieces and the second one is cut into five equal pieces.

Now suppose we want to add a piece from the first pie to a piece from the second pie. How much pie do we have?

As an equations this reads:

\[ \frac{1}{3} + \frac{1}{5} = ? \]

As a mathematical expression, there is nothing wrong with:

\[ \frac{1}{3} + \frac{1}{5}\]

but intuitively we don't really know how much pie this is.

We need to be able to compare \(\frac{1}{3}\) to \(\frac{1}{5}\) and to be able to add these two quantities together.

What we need is a **common denominator.**

What is Percent?

A percent is a special ratio or fraction that always has denominator equal to one hundred.

For example,

\[ \frac{1}{100},\]

\[ \frac{10}{100},\]

\[ \frac{50}{100},\]

are all percents, called one percent, ten percent, and fifty percent, respectively.

Changes in percent can be counter-intuitive.

For example, if the value of a stock falls by fifty percent on one day, and then rises by fifty percent on the next, it will not have recovered all of its value.

It will in fact be seventy-five percent of its original value.

It would need to increase by one hundred percent to recover all its value.