Say that the value of a certain stock is \(X\) dollars on Monday morning when the market opens. By the time the market closes it has lost half its value--it is down fifty percent--and so is \(\frac{X}{2}\) dollars.

On Tuesday morning the stock starts out at \(Y = \frac{X}{2}\) dollars. When the market closes on Tuesday the stock is up fifty precent (from where it was in the morning) and so is worth

\[ \frac{100 + 50}{100}Y = \frac{150}{100}Y = 1.5 Y \text{ dollars.} \]

But \(Y = \frac{X}{2},\) and so \(1.5 Y = \frac{1.5}{2} X = .75 X\)

The reason for the confusion is that when we say on Monday that a stock is "down fifty-percent," we mean down fifty-percent from its value that morning, that is it is down fifty-percent relative to \(X.\) When, on Tuesday afternoon, we say that the stock is "up fifty-percent," we mean up fifty-percent relative to the value on Tuesday morning, that is \(Y = \frac{X}{2}.\)

As another example, suppose a certain stock is worth \(X\) dollars on Monday morning, and suppose that its price falls by ten percent every day that week. How much will the stock be worth, as a percent of \(X\) on Friday afternoon?

On Monday afternoon, the stock is down ten percent and so is worth \( \frac{ 100 - 10}{100} X\) dollars.

That is

\[ \frac{100 -10}{100} X = \frac{ 90}{100} X = .9 X.\]

On Tuesday morning the stock is worth \(Y = .9 X\) dollars.

On Tuesday afternoon the stock is worth \(.9 Y = (.9)^{2} X.\)

By Friday afternoon, the stock has gone down ten percent for five days in a row, and so is now worth:

\[ (.9)^{5} X = .59 X = \frac{59}{100}X = \frac{100 -41}{100} X.\]

And so by Friday afternoon the stock is worth 59 percent of \(X,\) or is down by 41 percent from its value on Monday morning.