Company X’s stock price drops by p%, and then drops by p% again, where

If we let the original price = m, the,n after dropping by p% twice, we have:

(100 - p)/100 x (100 - p)/100 x m = new price

Let the percent increase needed to return to its original value be n percent. We can create the following equation:

(100 - p)/100 x (100 - p)/100 x m x (100 + n)/100 = m

Dividing both sides by m, we have:

(100 - p)/100 x (100 - p)/100 x (100 + n)/100 = 1

(100 - p)^2(100 + n)/100^3 = 1

Multiplying both sides by 100^3 and dividing both sides by (100 - p)^2, we have

100 + n = 100^3/(100 - p)^2

n = 100^3/(100 - p)^2 - 100

We obtain the common denominator of (100 - p)^2 on the right side of the equation:

n = 100^3/(100 - p)^2 - 100(100 - p)^2/(100 - p)^2

We factor the common 100 from both terms of the right side of the equation:

n = 100/(100 - p)^2 x (100^2 - (100 - p)^2)

Recognizing that (100^2 - (100 - p)^2) is a difference of squares, we factor and combine like terms:

n = 100/(100 - p)^2 x [(100 - (100 - p))(100 + (100 - p))]

n = 100/(100 - p)^2 x [p(200 - p)]

n = 100p(200 - p)/(100 - p)^2

Answer: E

Let, Initial stock price of \(X\) is \(100\).
After dropping \(p\%\), the value will be \(100-p\).
Again after dropping \(p\%\), the value will be \((100-p)(1-\frac{p}{100})\)
\((100-p)(1-\frac{p}{100})\)
\(=100-p-p+\frac{p^2}{100}\)
\(=100-2p+\frac{p^2}{100}\)
\(=\frac{(10000-200p+p^2)}{100}\)
\(=\frac{(100-p)^2}{100}\)
\(=\frac{(p-100)^2}{100}\)

∴ Net decrease in value is \(100-\frac{(p-100)^2}{100} = \frac{10000-(p-100)^2}{100}\)

Now the question becomes \(\frac{10000-(p-100)^2}{100}\) is what percent of \(\frac{(p-100)^2}{100}\)?

\(\frac{\frac{10000-(p-100)^2}{100}}{\frac{(p-100)^2}{100}}*100\)
\(=\frac{10000-(p-100)^2}{100}*\frac{100}{(p-100)^2}*100\)
\(=\frac{(10000-(p-100)^2)*100}{(p-100)^2}\)
\(=\frac{(10000-(p^2-2p.100+100^2))*100}{(p-100)^2}\)
\(=\frac{(10000-p^2+2p.100-10000)*100}{(p-100)^2}\)
\(=\frac{(200p-p^2)*100}{(p-100)^2}\)
\(=\frac{100p(200-p)}{(p-100)^2}\)

Answer: E. \(\frac{100p(200-p)}{(p-100)^2}\%\)

Plugged in values

P= 50%
original value =100.
First drop = 100*(50/100) = 50
Second drop = 50*(50/100) = 25

For the stock to return to original value, I used the following equation: New-Old/New

(25 -100)/25 = 300%

I then plugged in P=50 in answer E and got the answer matching 300%.

Answer E

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