Currently, \(\displaystyle 10\%\) of the employees have a college degree; that is \(\displaystyle 80\) out of the \(\displaystyle 800\) employees. Let \(\displaystyle x\) be the number of additional employees who have a college degree:

\(\displaystyle \frac{80+x}{800+100}=0.10\cdot 2\)

\(\displaystyle \frac{80+x}{900}=0.2\)

\(\displaystyle 80+x=0.2\cdot 900=180\)

\(\displaystyle x=180-80=100\)

All of the additional employees to be hired should have a college degree in order to double the percentage of employees with a college degree.

To begin with, we want to think about his problem conceptually. If Dmitri received a discount, we know our final answer will be higher than what he paid. Eliminate any choices that are less than \(\displaystyle \$436\). 

Next, we want to calculate how much the bike cost Dmitri before sales tax. To do this, we want to have a basic equation in front of us to ensure we are setting this up correctly.

\(\displaystyle Cost=Price*1.05\)

or

\(\displaystyle C=P*1.05\)

The cost is what Dmitri paid, and the price is the listed sale price of the bike. We get \(\displaystyle 1.05\) from knowing that sales tax is \(\displaystyle 5\%\) and that Dmitri paid \(\displaystyle 100\%\) of the price, plus \(\displaystyle 5\%\) for sales tax. Recall that to change from a percent to a decimal, we just move the decimal point two places to the left.

So, we can rearrange the equation to solve for the sale price of the bike before sales tax...

\(\displaystyle \frac{C}{1.05}=P=\frac{436}{1.05}=415.2381\)

We can round that to the nearest hundredth to get 

\(\displaystyle P=$415.24\)

So, the sale price of the bike was \(\displaystyle $415.24\); however, do not choose that as your answer, because we are not done yet! We still need to find the original price of the bike. To do that, another equation will be helpful.

\(\displaystyle Original \:Price*(1-0.35)=Sale \:Price\)

or

\(\displaystyle O*0.65=P\)

This means that the sale price is \(\displaystyle 65\%\) of the original price, which is the same as saying it is \(\displaystyle 35\%\) off.

We know \(\displaystyle P\) from above, so simply rearrange the equation, plug in the problem's values, and solve to find our final answer:

\(\displaystyle O=\frac{P}{0.65}=\frac{415.24}{0.65}=638.8308\)

Again, we can round to the nearest cent to get our final answer of \(\displaystyle \$638.83\).

If \(\displaystyle F\) is 40% of \(\displaystyle N\), then \(\displaystyle F = \frac{40}{100} N = \frac{2}{5} N\).

Cube both sides to get

\(\displaystyle F^{3} = \left (\frac{2}{5} N \right )^{3}\)

\(\displaystyle F^{3} = \frac{8}{125} N ^{3}\)

\(\displaystyle F^{3}\) is \(\displaystyle \frac{8}{125}\) of \(\displaystyle N ^{3}\), or equivalently, 

\(\displaystyle F^{3}\) is \(\displaystyle \frac{8}{125} \times 100 \% = 6 \frac{2}{5 } \%\) of \(\displaystyle N ^{3}\).

\(\displaystyle \sqrt{t}\) is 36% of \(\displaystyle \sqrt{r}\), so

\(\displaystyle \sqrt{t} = \frac{36}{100} \sqrt{r}\), or 

\(\displaystyle \sqrt{t} = \frac{9}{25} \sqrt{r}\)

Square both sides:

\(\displaystyle \left (\sqrt{t} \right )^{2}= \left (\frac{9}{25} \sqrt{r} \right )^{2}\)

\(\displaystyle t= \frac{81}{625} r\)

\(\displaystyle t\) is \(\displaystyle \frac{81}{625}\) of \(\displaystyle r\), or, equivalently, \(\displaystyle \frac{81}{625} \times 100 \% = 12 \frac{24}{25} \%\) of \(\displaystyle r\).

This answer is not among the given choices.

\(\displaystyle S\) is 40% of \(\displaystyle P\), so \(\displaystyle S = \frac{40}{100}P = \frac{2}{5}P\).

Eqivalently, 

\(\displaystyle P = \frac{5}{2}S\), and

 \(\displaystyle 2S + P = 2S +\frac{5}{2}S = \frac{9}{2}S\).

The question can be restated as

\(\displaystyle S\) is what percent of \(\displaystyle \frac{9}{2}S\) ?

This can be answered as 

\(\displaystyle \frac{S}{\frac{9}{2}S} \times 100 \% = \frac{2}{9} \times 100 \% = 22 \frac{2}{9} \%\).

\(\displaystyle M\) is 20% of \(\displaystyle N\), so \(\displaystyle M = \frac{20}{100}N= \frac{1}{5}N\).

\(\displaystyle N - M = N - \frac{1}{5}N = \frac{4}{5}N\)

\(\displaystyle N +M = N + \frac{1}{5}N = \frac{6}{5}N\)

The question can be rewritten as

\(\displaystyle \frac{4}{5}N\) is what percent of \(\displaystyle \frac{6}{5}N\)?

The answer is found as follows:

\(\displaystyle \frac{\frac{4}{5}N}{\frac{6}{5}N} \times 100 \% = \frac{4}{6} \times 100 \% = 66 \frac{2}{3} \%\)

\(\displaystyle M -N\) is 75% of \(\displaystyle M\), so 

\(\displaystyle M -N = \frac{3}{4}M\)

and 

\(\displaystyle \frac{1}{4}M = N\)

\(\displaystyle M + N = M +\frac{1}{4}M = \frac{5}{4}M\) 

The question can be rewritten as

\(\displaystyle M\) is what percent of \(\displaystyle \frac{5}{4}M\) ?

The answer is found as follows:

\(\displaystyle \frac{N}{\frac{5}{4}N} \times 100 \% = \frac{4}{5} \times 100 \% = 80 \%\).

To calculate percent, we need to take one number and divide it by the other. We need to plan out the order carefully, though, because doing the division backwards will yield a trap answer. 

We need to find what percent of 5 40 is. It needs to be greater than 100%, because 40 is larger than five. This clues us in that 40 needs to be in our numerator. Perform the following to solve:

\(\displaystyle \frac{40}{5}=8\)

Don't stop just yet though, because to convert from decimals to percents we need to multiply by 100. This makes our final answer 800%. So, we can say the 40 is 800% of 5.

To find percents, simply divide the part by the whole. In this case, we are asked to find what percent 69 is of 108. Take 69 over 108 to find the answer.

\(\displaystyle \small \frac{69}{108}=.638\bar{8}\)

Multiply by 100 and round to get our final answer

\(\displaystyle \small \small .63\bar{8}*100=63.8 \approx 64\)

So, the answer is 64%.

We can set up an equation to solve this problem where \(\displaystyle p\) represents the orignal price:

\(\displaystyle 0.25 \ p=68.75\)

That is, \(\displaystyle 25\)% of a number equals the \(\displaystyle \$ 68.75\) she saved. Now we just have to solve for \(\displaystyle p\): 

\(\displaystyle p=\frac{68.75}{0.25}= 275\)