1/2 is the least, then 2/3, then 3/4.

This question can be most easily answered by converting the fractions into decimals first.

$\displaystyle \frac2{}{5}=0.40$

$\displaystyle \frac{4}{8}=0.50$

$\displaystyle \frac{3}{12}=0.25$

$\displaystyle \frac{2}{10}=0.20$

$\displaystyle 0.50>0.40>0.25>0.20$

$\displaystyle \frac{4}{8}>\frac{2}{5}>\frac{3}{12}>\frac{2}{10}$

The most generic process of ordering fractions that can apply to all questions begins with finding the common denominator between all fractions, preferably the least common denominator. In this question, the LCD is 40. Therefore the fractions become: $\displaystyle \frac{25}{40},\frac{10}{40},\frac{36}{40},\frac{20}{40}$. From this point, we can simply arrange the fractions based upon their numerators. This becomes: $\displaystyle \frac{10}{40},\frac{20}{40},\frac{25}{40},\frac{36}{40}$. The final step is to reduce the fractions to their original denominators, which becomes: $\displaystyle \frac{1}{4},\frac{1}{2},\frac{5}{8},\frac{9}{10}$.

If you are comfortable and competent in quickly converting fractions to decimals, then there is a quicker method than what is described above. Simply convert each fraction to a decimal, organize the decimals, and then convert the decimals back to fractions. In this question we can convert $\displaystyle \frac{5}{8} , \frac{1}{4},\frac{9}{10},\frac{1}{2}$ to $\displaystyle 0.625,0.25,0.9,0.5$. Then we can reorganize this to $\displaystyle 0.25,0.5,0.625,0.9$ and convert back to $\displaystyle \frac{1}{4},\frac{1}{2},\frac{5}{8},\frac{9}{10}$.

When approaching this kind of problem, choose the method in which you are more comfortable working with. There is no sense in trying to save time by choosing a method that you are not comfortable with and increasing your risk of answering the question incorrectly.

To put these fractions in increasing order it is important to compare them as decimals, since fractions can often be deceiving. $\displaystyle \frac{2}{3}=0.6\bar{6};\; \frac{1}{2}=0.5;\; \frac{3}{8}=0.375;\; \frac{4}{5}=0.8;\; \frac{2}{15}=0.1\bar{3}$.

Now that the value of each fraction is obvious we can sort then in increasing order:

$\displaystyle \frac{2}{15},\frac{3}8{,}\frac{1}{2},\frac{}2{}3, \frac{4}{5}$

Divide the fraction out to get the decimal

5.5 / 10 = 0.55

To find the percentage, shift the decimal point two to the right = 55%

To find a percent from a fraction you can either divide the number on your calculator and multiply by 100:

$\displaystyle 100\cdot\frac{27}{25}=108$

or you can just multiply the numerator and denominator by the same number that makes the denominator equal to 100 (because a % is just #/100):

$\displaystyle \frac{27}{25}\cdot\frac{4}{4}=\frac{108}{100}$

To begin, remember with percentages that "of" means multiplication and "is" means "equals." Now, we know that one fourth of the remaining half are philosophers. This means that he philosophers really are:

$\displaystyle \frac{1}{4}*\frac{1}{2}=\frac{1}{8}$

So, there are $\displaystyle 1-\frac{1}{2}-\frac{1}{8}$ sociologists as a percentage. This is:

$\displaystyle 1-\frac{1}{2}-\frac{1}{8}=\frac{8}{8}-\frac{4}{8}-\frac{1}{8}=\frac{3}{8}$

Therefore, $\displaystyle \frac{3}{8}$ of the $\displaystyle 464$ are sociologists, or:

$\displaystyle \frac{3}{8}*464=174$ are sociologists.

To convert a fraction into a percent, make the denominator 100 and then the numerator will be the percent. Thus,

$\displaystyle \frac{2}{5}*\frac{20}{20}=\frac{40}{100}=40\%$

For percentages, it is always easiest to translate into language that uses "is" and "of." "Is" means "equals" and "of" means "multiply." So, we have:

What percentage of $\displaystyle 15,000$ is $\displaystyle 7$? This is the same as:

$\displaystyle x * 15,000=7$

Solving for $\displaystyle x$, you get:

$\displaystyle x=\frac{7}{15000}=0.00046666666667$

Now, as a percent, this is equal to:

$\displaystyle 0.00046666666667 * 100= 0.046666666667\%$

Rounding to the nearest hundredth, you have $\displaystyle 0.05\%$.

$\displaystyle Y$  is 35% of $\displaystyle Z$, so $\displaystyle Y = \frac{35}{100}Z = \frac{7}{20}Z$. 

$\displaystyle X$ is $\displaystyle 8\frac{1}{3} \%$ of $\displaystyle Y$, so $\displaystyle X = \frac{8\frac{1}{3}}{100}Y = \frac{8\frac{1}{3}\times 3}{100 \times 3}Y = \frac{25}{300 }Y = \frac{1}{12}Y$.

Consequently, 

$\displaystyle X = \frac{1}{12}Y = \frac{1}{12} \cdot \frac{7}{20}Z = \frac{7}{240}Z$

The smallest integer $\displaystyle Z$ can be is 240. If this happens, 

$\displaystyle Y = \frac{7}{20}Z = \frac{7}{20} \cdot 240 = 84$

$\displaystyle X = \frac{7}{240} \cdot 240 = 7$

Their sum is

$\displaystyle X + Y + Z = 7 + 84 + 240 = 331$

The correct choice is that the sum is between 300 and 400.