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.

$\displaystyle A$ is $\displaystyle 16\frac{2}{3} \%$ of $\displaystyle B$ and $\displaystyle 62\frac{1}{2} \%$ of $\displaystyle C$, so 

$\displaystyle A = \frac{16\frac{2}{3} }{100}B = \frac{16\frac{2}{3}\cdot 3 }{100 \cdot 3}B = \frac{50}{300}B = \frac{1}{6} B$,

so $\displaystyle B = 6A$.

$\displaystyle A = \frac{62\frac{1}{2} }{100}C = \frac{62\frac{1}{2} \cdot 2 }{100 \cdot 2}C= \frac{125}{200}C = \frac{5}{8} C$

so $\displaystyle C = \frac{8} {5}A$

$\displaystyle A$ must be divisible by 5, so the least possible value is 5 itself.

$\displaystyle M$ is $\displaystyle 7\frac{1}{2} \%$ of $\displaystyle N$, so

$\displaystyle M = \frac{7\frac{1}{2}}{100} \cdot N = \frac{7\frac{1}{2} \cdot 2}{100\cdot 2} \cdot N = \frac{15}{200} N = \frac{3}{40} N$

$\displaystyle N = \frac{40}{3}M = \frac{40M}{3}$

$\displaystyle N$ is 80% of $\displaystyle P$, so $\displaystyle N = \frac{80 }{100}P = \frac{4}{5}P$.

Consequently, $\displaystyle P = \frac{5}{4}N$

$\displaystyle M$ is $\displaystyle 15 \frac{5}{8} \%$ of $\displaystyle N$, so $\displaystyle M = \frac{15 \frac{5}{8} }{100} N = \frac{15 \frac{5}{8} \cdot 8 }{100\cdot 8} N = \frac{5 }{32} N$

Consequently, $\displaystyle N = \frac{32} {5 }M$.

$\displaystyle P = \frac{5}{4} \cdot\frac{32} {5 }M = \frac{32}{4}M = 8M$, so 

$\displaystyle P$is 800% of $\displaystyle M$. This is not one of the choices.

$\displaystyle M$ is $\displaystyle 22\frac{1}{2} \%$ of $\displaystyle N$, so 

$\displaystyle M = \frac{22\frac{1}{2}}{100} N= \frac{22\frac{1}{2}\cdot 2}{100\cdot 2} N= \frac{45}{200 } N= \frac{9}{40} N=\frac{9N}{40}$,

so 

$\displaystyle \frac{1}{M} = \frac{40}{9N}$

and 

$\displaystyle \frac{2}{M} = \frac{80}{9N}$

The question can be rewritten as

$\displaystyle \frac{80}{9N}$ is what percent of $\displaystyle \frac{5}{N}$ ?

The answer is 

$\displaystyle \frac{\frac{80}{9N}}{\frac{5}{N}} \times 100 \%$

$\displaystyle =\left ( \frac{80}{9N} \div \frac{5}{N}} \right ) \times 100 \%$

$\displaystyle =\left ( \frac{80}{9N} \cdot \frac{N}{5} \right ) \times 100 \%$

$\displaystyle = \frac{16}{9 } \times 100 \%$

$\displaystyle = 177 \frac{7}{9} \%$