In order to convert from a percentage to a decimal, simply remove the percentage and move the decimal place back two spaces or divide by 100.

$\displaystyle 23.2\% = \frac{232}{100}= 0.232$

The decimal conversion is:  $\displaystyle 0.232$

To increase a number by 25% is to take

$\displaystyle (100 + 25) \% = 125 \%$ of the number, or, equivalently, multiply it by

$\displaystyle \frac{125}{100} = \frac{125 \div 25 }{100 \div 25 } = \frac{5}{4}$.

Therefore, we divide 7 by $\displaystyle \frac{5}{4}$:

$\displaystyle 7 \div \frac{5}{4} = \frac{7 }{1}\div \frac{5}{4} = \frac{7 }{1}\times \frac{4} {5}= \frac{7\times 4 }{1\times 5} = \frac{28}{5}$

$\displaystyle 28 \div 5 = 5 \textup{ R }3$, so

$\displaystyle \frac{28}{5} = 5 \frac{3}{5}$.

To decrease a number by 25% is to take

$\displaystyle (100 - 25) \% = 75 \%$ of the number, or, equivalently, multiply it by

$\displaystyle \frac{75}{100} = \frac{75 \div 25 }{100 \div 25 } = \frac{3}{4}$.

Therefore, we divide 7 by $\displaystyle \frac{3}{4}$:

$\displaystyle 7 \div \frac{3}{4} = \frac{7 }{1}\div \frac{3}{4} = \frac{7 }{1}\times \frac{4} {3}= \frac{7\times 4 }{1\times 3} = \frac{28}{3}$

$\displaystyle 28 \div 3 = 9 \textup{ R }1$, so

$\displaystyle \frac{28}{3} = 9 \frac{1}{3}$.

Increasing a number by 30% is equivalent to taking 130% of a number, which in turn is equivalent to multiplying it by 

$\displaystyle \frac{130}{100} = \frac{130 \div 10}{100 \div 10} = \frac{13}{10}$. 

$\displaystyle 3 \frac{1}{2}$ increased by 30% is the product of $\displaystyle 3 \frac{1}{2}$ and $\displaystyle \frac{13}{10}$:

$\displaystyle 3 \frac{1}{2} \times \frac{13}{10}$

$\displaystyle = \frac{3 \times 2 + 1}{2} \times \frac{13}{10}$

$\displaystyle = \frac{7}{2} \times \frac{13}{10}$

$\displaystyle = \frac{7 \times 13}{2 \times 10}$

$\displaystyle = \frac{91}{2 0}$

$\displaystyle 91 \div 20 = 4 \textup{ R }11$, so this quantity is equal to $\displaystyle 4 \frac{11}{2 0}$, the correct answer.

Decreasing a number by 40% is equivalent to taking 60% of a number, which in turn is equivalent to multiplying it by 

$\displaystyle \frac{60}{100} = \frac{60 \div 20}{100 \div 20} = \frac{3}{5}$. 

$\displaystyle 3 \frac{1}{2}$ decreased by 40% is the product of $\displaystyle 3 \frac{1}{2}$ and $\displaystyle \frac{3}{5}$:

$\displaystyle 3 \frac{1}{2} \times \frac{3}{5}$

$\displaystyle = \frac{3 \times 2 + 1}{2} \times \frac{3}{5}$

$\displaystyle = \frac{7}{2} \times \frac{3}{5}$

$\displaystyle = \frac{7\times 3 }{2\times 5}$

$\displaystyle = \frac{21 }{10}$

$\displaystyle 21 \div 10 = 2 \textup{ R }1$, so the correct response is $\displaystyle 2 \frac{1 }{10}$.

The difference of 6 and $\displaystyle 4\frac{2}{3}$ is 

$\displaystyle 6 - 4\frac{2}{3}$

$\displaystyle = \frac{6 \times 3}{3} - \frac{4 \times 3+ 2}{3}$

$\displaystyle = \frac{18}{3} - \frac{14}{3}$

$\displaystyle = \frac{18-14}{3}$

$\displaystyle = \frac{ 4}{3}$.

Therefore, 6 is being decreased by $\displaystyle \frac{ 4}{3}$; to find out what percent this is of 6, we calculate:

$\displaystyle \frac{\frac{ 4}{3}}{6} \times 100 \%$

$\displaystyle =\left ( \frac{ 4}{3} \div 6 \right ) \times 100 \%$

$\displaystyle =\left ( \frac{ 4}{3} \times \frac{1}{6} \right ) \times 100 \%$

$\displaystyle =\left ( \frac{ 2}{3} \times \frac{1}{3} \right ) \times \frac{100 }{1}\%$

$\displaystyle =\left ( \frac{ 2}{3} \times \frac{1}{3} \right ) \times \frac{100 }{1}\%$

$\displaystyle =\left ( \frac{ 2 \times 1 \times 100}{3 \times 3 \times 1} \right ) \%$

$\displaystyle = \frac{ 2 00}{9} \%$

$\displaystyle 2 00 \div 9 = 22 \textup{ R 2}$, so

$\displaystyle \frac{ 2 00}{9} \% = 22\frac{ 2 }{9} \%$.

The answer that comes closest among the four choices is 20%.

The difference of $\displaystyle 15\frac{1}{2}$ and 12 is

$\displaystyle 15\frac{1}{2} - 12= 3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}$.

This represents an increase of 

$\displaystyle \frac{\frac{7}{2}}{12} \times 100 \%$

$\displaystyle = \left ( \frac{7}{2} \div 12 \right ) \times 100 \%$

$\displaystyle = \left ( \frac{7}{2} \times \frac{1}{12 } \right ) \times \frac{100 }{1}\%$

$\displaystyle = \left ( \frac{7}{2} \times \frac{1}{3} \right ) \times \frac{25}{1}\%$

$\displaystyle = \frac{175}{6}\%$

$\displaystyle 175 \div 6 = 29 \textup{ R }1$, so this is $\displaystyle 29 \frac{1}{6}\%$.

Tthe response closest to the correct percent is 30%.

$\displaystyle X$ is 40% of 150, which is

$\displaystyle 150 \times 0.40 = 60$

$\displaystyle Y$ is 30% of 200, which is 

$\displaystyle 200 \times 0.30 = 60$

$\displaystyle Z$ is 20% of 225, which is 

$\displaystyle 225 \times 0.20 = 45$

The correct response is $\displaystyle Z < X = Y$.

First you want to make a proportion so $\displaystyle \frac{15}{20}$ of the class are male.  

You can reduce this fraction by $\displaystyle 5$ since both the numerator and denominator are divisible by it.  

This gives you $\displaystyle \frac{3}{4}$ which as a decimal is $\displaystyle 0.75$.  

To make that into a percentage, you multiple by $\displaystyle 100$ and add a "%" symbol.  

So that gives you $\displaystyle 75\%$.

The reasoning is independent of the value of $\displaystyle B$, so assume that $\displaystyle B = 100$. 

$\displaystyle A$ is 85% of $\displaystyle B$, so $\displaystyle A = 85$.

To find out by what percent $\displaystyle A$ must be increased by to obtain $\displaystyle B$, calculate

$\displaystyle \frac{B - A}{A} \cdot 100 \%$

This is

$\displaystyle \frac{100 - 85 }{85} \cdot 100 \%$

$\displaystyle \frac{1 5 }{85} \cdot 100 \%$

$\displaystyle \approx 17.6 \%$

The correct choice is 18%.