Division is the same as multiplying by the reciprocal.  Thus, a/b ÷ c/d = a/b x d/c = ad/bc

Remember that if x has a remainder of 4 when divided by 5, x minus 4 must be divisible by 5. We are therefore looking for a number p such that p - 8 - 4 is divisible by 5. The only answer choice that fits this description is 17. 

Dividing by a number (in this case $\displaystyle \dpi{100} \small \frac {3}{4}$) is equivalent to multiplying by its reciprocal (in this case $\displaystyle \dpi{100} \small \frac {4}{3}$).  Therefore:

$\displaystyle \dpi{100} \small \frac {2}{3}\div \frac{3}{4} = \frac{2}{3}\times \frac{4}{3} = \frac{8}{9}$

$\displaystyle \left ( \frac{5}{6}\times \frac{12}{13} \right )^{2}\div \frac{3}{4}$

First we will evaluate the terms in the parentheses:

$\displaystyle \left ( \frac{5}{6}\times \frac{2\times 6}{13} \right )^{2}\div \frac{3}{4}$

$\displaystyle \left ( \frac{5}{1}\times \frac{2}{13} \right )^{2}\div \frac{3}{4}$

$\displaystyle \left ( \frac{10}{13} \right )^{2}\div \frac{3}{4}$

Next, we will square the first fraction:

$\displaystyle \left ( \frac{10^{2}}{13^{2}} \right )\div \frac{3}{4}$

$\displaystyle \frac{100}{169}\div \frac{3}{4}$

We can evaluate the division as such:

$\displaystyle \frac{100}{169}\times\frac{4}{3}=\frac{400}{507}$

Start by rewriting this fraction as a division problem:

$\displaystyle \frac{3x}{2}\div \frac{5x}{4}$

When dividing fractions, you multiply by the reciprocal of the second fraction, so you can rewrite your problem like this:

$\displaystyle \frac{3x}{2}\times \frac{4}{5x}$

Multiply across the numerators and then across the denominators to get $\displaystyle \frac{12x}{10x}$. The x's cancel, and you can reduce the fraction to be $\displaystyle \frac{6}{5}$.

$\displaystyle a \otimes b = \frac{4-a}{b}$, or, equivalently,

$\displaystyle a \otimes b =( 4-a ) \div b$

$\displaystyle 1 \frac{1}{2} \otimes 2 \frac{1}{3} =\left ( 4-1 \frac{1}{2} \right ) \div 2 \frac{1}{3}$

$\displaystyle = 2 \frac{1}{2} \div 2 \frac{1}{3}$

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

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

$\displaystyle = \frac{15}{14}$

$\displaystyle = 1 \frac{1}{14}$

$\displaystyle a \odot b = \frac{a}{b+2}$,

or, equivalently, 

$\displaystyle a \odot b = a \div\left ( b+2 \right )$

$\displaystyle \frac{3}{5} \odot \frac{1}{5}= \frac{3}{5} \div \left (\frac{1}{5}+2 \right )$

$\displaystyle = \frac{3}{5} \div \frac{11}{5}$

$\displaystyle = \frac{3}{5} \cdot \frac{5}{11}$

$\displaystyle = \frac{3}{1} \cdot \frac{1}{11}$

$\displaystyle = \frac{3} {11}$

$\displaystyle a \; \blacklozenge \; b = a \div b^{2}$

$\displaystyle 25 \; \blacklozenge \; 6\frac{1}{4}= 25 \div \left ( 6\frac{1}{4} \right ) ^{2}$

$\displaystyle = 25 \div \left ( \frac{25}{4} \right ) ^{2}$

$\displaystyle = 25 \div \left ( \frac{625}{16} \right )$

$\displaystyle = \frac{25 }{1}\times \left ( \frac{16} {625}\right )$

$\displaystyle = \frac{1}{1}\times \left ( \frac{16} {25}\right )$

$\displaystyle = \frac{16} {25}$

$\displaystyle a \; \ominus \; b = a^{2} \div b$

$\displaystyle 16 \; \ominus \; 3\frac{1}{5} = 16^{2} \div 3\frac{1}{5}$

$\displaystyle = 256 \div \frac{16}{5}$

$\displaystyle = \frac{256 }{1}\times \frac{5}{16}$

$\displaystyle = \frac{16}{1}\times \frac{5}{1}$

$\displaystyle = 16 \times 5 = 80$

$\displaystyle a \amalg b = 5 - \frac{a}{b}$

or, equivalently,

$\displaystyle a \amalg b = 5 - a \div b$

$\displaystyle 1 \frac{1}{2}\; \amalg \; 2 \frac{1}{3}= 5 - 1 \frac{1}{2} \div 2 \frac{1}{3}$

$\displaystyle = 5 - \frac{3}{2} \div \frac{7}{3}$

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

$\displaystyle = 5 - \frac{9}{14}$

$\displaystyle = 4 \frac{5}{14}$