We can set up the following equation from what we are told in the question: \(\displaystyle n = 16q + 15\) where \(\displaystyle q\) is the quotient, then we divide \(\displaystyle n\) by 8: \(\displaystyle \frac{16q+7}{8}\) or \(\displaystyle 2q+\frac{7}{8}\). From there we can  see that \(\displaystyle \frac{7}{8}\) will yield a remainder of \(\displaystyle 7\), which is our final answer.

In order to simplify the expression \(\displaystyle \frac{3}{4}-\frac{5}{8}+\frac{3}{16}\), let's first change the terms to reflect a common denominator:

\(\displaystyle \frac{3\times 4}{4\times 4}-\frac{5\times 2}{8\times 2}+\frac{3}{16}\)

\(\displaystyle =\frac{12}{16}-\frac{10}{16}+\frac{3}{16}\)

\(\displaystyle =\frac{12-10+3}{16}\)

\(\displaystyle =\frac{5}{16}\)

The remainder cannot be greater or equal to the divisor, so we can already eliminate 3, 4 and 5. Then, we can set up an equation with the given information. We know that when \(\displaystyle n\) is divided by 12, the remainder is 7 : \(\displaystyle n= 12q + 7\), where \(\displaystyle q\) is the quotient. So, let's try to divide \(\displaystyle n\) by 3 and we get : \(\displaystyle \frac{12q+7}{3}\) or \(\displaystyle 4q + \frac{7}{3}\). Therefore, the remainder must be one, since when 7 is divided by 3, the remainder is \(\displaystyle 1\).

We are told that \(\displaystyle \frac{a}{b}=36+0.24\). In other words, the remainder \(\displaystyle r\) can be expressed as follows: 

\(\displaystyle \frac{r}{b}=0.24\) or \(\displaystyle \frac{r}{b}= \frac{24}{100}\)

If we simplify, we get \(\displaystyle \frac{r}{b}= \frac{6}{25}\).

Therefore, we can see that \(\displaystyle r\) is a multiple of \(\displaystyle 6\). The only possible multiple of \(\displaystyle 6\) in the answer choice is \(\displaystyle 42\).

Here we can be tempted to answer that the answer does not exist since there can be no division by 0; however, \(\displaystyle (3-3)!=0!=1\), or in other words, the factorial of 0 is 1. Therefore, the final answer is given by \(\displaystyle 3!\) or \(\displaystyle 6\).

We can solve this problem by setting up our equation and solving for the number, \(\displaystyle n\):

\(\displaystyle \frac{3}{8}n=24\)

\(\displaystyle 3n=(8)(24)\)

\(\displaystyle 3n=192\)

\(\displaystyle n=64\)

\(\displaystyle \frac{5}{16}n=125\)

\(\displaystyle 5n=(125)(16)\)

\(\displaystyle 5n=2000\)

\(\displaystyle n=400\)

In order to add the two fractions, we must find the lowest common denominator. To do this, we simply multiply  each fraction by the denominator of the opposite over itself:

\(\displaystyle \frac{9}{9}\cdot\frac{3}{8} + \frac{8}{8}\cdot \frac{2}{9} = \frac{27}{72} + \frac{16}{72} = \frac{43}{72}\)

To simply, we must first find the common denominator of the two fractions. That would be \(\displaystyle b \times 7c\) or \(\displaystyle 7bc\)

Hence we multiply the first fraction by \(\displaystyle \frac{7c}{7c}\) and the second fraction by \(\displaystyle \frac{b}{b}\), and we will have.

\(\displaystyle \frac{7ac}{7bc} + \frac{6ab}{7bc}\).

Now that the denominators match, we can add the fractions. The denominator stays the same after this, only the numerators add together.

Then factor out an \(\displaystyle a\) from the numerator to get the final answer.