First, lets convert the exponent into a fraction. Any negative exponent means it's the reciprocal of the positive exponent. So \(\displaystyle 10^{-4}\) means \(\displaystyle \frac{1}{10^{4}}\). Now lets multiply it with the \(\displaystyle 3\). This means \(\displaystyle 3*\frac{1}{10^4}\) or \(\displaystyle 3*\frac{1}{10000}\) or \(\displaystyle \frac{3}{10000}\).

Let \(\displaystyle 0.33333333333....\) be \(\displaystyle x\). \(\displaystyle (0.33333333333=x)\) 

Lets multiply \(\displaystyle x\) by \(\displaystyle 10\). Now we have:

\(\displaystyle 3.33333333333= 10x\) 

Lets subtract this equation with the first one and we get:

\(\displaystyle 9x=3\) We do this because we want to get rid of the repeating decimals and now we have a simple equation, isolate \(\displaystyle x\) and we arrive at the final answer. 

Let \(\displaystyle 0.857857857...\) be \(\displaystyle x\). \(\displaystyle (0.857857857...=x)\) 

Lets multiply \(\displaystyle x\) by \(\displaystyle 1000\). I chose \(\displaystyle 1000\), because there is a set of \(\displaystyle 3\) numbers that make the repeating decimal. To determine the value to multiply the repeating decimal, we do \(\displaystyle 10\) to the power of number in a set before it repeats.

Now we have:

\(\displaystyle 857.857857857...=1000x\) 

Lets subtract this equation with the first one and we get:

\(\displaystyle 999x=857\) We do this because we want to get rid of the repeating decimals and now we have a simple equation, isolate \(\displaystyle x\) and we arrive at the final answer. 

\(\displaystyle \frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5}\), so \(\displaystyle 6\frac{12}{15}\) simplifies to \(\displaystyle 6\frac{4}{5}\).

The numerator of the improper form of a mixed fraction is the original numerator added to the product of the integer and the original denominator. The new denominator is the same as the old one. Therefore, 

\(\displaystyle 6\frac{4}{5} = \frac{6 \times 5 + 4}{5} = \frac{30+4}{5} = \frac{34}{5}\) 

Multiply the numerator and the denominator: \(\displaystyle 34 \times 5 = 170\),

a three-digit number.

\(\displaystyle \frac{16}{60} = \frac{16 \div 4}{60 \div 4} = \frac{4}{15}\)

Add the numerator and the denominator: \(\displaystyle 4+15= 19\)

The correct response is therefore \(\displaystyle 15 < N \le 20\).

The prime factorization of \(\displaystyle 65\) is \(\displaystyle 5 \times 13\), so the fraction \(\displaystyle \frac{N}{65}\) is reducible if and only if \(\displaystyle N\) is a multiple of 5 or 13.

We can immediately tell that 115 is the only multiple of 5, so we test the other numbers to see if there is a multiple of 13. We soon see that

\(\displaystyle 117 \div 13 = 9\), so 116 and 118 cannot be multples of 13.

115 and 117 are the only values of \(\displaystyle N\) that yield reducible fractions, so the correct response is two.

\(\displaystyle \frac{39 \div 3}{111 \div 3} = \frac{13}{37}\)

Multiply the numerator and the denominator: \(\displaystyle 13 \times 37 = 481\)

The product is a three-digit number.

To put a fraction in simplest form, keep dividing the numerator and denominator by the same number until you cannot go any further.

\(\displaystyle \frac{24\div2}{36\div2}=\frac{12}{18}\)

\(\displaystyle \frac{12\div2}{18\div2}=\frac{6}{9}\)

\(\displaystyle \frac{6\div3}{9\div3}=\frac{2}{3}\)

To simplify a fraction, divide both the numerator and the denominator by the same numbers until there is no number that can divide them both without resulting in a remainder.

\(\displaystyle \frac{224\div4}{400\div4}=\frac{56\div2}{100\div2}=\frac{28\div2}{50\div2}=\frac{14}{25}\)

To simplify a fraction, divide both the numerator and the denominator by the same numbers until there is no number that can divide them both without resulting in a remainder.

\(\displaystyle \frac{36 \div 12}{120 \div 12}=\frac{3}{10}\)