To divide a term by a fraction, take the reciprocal of the fraction.  

Then mutiply both terms.

\(\displaystyle \frac{3}{7} \div \frac{7}{3}= \frac{3}{7} \cdot \frac{3}{7} = \frac{9}{49}\)

To find the reciprocal of a fraction, flip the numerator and the denominator.

Thus, the reciprocal of \(\displaystyle \frac{1}{4}\) is \(\displaystyle \frac{4}{1}=4\).

Then we need to find the sum of 4 and 7, which is 11.

To find the reciprocal of a fraction, we simply need to switch the numerator and the denominator: for example, the reciprocal of a fraction \(\displaystyle \small \frac{a}{b}\) is \(\displaystyle \small \frac{b}{a}\).

With integers, it helps to remember that all integers are really fractions with a denominator of \(\displaystyle \small 1\):

\(\displaystyle \small 4=\frac{4}{1}\), \(\displaystyle \small 2=\frac{2}{1}\), and \(\displaystyle \small 17=\frac{17}{1}\)

The reciprocals of these numbers are \(\displaystyle \small \frac{1}{4}, \frac{1}{2},\) and \(\displaystyle \small \frac{1}{17}\) respectively. 

Therefore, to solve the problem, we first need to find the reciprocals of \(\displaystyle \small 8\) and \(\displaystyle \small \frac{3}{2}\). If we keep in mind that \(\displaystyle \small 8=\frac{8}{1}\), we can determine that the reciprocals are \(\displaystyle \small \frac{1}{8}\) and \(\displaystyle \small \frac{2}{3}\), respectively. The product of these two numbers is:

\(\displaystyle \small \frac{1}{8}\times\frac{2}{3}=\frac{1\times2}{8\times3}=\frac{2}{24}=\frac{1}{12}\)

\(\displaystyle \small \frac{1}{12}\) is our final answer.

To rewrite the mixed fractions as improper fractions we multiply the whole number by the denominator and add the numerator to get our new numerator. Our denominator stays the same. With this in mind we get the following:

\(\displaystyle 1\tfrac{2}{3}= \frac{(1 \cdot 3) +2}{3}=\frac{3+2}{3}=\frac{5}{3}\) and

 \(\displaystyle 3\tfrac{2}{3}=\frac{(3 \cdot 3)+2}{3}=\frac{9+2}{3}=\frac{11}{3}\).

Adding them gives us 

\(\displaystyle \frac{5}{3}+\frac{11}{3}=\frac{16}{3}\).

The correct answer is \(\displaystyle \frac{13}{4}\).

The mixed number can be broken down as follows: 3 is the whole number, 1 is the numerator, and 4 is the denominator.

In order to convert the mixed number to an improper fraction, you must first multiply the denominator with the whole number and then add the numerator to it.

This answer is the numerator of the improper fraction:

\(\displaystyle 4\cdot 3+1 = 12+1 = 13\).

The denominator of the fraction remains the same and the final answer is \(\displaystyle \frac{13}{4}\).

This question is asking you to change an improper fraction into a mixed number. To do that we will need to divide 75 by 6.

\(\displaystyle 75\div6=12 \text{ remainder }3\)

Now, we place the remainder as the numerator of a fraction with a denominator that is 6 because we just divided by six.

So then, 12 remainder 3 becomes \(\displaystyle 12\frac{3}{6}\).

Simplify this mixed fraction.

\(\displaystyle 12\frac{3}{6}=12\frac{1}{2}\)

To add mixed numbers, you need to first change both of them into improper fractions.

\(\displaystyle 9\frac{2}{3}=\frac{29}{3}\)

\(\displaystyle 4\frac{1}{27}=\frac{109}{27}\)

Then, add the two improper fractions like you would any other fraction.

We need both fractions to have the same denominator before we can add them. Since 27 is a multiple of 3, we only need to change 1 fraction.

\(\displaystyle \frac{29}{3}\times\frac{9}{9}=\frac{261}{27}\)

Now, we can add these fractions.

\(\displaystyle \frac{261}{27}+\frac{109}{27}=\frac{370}{27}\)

Finally, change the improper fraction back into a mixed number.

\(\displaystyle \frac{370}{27}=13\frac{19}{27}\)

So we first multiply the numerators:

\(\displaystyle 2\cdot1=2\)

We then multiply the denominators

\(\displaystyle 7\cdot4=28\)

Our resulting fraction is:

\(\displaystyle \frac{2}{28}\)

We reduce by dividing by 2 from the numerator and denominator.

For the numerator:

\(\displaystyle \frac{2}{2}=1\)

For the denominator:

\(\displaystyle \frac{28}{2}=14\)

The resulting fraction is:

\(\displaystyle \frac{1}{14}\)

We first multiply the numerators:

\(\displaystyle 3 \cdot4=12\)

Then we multiply the denominators:

\(\displaystyle 4\cdot6=24\)

The resulting fraction is:

\(\displaystyle \frac{12}{24}\)

We divide the numerator and denominator by 12.

The numerator is:

\(\displaystyle \frac{12}{12}=1\)

The denominator results in:

\(\displaystyle \frac{24}{12}=2\)

Our resulting fraction is:

\(\displaystyle \frac{1}{2}\)

First, multiply the numerators of each term:

\(\displaystyle 7\times4=28\)

Then, multiply the denominators:

\(\displaystyle 12\times5=60\)

This results in a fraction of \(\displaystyle \frac{28}{60}\). Since the numerator and denominator both have a common factor of 4, reduce both the numerator and denominator by dividing by 4 to get the simplest form fraction \(\displaystyle \frac{7}{15}\).