The reciprocal of any fraction can be found by switching numerator and denominator. Since both numbers are negative, both reciprocals will be negative.

(a) \(\displaystyle -\frac{5}{6}\) will have reciprocal \(\displaystyle -\frac{6}{5}\)

(b) \(\displaystyle -\frac{6}{7}\) will have reciprocal \(\displaystyle -\frac{7}{6}\)

We can compare these by writing them both with common denominator \(\displaystyle LCM (5,6) = 30\).

\(\displaystyle -\frac{6}{5}= -\frac{6 \times 6}{5\times 6} = -\frac{36}{30}\)

\(\displaystyle -\frac{7}{6}= -\frac{5 \times 7}{5\times 6} = -\frac{35}{30}\)

\(\displaystyle -\frac{6}{5} = -\frac{36}{30} < -\frac{35}{30}-\frac{7}{6}\)

making (b) greater

The reciprocal of any fraction can be found by switching numerator and denominator.

(a) \(\displaystyle \frac{3}{4}\) will have reciprocal \(\displaystyle \frac{4}{3}\)

(b) \(\displaystyle 0.7 = \frac{7}{10}\) and will have reciprocal \(\displaystyle \frac{10}{7}\)

We can compare these by writing them both with common denominator \(\displaystyle LCM (3,7) = 21\):

\(\displaystyle \frac{4}{3}= \frac{4\times 7}{3\times 7} = \frac{28}{21}\)

\(\displaystyle \frac{10}{7}= \frac{3\times10}{3\times 7} = \frac{30}{21}\)

\(\displaystyle \frac{4}{3} = \frac{28}{21} < \frac{30}{21} = \frac{10}{7}\)

making (b) greater

When dealing with math problems that involve fractions and whole numbers or mixed numbers, you should first convert the non-fraction into a fraction:

\(\displaystyle 3\rightarrow \frac{3}{1}\)

The problem should look like this:

\(\displaystyle \frac{3}{1}\div \frac{1}{2}\)

When dividing with fractions, you need to find the reciprocal of the second fraction before you can do anything else. In other words, flip the second number.

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

When you find the reciprocal of the second number, change the problem from division to multiplication. The new problem should look like this:

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

The answer is 6.

When dividing fractions, you must first find the reciprocal of the second number in the operation. In other words, flip the second fraction.

\(\displaystyle \frac{1}{8}\rightarrow\frac{8}{1}\)

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

\(\displaystyle \frac{1}{2} \times \frac{8}{1} =\)

Then multiply both the numerators and denominators.

\(\displaystyle \frac{1}{2} \times \frac{8}{1} =\frac{1\times8}{2\times1}=\frac{8}{2}\)

When possible, always reduce the fraction. In this case, both 2 and 8 are divisible by 2.

\(\displaystyle \tfrac{8}{2}\rightarrow\tfrac{4}{1}\rightarrow4\)

The result is your answer.

When dividing fractions, you must first find the reciprocal of the second fraction. In other words, flip the second fraction.

\(\displaystyle \frac{3}{5} \rightarrow\frac{5}{3}\)

When you do this, the operation also changes from division to multiplication. The problem should now look like this:

\(\displaystyle \frac{1}{3}\times\frac{5}{3} =\)

Multiply the numerators and denominators. The result is your answer.

\(\displaystyle \frac{1}{3}\times\frac{5}{3} =\frac{1\times5}{3\times3}=\frac{5}{9}\)

When dealing with fractions and whole numbers, first convert the whole number to a fraction. This is easily done by putting the whole number over 1.

\(\displaystyle 4\rightarrow\tfrac{4}{1}\)

When dividing fractions, you must first find the reciprocal of the second fraction in the operation. In other words, flip the second fraction. 

\(\displaystyle \tfrac{1}{4}\rightarrow \tfrac{4}{1}\)

When you do this, the operation changes from division to multiplication. The problem should now look like this:

\(\displaystyle \tfrac{4}{1}\times\tfrac{1}{4}=\)

Solve the multiplication by multiplying the numerators and the denominators.

\(\displaystyle \tfrac{4}{1}\times\tfrac{4}{1} =\tfrac{4\times4}{1\times1}= \tfrac{16}{1}\rightarrow16\)

Since the result is \(\displaystyle \tfrac{16}{1}\), it reduces to 16 as any fraction with a denominator of 1 is equal to the value of its numerator. 

Candy has \(\displaystyle \small \frac{3}{4}\) cups of trail mix. If she divides it amoung herself and seven of her friends, she is dividing it by \(\displaystyle \small \frac{8}{1}\).  In order to solve this problem, we simply find the reciprocal of the second fraction and multiply.  

To find the reciprical of a fraction, we switch the numerator and denominator.

\(\displaystyle \small \frac{8}{1}\) becomes \(\displaystyle \small \frac{1}{8}\).

Now we multiply \(\displaystyle \small \frac{3}{4}\) by \(\displaystyle \small \frac{1}{8}\).

\(\displaystyle \small \frac{3}{4}\cdot \frac{1}{8}=\frac{3}{32}\)

Each person gets \(\displaystyle \small \small \frac{3}{32}\) of the trail mix.

Division by a fraction is equivalent to multiplication by its reciprocal, so

\(\displaystyle A \div \frac{1}{3}\) can be rewritten as \(\displaystyle A \cdot \frac{3}{1} = A \cdot 3\)

\(\displaystyle 3 > \frac{1}{3}\), and \(\displaystyle A\) is positive, so, by the multiplication property of inequality, 

\(\displaystyle A \cdot 3 >A \cdot \frac{1}{3}\)

or

\(\displaystyle A \div \frac{1}{3}>A \cdot \frac{1}{3}\).

If \(\displaystyle X\) is the reciprocal of \(\displaystyle Y\), then 

\(\displaystyle X = \frac{1}{Y}\)

\(\displaystyle X + 1 = \frac{1}{Y} + 1 = \frac{1}{Y} + \frac{Y}{Y} = \frac{Y+1}{Y}\)

The reciprocal of \(\displaystyle X + 1\) is \(\displaystyle \frac{Y} {Y+1}\).

 The reciprocal of \(\displaystyle Y\) is \(\displaystyle 1 \frac{2}{3} = \frac{1 \times 3+ 2}{3} = \frac{5}{3}\), so \(\displaystyle Y\) is the reciprocal of this, or 

\(\displaystyle Y = \frac{3}{5}\)

\(\displaystyle Y - 1 = \frac{3}{5} - 1 = \frac{3}{5} - \frac{5}{5} = - \frac{2}{5}\)

The reciprocal of \(\displaystyle Y - 1\) is \(\displaystyle - \frac{5}{2}\), which is less than 0.