Basic Math - Reciprocals

What is the sum of the reciprocal of $\frac{1}{4}$ and ?

$7\frac{1}{4}$

$\frac{7}{4}$

Explanation

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

Thus, the reciprocal of $\frac{1}{4}$ is $\frac{4}{1}=4$ .

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

What is the reciprocal of $\small 8$ multiplied by the reciprocal of $\small \small \frac{3}{2}$ ?

$\small \frac{1}{12}$

$\small 12$

$\small \frac{3}{16}$

$\small \frac{16}{3}$

$\small 6$

Explanation

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

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

$\small 4=\frac{4}{1}$ , $\small 2=\frac{2}{1}$ , and $\small 17=\frac{17}{1}$

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

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

$\small \frac{1}{8}\times\frac{2}{3}=\frac{1\times2}{8\times3}=\frac{2}{24}=\frac{1}{12}$

$\small \frac{1}{12}$ is our final answer.

Evaluate:

$\frac{3}{7} \div \frac{7}{3}$

$\frac{9}{49}$

$\frac{1}{7}$

$\frac{1}{21}$

Explanation

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

Then mutiply both terms.

$\frac{3}{7} \div \frac{7}{3}= \frac{3}{7} \cdot \frac{3}{7} = \frac{9}{49}$

Compute: $\frac{x}{\left(2x)^{-2}\right }$

$4x^{3}$

$\frac{4}{x}$

$\frac{1}{4x}$

$\frac{4}{x^3}$

$\frac{1}{4}$

Explanation

We will need to rewrite this in order to eliminate the negative exponent in the problem.

Because the denominator has a negative exponent, that is the same as having a positive exponent in the numerator. Therefore we can rewrite the problem as follows and then multiply.

$\frac{x}{\left(2x)^{-2}\right } = x\cdot (2x)^2 = x\cdot4x^2 = 4x^3$