Chapter 0.5, Problem 27E

To determine

To find: The reciprocal of the number $-\frac{14}{23}$ .

Answer to Problem 27E

The reciprocal of the number $-\frac{14}{23}$ is $-\frac{23}{14}$ .

Explanation of Solution

Given information:

The number $-\frac{14}{23}$ .

Formula used:

Reciprocal of any real number (except 0) is obtained when 1 is divided by the number.

  $\begin{array}{c}\frac{a}{b}=1÷\frac{a}{b}\\ =1\cdot \frac{b}{a}\\ =\frac{b}{a}\end{array}$

Here, $\frac{b}{a}$ is called the multiplicative inverse of $\frac{a}{b}$ .

When a number is multiplied with its reciprocal, then the result is always 1.

  $\frac{a}{b}\cdot \frac{b}{a}=1$

Product of two negative integers is always 1.

Calculation:

Consider the expression $-\frac{14}{23}$ .

Reciprocal of any real number (except 0) is obtained when 1 is divided by the number.

  $\begin{array}{c}\frac{a}{b}=1÷\frac{a}{b}\\ =1\cdot \frac{b}{a}\\ =\frac{b}{a}\end{array}$

Here, $\frac{b}{a}$ is called the multiplicative inverse of $\frac{a}{b}$ .

So, the reciprocal of $-\frac{14}{23}$ is evaluated as,

  $\begin{array}{c}-\frac{14}{23}=1÷\left(-\frac{14}{23}\right)\\ =1\cdot \left(-\frac{23}{14}\right)\\ =-\frac{23}{14}\end{array}$

Recall when a number is multiplied with its reciprocal, then the result is always 1.

  $\frac{a}{b}\cdot \frac{b}{a}=1$

So, when $-\frac{14}{23}$ is multiplied by $-\frac{23}{14}$ , the result should be 1.

  $\begin{array}{c}-\frac{14}{23}\cdot -\frac{23}{14}=+\left(\frac{14}{23}\cdot \frac{23}{14}\right)\\ =1\end{array}$

Thus, the reciprocal of the number $-\frac{14}{23}$ is $-\frac{23}{14}$ .