Chapter 0.5, Problem 29E

To determine

To find: The reciprocal of the number $-5\frac{1}{3}$ .

Answer to Problem 29E

The reciprocal of the number $-5\frac{1}{3}$ is $-\frac{3}{16}$ .

Explanation of Solution

Given information:

The number $-5\frac{1}{3}$ .

Formula used:

Mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

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 positive.

Calculation:

Consider the expression $-5\frac{1}{3}$ .

Recall that mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

So, $-5\frac{1}{3}$ will be converted as,

  $\begin{array}{c}-5\frac{1}{3}=-\frac{\left(5\times 3\right)+1}{3}\\ =-\frac{16}{3}\end{array}$

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 $-\frac{16}{3}$ is evaluated as,

  $\begin{array}{c}-\frac{16}{3}=1÷\left(-\frac{16}{3}\right)\\ =1\cdot \left(-\frac{3}{16}\right)\\ =-\frac{3}{16}\end{array}$

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 $-\frac{16}{3}$ is multiplied by $-\frac{3}{16}$ , the result should be 1.

  $-\frac{16}{3}\cdot \left(-\frac{3}{16}\right)=1$

Thus, the reciprocal of the number $-\frac{16}{3}$ is $-\frac{3}{16}$ .