Chapter 0.5, Problem 41E

To determine

The quotient of the expression $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ .

Answer to Problem 41E

The quotient of the expression $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ is $\frac{5}{18}$ .

Explanation of Solution

Given information:

The expression $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ .

Formula used:

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

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

Product of two real numbers is obtained when they are multiplied together. That is two or more numbers are separated by multiplication signs $\left(\right),\times \text{ or }\cdot$ .

Product of two negative integers is always positive.

Division in real numbers is multiplication by reciprocal. If $b\ne 0$ , then,

  $\begin{array}{c}a÷b=a\left(\frac{1}{b}\right)\\ =\frac{a}{b}\end{array}$

Here, $\frac{1}{b}$ is the multiplicative inverse.

Calculation:

Consider the expression $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ .

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, $-1\frac{1}{5}$ can be written as,

  $\begin{array}{c}-1\frac{1}{5}=-\frac{\left(1\times 5\right)+1}{5}\\ =-\frac{6}{5}\end{array}$

Simplify $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ further as,

  $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)=-\frac{1}{3}÷\left(-\frac{6}{5}\right)$

Recall that division in real numbers is multiplication by reciprocal. If $b\ne 0$ , then,

  $\begin{array}{c}a÷b=a\left(\frac{1}{b}\right)\\ =\frac{a}{b}\end{array}$

Here, $\frac{1}{b}$ is the multiplicative inverse.

Apply it,

  $\begin{array}{c}-\frac{1}{3}÷\left(-1\frac{1}{5}\right)=-\frac{1}{3}÷\left(-\frac{6}{5}\right)\\ =-\frac{1}{3}\cdot \left(-\frac{5}{6}\right)\\ =+\left(\frac{1\cdot 5}{3\cdot 6}\right)\end{array}$

Recall that product of two real numbers is obtained when they are multiplied together. That is two or more numbers are separated by multiplication signs $\left(\right),\times \text{ or }\cdot$ .

Simplify it,

  $\begin{array}{c}-\frac{1}{3}÷\left(-1\frac{1}{5}\right)=+\left(\frac{1\cdot 5}{3\cdot 6}\right)\\ =\frac{5}{18}\end{array}$

Thus, the quotient of the expression $-\frac{1}{3}÷\left(-1\frac{1}{5}\right)$ is $\frac{5}{18}$ .