Chapter 0.5, Problem 23E

To determine

The simplified value of the expression $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ .

Answer to Problem 23E

The simplified value of the expression $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ is $\frac{5}{2}$ .

Explanation of Solution

Given information:

The expression $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ .

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.

Calculation:

Consider the expression $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ .

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, $-7\frac{1}{2}$ can be converted into proper fraction as,

  $\begin{array}{c}-7\frac{1}{2}=-\frac{\left(7\times 2\right)+1}{2}\\ =-\frac{15}{2}\end{array}$

Now, 15 can be written as the product of 3 and 5.

So, $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ can be simplified as,

  $\begin{array}{c}\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)=\left(-\frac{1}{3}\right)\cdot \left(-\frac{15}{2}\right)\\ =\frac{1\cdot 3\cdot 5}{3\cdot 2}\end{array}$

Striking off the common terms, we get,

  $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)=\frac{1\cdot 5}{2}$

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,

  $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)=\frac{5}{2}$

Thus, the simplified value of the expression $\left(-\frac{1}{3}\right)\cdot \left(-7\frac{1}{2}\right)$ is $\frac{5}{2}$ .