Chapter 1.1, Problem 29E

(a)

To determine

To find: the simplest value of $\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}$ .

Answer to Problem 29E

  $\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}=\frac{8}{3}$

Explanation of Solution

Given information:

An expression is given as $\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}$ .

Formula used:

Difference of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{a}{b}-\frac{c}{d}=\frac{ab-cb}{bd}$

Product of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$

Quotient of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}$

Calculation:

Consider the given expression.

  $\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}$

Now, simplify the expression as shown:

  $\begin{array}{c}\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}=\frac{2\cdot 2-\frac{2}{3}\cdot \frac{2}{3}}{\frac{2\cdot 2}{3}}\\ =\frac{4-\frac{4}{9}}{\frac{4}{3}}\\ =\frac{4\cdot 9-4}{9}\cdot \frac{3}{4}\\ =\frac{36-4}{9}\cdot \frac{3}{4}\\ =\frac{32}{9}\cdot \frac{3}{4}\\ =\frac{8}{3}\end{array}$

Here, the simplified value is $\frac{8}{3}$ .

(b)

To determine

To find: the simplest value of $\frac{\frac{1}{12}}{\frac{1}{8}-\frac{1}{9}}$ .

Answer to Problem 29E

  $\frac{\frac{1}{12}}{\frac{1}{8}-\frac{1}{9}}=6$

Explanation of Solution

Given information:

An expression is given as $\frac{\frac{1}{12}}{\frac{1}{8}-\frac{1}{9}}$ .

Formula used:

Difference of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{a}{b}-\frac{c}{d}=\frac{ab-cb}{bd}$

Product of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$

Quotient of two fraction $\frac{a}{b}$ and $\frac{c}{d}$ can be performed by

  $\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}$

Calculation:

Consider the given expression.

Now, simplify the expression as shown:

  $\begin{array}{c}\frac{\frac{1}{12}}{\frac{1}{8}-\frac{1}{9}}=\frac{\frac{1}{12}}{\frac{9-8}{8\cdot 9}}\\ =\frac{\frac{1}{12}}{\frac{1}{72}}\\ =\frac{1}{12}\cdot \frac{72}{1}\\ =6\end{array}$

Here, the simplified value is $6$ .