Chapter 3.2, Problem 6CYU

To determine

To write: The decimal $0.\overline{78}$ as a fraction in its simplest form.

Answer to Problem 6CYU

The decimal $0.\overline{78}$ as a fraction in its simplest form is $\frac{78}{99}$ .

Explanation of Solution

Given:

The decimal is $0.\overline{78}$ .

Calculation:

Consider the given decimal as $x=0.787878\dots$ .

Multiply both sides of the given fraction by 100 to convert it in fraction.

  $\begin{array}{c}x\times 100=\left(0.787878\dots \right)\times 100\\ 100x=78.787878\dots \end{array}$

Subtract the equation $x=0.787878\dots$ from $100x=78.787878\dots$ as follows:

  $\begin{array}{l}100x=78.787878\dots \\ \underset{\_}{\begin{array}{l}x=0.787878\dots \\ --\end{array}}\\ 99x=78\end{array}$

Simplify the equation $99x=78$ as follows:

  $\begin{array}{c}99x=78\\ x=\frac{78}{99}\end{array}$

Thus, the decimal $0.\overline{78}$ as a fraction in its simplest form is $\frac{78}{99}$ .