Chapter 12, Problem 78CLR

To determine

To write: The number $0.4\overline{25}$ as a fraction in the simplest form.

Answer to Problem 78CLR

The fraction is $\frac{421}{900}$ .

Explanation of Solution

Calculation:

Let $p=0.4\overline{25}$ .

Since there are 2 digits repeating in the number p , multiply $p=0.4\overline{25}$ by 100 on both sides.

  $\begin{array}{c}100\left(p\right)=100\left(0.4\overline{25}\right)\\ 100p=42.5\overline{25}\end{array}$

Now subtract $p=0.4\overline{25}$ from $100p=42.5\overline{25}$ to obtain the following result.

  $\begin{array}{l}99p=42.1\\ p=\frac{42.1}{99}\left[\text{Dividing both sides by 99}\right]\\ p=\frac{421}{990}\left[\begin{array}{l}\text{Multiplying both the numerator and denominator}\\ \text{on the right by 10}\end{array}\right]\end{array}$

So, the simplest form is $\frac{421}{990}$ .