Question

Express the number as a ratio of integers.

\(\)\(0.\overline {46} = 0.46464646...\)

Short answer

The ratio of the integer is given by \(\frac{{46}}{{99}}\).

Step-by-step solution

Identify the repeating decimal numbers

A rational number can be written as a ratio of two integers. Here, we have to convert the given decimal number into a ratio of integers.

Figure out the repeating decimal numbers

In the decimal number\(0.464646..\), there is no whole number and the decimal number as\(464646\). In this decimal number, we can figure out that the number\(46\) is repeating.

Geometric series

Here, we have to sum all the repeated decimal value with their equivalent fraction.

A repeating decimal can be thought of as an infinite geometric series.

Here, we have to use the geometric series formula to find the infinite series sum

\(\sum\limits_{n = 1}^\infty {a{r^{n - 1}}} = \frac{a}{{1 - r}}\)

Now, we have to compare the expression with \(\sum\limits_{n = 1}^\infty {a{r^{n - 1}}} \)

Here, \(a = 0.46\)and \(r = {10^{ - 2}}\)

\( \Rightarrow r = 0.01\)
Thus, the sum is

Therefore, the number\(0.\overline {46} = 0.46464646...\)is expressed in ratio of integers as\(\frac{{46}}{{99}}\).