Question

Express the number as a ratio of integers. i) 10.135=10.135353535….

Short answer

The ratio of the integer is given by \(\frac{{5017}}{{495}}\).

Step-by-step solution

Indentify 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.

First, we have to separate the numbers that does not repeat from the repeating decimals.

Figure out the repeating decimal numbers.

In the decimal number 10.135353535, we have the whole number 10 and the decimal number as 135353535. In this decimal number, we can figure out that the number35 is repeating.

Sum of geometric progression series.

Definition: The n-th term from the end of the geometric progression with the infinite terms is given by\(S = \frac{a}{{1 - r}}\)such that 0<r<1.

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

\( = 10.1 + (\frac{{35}}{{1000}} + \frac{{35}}{{100000}} + \frac{{35}}{{10000000}} + \frac{{35}}{{1000000000}} + ....)\)

Now, write the 1000, 100000, 10000000 and 1000000000 as \({10^3},{10^5},{10^7}\)and \({10^9}\)respectively.

\( = 10.1 + (\frac{{35}}{{{{10}^3}}} + \frac{{35}}{{{{10}^5}}} + \frac{{35}}{{{{10}^7}}} + \frac{{35}}{{{{10}^9}}} + ....)\)

Here,

First term\(\frac{{35}}{{{{10}^3}}}\)is the value of a.

And, the value of r is given the common ratio. Here, the denominator values are increased with the multiply of 100. Thus, the value of r is \({10^2}\).

Substitute the values of a and r into the formula, we get

\( = 10.1 + (\frac{{\frac{{35}}{{{{10}^3}}}}}{{1 - \frac{1}{{{{10}^2}}}}})\)

Multiply 10^3 on both the numerator and denominator to simplify the numerator term.

\(\begin{aligned} = 10.1 + (\frac{{35}}{{{{10}^3} - 10}})\\ = 10.1 + (\frac{{35}}{{1000 - 10}})\\ = 10.1 + (\frac{{35}}{{990}})\end{aligned}\)

Simplify the fractions by taking the common denominator of 990, we get

\(\begin{aligned} = \frac{{9999}}{{990}} + \frac{{35}}{{990}}\\ = \frac{{10034}}{{990}}\end{aligned}\)

Divide the fraction by 2 on both and denominator to simplify it to their lowest term.

\( = \frac{{5017}}{{495}}\)

Therefore, the number 10.135353535…. is expressed as a ratio of integers as\( = \frac{{5017}}{{495}}\).