Question

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{27}=0.272727 \ldots$$

Short answer

Answer: The fraction representation of the repeating decimal \(0.\overline{27}=0.272727\ldots\) is \(\frac{3}{11}\).

Step-by-step solution

Identify the repeating digits and period

In this exercise, the repeating decimal is \(0.\overline{27}=0.272727\ldots\). The repeating digits are "27", and the period (the number of digits in one repetition) is 2.

Write the decimal as a geometric series

We can write the repeating decimal as a geometric series with the common ratio \(r=10^{-2}=0.01\): $$0.272727\ldots = 0.27 + 0.27(0.01) + 0.27(0.01)^2 + \ldots$$ This can also be written as: $$0.272727\ldots = \sum_{n=0}^{\infty} 0.27(0.01)^n$$

Convert the geometric series into a fraction

To convert the geometric series into a fraction, we can use the formula for the sum of an infinite geometric series: $$\frac{a}{1-r}$$ In our case, the first term \(a=0.27\) and the common ratio \(r=0.01\). Plugging in these values, we get: $$\frac{0.27}{1-0.01}$$

Simplify and express the fraction as a ratio of integers

Now, we will simplify the fraction and express it as a ratio of integers: $$\frac{0.27}{0.99} = \frac{27}{99} = \frac{3 \times 9}{11 \times 9} = \frac{3}{11}$$ So, the fraction representation of the repeating decimal \(0.\overline{27}=0.272727\ldots\) is \(\frac{3}{11}\).