Question

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{5}=0.555 \ldots$$

Short answer

Question: Express the repeating decimal 0.555... as a fraction. Answer: The repeating decimal 0.555... can be expressed as the fraction 5/9.

Step-by-step solution

Identify the Repeating Decimal Pattern

In the given repeating decimal, we observe that the digit \(5\) repeats itself indefinitely: $$0 . \overline{5}=0.555 \ldots$$

Express the Repeating Decimal as a Geometric Series

To express the repeating decimal as a geometric series, we can start by representing the decimal with repeating digits as an infinite sum. Each term in this sum will be the digit multiplied by the common ratio, which is the place value of the digit (in this case, tenths, hundredths, and so on): $$0 . \overline{5}= 0.5 + 0.05 + 0.005 + \ldots$$ This can be written as a geometric series with the first term \(a\) as \(0.5\) and the common ratio \(r\) as \(0.1\): $$0 . \overline{5}= \sum_{n=0}^{\infty} a \times r^n = 0.5 \times (1 + 0.1 + 0.01 + \ldots)$$

Find the Sum of the Geometric Series

To find the sum of the geometric series, we will use the formula: $$S = \frac{a}{1-r}$$ In our case, \(a = 0.5\) and \(r = 0.1\). Plugging in these values: $$S = \frac{0.5}{1 - 0.1}$$

Simplify the Fraction

Now, we can simplify the fraction: $$S = \frac{0.5}{0.9}$$ Since we want to express the result as a ratio of two integers, we can multiply both the numerator and the denominator by \(10\), which will eliminate the decimals: $$S = \frac{0.5 \times 10}{0.9 \times 10} = \frac{5}{9}$$ So, the repeating decimal \(0 . \overline{5}\) can be expressed as a fraction, which is \(\frac{5}{9}\).