Question

Write each of the repeating decimals as a fraction using the following technique. To express \(0.232323 \ldots\) as a fraction, write it as a geometric series $$ 0.232323 \ldots=0.23+0.23(0.01)+0.23(0.01)^{2}+\cdots $$ with \(a=0.23\) and \(r=0.01\). Use the formula for the sum of an infinite geometric series to find $$ S=\frac{0.23}{1-0.01}=\frac{0.23}{0.99}=\frac{23}{99} $$. $$ 0.12222222 \ldots $$

Short answer

Question: Write the repeating decimal 0.12222222... as a fraction. Answer: The repeating decimal 0.12222222... can be written as a fraction: \(\frac{11}{90}\).

Step-by-step solution

Express the decimal as a geometric series

We are given the repeating decimal \(0.12222222 \ldots\). To express it as a geometric series, we break it up into the non-repeating part and the repeating part: $$0.12222222 \ldots = 0.1 + 0.02222222 \ldots$$ Next, we express the repeating part as a geometric series: $$0.022222 \ldots = 0.02 + 0.02(0.1)+0.02(0.1)^{2}+\cdots$$ Here, the first term of the series \(a=0.02\) and the common ratio \(r=0.1\).

Use the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is: $$S=\frac{a}{1-r}$$ We will use the first term \(a=0.02\) and the common ratio \(r=0.1\) found in step 1 to calculate the sum of the repeating part: $$S=\frac{0.02}{1-0.1}=\frac{0.02}{0.9}$$

Combine the non-repeating part and the sum of the repeating part

Now we combine the non-repeating part \((0.1)\) and the sum of the repeating part \((\frac{0.02}{0.9})\) to write the entire decimal as a fraction: $$0.12222222 \ldots =0.1+\frac{0.02}{0.9}=\frac{1}{10}+\frac{0.02}{0.9}$$

Simplify the fraction

In order to simplify the fraction, we need to have the same denominator for both fractions: $$\frac{1}{10} + \frac{0.02}{0.9} = \frac{1}{10} + \frac{2}{90}$$ Now we can add the fractions: $$=\frac{9+2}{90}=\frac{11}{90}$$ So the repeating decimal \(0.12222222 \ldots\) can be written as a fraction $$\frac{11}{90}.$$

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is essential in converting repeating decimals into fractions because the repeating part of the decimal can be seen as a geometric series.

In our example, to convert the repeating decimal \(0.12222222 \ldots\), the part that repeats is \(0.02222222 \ldots\). This repeating part can be expressed as a geometric series: \(0.02 + 0.02(0.1) + 0.02(0.1)^2 + \cdots\). Here:

• First term \(a = 0.02\)
• Common ratio \(r = 0.1\)

Understanding this setup allows us to evaluate the sum using a specific formula designed for infinite series.

Infinite Series Sum

When dealing with infinite geometric series, where each successive term gets smaller, the series converges to a particular sum. This is calculated using the formula: \(S = \frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio.

For the repeating decimal \(0.12222222 \ldots\), its repeating part \(0.02222222 \ldots\) forms such a series. Substituting the known values into the formula, we find:

• \(S = \frac{0.02}{1-0.1} = \frac{0.02}{0.9}\)

This calculation gives us the sum of the repeating segment of the decimal as a fraction, which is then combined with the non-repeating part to convert the entire decimal into a fraction.

Fraction Conversion

Fraction conversion involves assembling all parts of the decimal into a single fraction. After deducing the sum of the repeating series, you must add it to the non-repeating part and work towards the simplest form of a fraction by finding common denominators.

In our example, the non-repeating part of \(0.12222222 \ldots\) is \(0.1\), which can be expressed as \(\frac{1}{10}\). Adding this to the fraction we found from the sum of the series \(\frac{0.02}{0.9}\) becomes:

• \(\frac{1}{10} + \frac{0.02}{0.9} = \frac{1}{10} + \frac{2}{90}\)
• Converting both to a common denominator yields \(\frac{9}{90} + \frac{2}{90} = \frac{11}{90}\)

Thus, the repeating decimal \(0.12222222 \ldots\) is converted into the fraction \(\frac{11}{90}\). Understanding how to manipulate these fraction forms is critical in simplifying and correctly converting repeating decimals.