Question
(b) Convert the each of the following recurring decimals into a common fraction, by first writing it as an infinite series: \( \begin{array}{lll}\text { (1) } 0 . \dot{4} & \text { (2) } 0, \dot{7} & \text { (3) } 0,25 \dot{5} \\ \text { (4) } 0, \dot{4} \dot{2} & \text { (5) } & -2, \dot{6}\end{array} \)
Ask by Savage Campbell. in South Africa
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\begin{array}{lll}
1. & 0.\overline{4} = \frac{4}{9} & 2. & 0.\overline{7} = \frac{7}{9} \\
3. & 0.25\overline{5} = \frac{23}{90} & 4. & 0.\overline{42} = \frac{14}{33} \\
5. & -2.\overline{6} = -\frac{8}{3}
\end{array}
\]
Solution
To convert recurring decimals into common fractions, we can express each decimal as an infinite geometric series and then find the sum of that series. Below are the step-by-step conversions for each given recurring decimal:
### **(1) \(0.\overline{4}\)**
**Step 1: Express as an Infinite Series**
\[
0.\overline{4} = 0.4444\ldots = \frac{4}{10} + \frac{4}{100} + \frac{4}{1000} + \frac{4}{10000} + \cdots
\]
**Step 2: Identify the Geometric Series Parameters**
- **First term (\(a\))**: \(\frac{4}{10}\)
- **Common ratio (\(r\))**: \(\frac{1}{10}\)
**Step 3: Calculate the Sum of the Infinite Series**
\[
\text{Sum} = \frac{a}{1 - r} = \frac{\frac{4}{10}}{1 - \frac{1}{10}} = \frac{\frac{4}{10}}{\frac{9}{10}} = \frac{4}{9}
\]
**Final Answer:**
\[
0.\overline{4} = \frac{4}{9}
\]
---
### **(2) \(0.\overline{7}\)**
**Step 1: Express as an Infinite Series**
\[
0.\overline{7} = 0.7777\ldots = \frac{7}{10} + \frac{7}{100} + \frac{7}{1000} + \frac{7}{10000} + \cdots
\]
**Step 2: Identify the Geometric Series Parameters**
- **First term (\(a\))**: \(\frac{7}{10}\)
- **Common ratio (\(r\))**: \(\frac{1}{10}\)
**Step 3: Calculate the Sum of the Infinite Series**
\[
\text{Sum} = \frac{a}{1 - r} = \frac{\frac{7}{10}}{1 - \frac{1}{10}} = \frac{\frac{7}{10}}{\frac{9}{10}} = \frac{7}{9}
\]
**Final Answer:**
\[
0.\overline{7} = \frac{7}{9}
\]
---
### **(3) \(0.25\overline{5}\)**
**Step 1: Express as an Infinite Series**
\[
0.25\overline{5} = 0.25555\ldots = 0.25 + 0.005 + 0.0005 + 0.00005 + \cdots
\]
**Step 2: Separate the Non-Repeating and Repeating Parts**
- **Non-repeating part**: \(0.25\)
- **Repeating part**: \(0.005 + 0.0005 + 0.00005 + \cdots\)
**Step 3: Identify the Geometric Series Parameters for the Repeating Part**
- **First term (\(a\))**: \(\frac{5}{1000}\) (which is \(0.005\))
- **Common ratio (\(r\))**: \(\frac{1}{10}\)
**Step 4: Calculate the Sum of the Repeating Infinite Series**
\[
\text{Sum of repeating part} = \frac{a}{1 - r} = \frac{\frac{5}{1000}}{1 - \frac{1}{10}} = \frac{\frac{5}{1000}}{\frac{9}{10}} = \frac{5}{900} = \frac{1}{180}
\]
**Step 5: Add the Non-Repeating Part**
\[
\text{Total Sum} = 0.25 + \frac{1}{180} = \frac{25}{100} + \frac{1}{180} = \frac{45}{180} + \frac{1}{180} = \frac{46}{180} = \frac{23}{90}
\]
**Final Answer:**
\[
0.25\overline{5} = \frac{23}{90}
\]
---
### **(4) \(0.\overline{42}\)**
**Step 1: Express as an Infinite Series**
\[
0.\overline{42} = 0.424242\ldots = \frac{42}{100} + \frac{42}{10000} + \frac{42}{1000000} + \cdots
\]
**Step 2: Identify the Geometric Series Parameters**
- **First term (\(a\))**: \(\frac{42}{100}\)
- **Common ratio (\(r\))**: \(\frac{1}{100}\)
**Step 3: Calculate the Sum of the Infinite Series**
\[
\text{Sum} = \frac{a}{1 - r} = \frac{\frac{42}{100}}{1 - \frac{1}{100}} = \frac{\frac{42}{100}}{\frac{99}{100}} = \frac{42}{99} = \frac{14}{33}
\]
**Final Answer:**
\[
0.\overline{42} = \frac{14}{33}
\]
---
### **(5) \(-2.\overline{6}\)**
**Step 1: Express the Decimal Part as an Infinite Series**
\[
-2.\overline{6} = -2.6666\ldots = -2 - 0.6666\ldots
\]
\[
0.\overline{6} = \frac{6}{10} + \frac{6}{100} + \frac{6}{1000} + \frac{6}{10000} + \cdots
\]
**Step 2: Identify the Geometric Series Parameters for the Decimal Part**
- **First term (\(a\))**: \(\frac{6}{10}\)
- **Common ratio (\(r\))**: \(\frac{1}{10}\)
**Step 3: Calculate the Sum of the Decimal Infinite Series**
\[
\text{Sum of decimal part} = \frac{a}{1 - r} = \frac{\frac{6}{10}}{1 - \frac{1}{10}} = \frac{\frac{6}{10}}{\frac{9}{10}} = \frac{6}{9} = \frac{2}{3}
\]
**Step 4: Combine with the Integer Part**
\[
-2.\overline{6} = -2 - \frac{2}{3} = -\frac{6}{3} - \frac{2}{3} = -\frac{8}{3}
\]
**Final Answer:**
\[
-2.\overline{6} = -\frac{8}{3}
\]
---
### **Summary of Conversions**
1. \(0.\overline{4} = \frac{4}{9}\)
2. \(0.\overline{7} = \frac{7}{9}\)
3. \(0.25\overline{5} = \frac{23}{90}\)
4. \(0.\overline{42} = \frac{14}{33}\)
5. \(-2.\overline{6} = -\frac{8}{3}\)
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Simplify this solution Bonus Knowledge
To convert the recurring decimals into fractions using the infinite series method, we can express each recurring decimal as an infinite geometric series. First Line: For \(0.\dot{4}\), we express it as \(0.4444...\). This can be rewritten as \(4/10 + 4/100 + 4/1000 + \ldots\). This series has the first term \(a = 4/10\) and a common ratio \(r = 1/10\). The sum of this infinite series is given by \(S = \frac{a}{1 - r} = \frac{4/10}{1 - 1/10} = \frac{4/10}{9/10} = \frac{4}{9}\). Second Line: For \(0,\dot{7}\), visualize it as \(0.7777...\) which equals \(7/10 + 7/100 + 7/1000 + ...\). Here, we have \(a = 7/10\) and \(r = 1/10\). Thus, the sum becomes \(S = \frac{7/10}{1 - 1/10} = \frac{7/10}{9/10} = \frac{7}{9}\). For the other decimals, you can use the same method to find their fractions!
