Introduction
Repeating decimals can be a challenging concept for many students to grasp. However, with the right approach and techniques, converting repeating decimals to fractions can be a straightforward process. In this article, we will explore how to write the repeating decimal $-0.1 \overline{6}$ as a fraction.
Understanding Repeating Decimals
A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $0.1 \overline{6}$ is a repeating decimal because the digit 6 repeats indefinitely. Repeating decimals can be represented as fractions using a specific technique.
Converting Repeating Decimals to Fractions
To convert a repeating decimal to a fraction, we need to follow these steps:
- Identify the repeating block: Identify the block of digits that repeats indefinitely.
- Let x equal the repeating decimal: Let x equal the repeating decimal.
- Multiply x by a power of 10: Multiply x by a power of 10 that is equal to the number of digits in the repeating block.
- Subtract the original decimal from the new decimal: Subtract the original decimal from the new decimal.
- Solve for x: Solve for x to find the fraction equivalent of the repeating decimal.
Converting $-0.1 \overline{6}$ to a Fraction
Now, let's apply these steps to convert $-0.1 \overline{6}$ to a fraction.
Step 1: Identify the Repeating Block
The repeating block in $-0.1 \overline{6}$ is 6.
Step 2: Let x Equal the Repeating Decimal
Let x equal $-0.1 \overline{6}$.
Step 3: Multiply x by a Power of 10
Multiply x by 10 to get:
10x=−1.166666...
Step 4: Subtract the Original Decimal from the New Decimal
Subtract $-0.1 \overline{6}$ from $-1.166666...$ to get:
10x−x=−1.166666...−(−0.16)
9x=−1.066666...
Step 5: Solve for x
Solve for x to get:
x=9−1.066666...
x=−90000106666...
x=−90000106666...×11
x=−90000106666...
x=−90000106666...×11
x=−90000106666...
x=−90000106666...×11
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**Converting Repeating Decimals to Fractions: A Q&A Guide**
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**Introduction**
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Converting repeating decimals to fractions can be a challenging concept for many students to grasp. However, with the right approach and techniques, it can be a straightforward process. In this article, we will explore some common questions and answers related to converting repeating decimals to fractions.
**Q: What is a repeating decimal?**
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A: A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, $0.1 \overline{6}$ is a repeating decimal because the digit 6 repeats indefinitely.
**Q: How do I convert a repeating decimal to a fraction?**
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A: To convert a repeating decimal to a fraction, you need to follow these steps:
1. **Identify the repeating block**: Identify the block of digits that repeats indefinitely.
2. **Let x equal the repeating decimal**: Let x equal the repeating decimal.
3. **Multiply x by a power of 10**: Multiply x by a power of 10 that is equal to the number of digits in the repeating block.
4. **Subtract the original decimal from the new decimal**: Subtract the original decimal from the new decimal.
5. **Solve for x**: Solve for x to find the fraction equivalent of the repeating decimal.
**Q: Can you give an example of converting a repeating decimal to a fraction?**
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A: Let's consider the repeating decimal $-0.1 \overline{6}$. To convert this to a fraction, we need to follow the steps above.
### Step 1: Identify the Repeating Block
The repeating block in $-0.1 \overline{6}$ is 6.
### Step 2: Let x Equal the Repeating Decimal
Let x equal $-0.1 \overline{6}$.
### Step 3: Multiply x by a Power of 10
Multiply x by 10 to get:
$10x = -1.166666...
Step 4: Subtract the Original Decimal from the New Decimal
Subtract $-0.1 \overline{6}$ from $-1.166666...$ to get:
10x−x=−1.166666...−(−0.16)
9x=−1.066666...
Step 5: Solve for x
Solve for x to get:
x=9−1.066666...
x=−90000106666...
Q: What if the repeating decimal has a negative sign?
A: If the repeating decimal has a negative sign, you can simply multiply the fraction by -1 to get the correct answer. For example, if you have the repeating decimal $-0.1 \overline{6}$, you can multiply the fraction by -1 to get:
x=−90000106666...
Q: Can you give an example of a repeating decimal with a negative sign?
A: Let's consider the repeating decimal $-0.2 \overline{5}$. To convert this to a fraction, we need to follow the steps above.
Step 1: Identify the Repeating Block
The repeating block in $-0.2 \overline{5}$ is 5.
Step 2: Let x Equal the Repeating Decimal
Let x equal $-0.2 \overline{5}$.
Step 3: Multiply x by a Power of 10
Multiply x by 10 to get:
10x=−2.555555...
Step 4: Subtract the Original Decimal from the New Decimal
Subtract $-0.2 \overline{5}$ from $-2.555555...$ to get:
10x−x=−2.555555...−(−0.25)
9x=−2.355555...
Step 5: Solve for x
Solve for x to get:
x=9−2.355555...
x=−90000235555...
Conclusion
Converting repeating decimals to fractions can be a challenging concept for many students to grasp. However, with the right approach and techniques, it can be a straightforward process. By following the steps outlined in this article, you can convert any repeating decimal to a fraction. Remember to identify the repeating block, let x equal the repeating decimal, multiply x by a power of 10, subtract the original decimal from the new decimal, and solve for x. With practice, you will become proficient in converting repeating decimals to fractions.