Convert each of the following into a vulgar fraction :
`0.1bar6`

Answer

To convert the repeating decimal \(0.1\bar{6}\) into a vulgar fraction, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.1\bar{6} \). This means \( x = 0.166666...\) where the digit '6' repeats indefinitely.

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \( x \) by 10 to shift the decimal point:
\[ 10x = 1.66666... \]
This can also be written as:
\[ 10x = 1.6\bar{6} \]

### Step 3: Create a second equation
Now, multiply the original \( x \) by 100 to shift the decimal point two places:
\[ 100x = 16.6666... \]
This can also be written as:
\[ 100x = 16.6\bar{6} \]

### Step 4: Set up the equations
Now we have two equations:
1. \( 10x = 1.6\bar{6} \) (Equation 1)
2. \( 100x = 16.6\bar{6} \) (Equation 2)

### Step 5: Subtract the first equation from the second
Subtract Equation 1 from Equation 2 to eliminate the repeating part:
\[ 100x - 10x = 16.6\bar{6} - 1.6\bar{6} \]
This simplifies to:
\[ 90x = 15 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 90:
\[ x = \frac{15}{90} \]

### Step 7: Simplify the fraction
Now simplify \( \frac{15}{90} \):
\[ x = \frac{1}{6} \]

### Conclusion
Thus, the repeating decimal \( 0.1\bar{6} \) can be expressed as the vulgar fraction \( \frac{1}{6} \).

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