The simplest form of `1.bar6` is

Options

A. `(833)/(500)`
B. `(8)/(5)`
C. `(5)/(3)`
D. None of these

Answer

To find the simplest form of \(1.\overline{6}\), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 1.\overline{6} \). This means that \( x = 1.66666...\) where the digit 6 repeats indefinitely.

### Step 2: Multiply by 10
To eliminate the repeating part, multiply both sides of the equation by 10:
\[ 10x = 16.66666... \]
This can be rewritten as:
\[ 10x = 16.\overline{6} \]

### Step 3: Set up the equation
Now we have two equations:
1. \( x = 1.\overline{6} \) (Equation 1)
2. \( 10x = 16.\overline{6} \) (Equation 2)

### Step 4: Subtract the first equation from the second
Now, subtract Equation 1 from Equation 2:
\[ 10x - x = 16.\overline{6} - 1.\overline{6} \]
This simplifies to:
\[ 9x = 15 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 9:
\[ x = \frac{15}{9} \]

### Step 6: Simplify the fraction
Now, simplify \(\frac{15}{9}\):
\[ x = \frac{15 \div 3}{9 \div 3} = \frac{5}{3} \]

### Conclusion
Thus, the simplest form of \(1.\overline{6}\) is \(\frac{5}{3}\).

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