Express `0.bar(7)` in the simplest form.

Answer

To express \( 0.\overline{7} \) in its simplest form, we can follow these steps:

### Step 1: Let \( x = 0.\overline{7} \)
We start by defining \( x \) as the repeating decimal we want to convert.

### Step 2: Write the repeating decimal
This means \( x = 0.7777\ldots \) (where the 7 repeats infinitely).

### Step 3: Multiply both sides by 10
To eliminate the repeating part, we multiply both sides by 10:
\[ 10x = 7.7777\ldots \]

### Step 4: Set up the equations
Now we have two equations:
1. \( x = 0.7777\ldots \) (Equation 1)
2. \( 10x = 7.7777\ldots \) (Equation 2)

### Step 5: Subtract Equation 1 from Equation 2
Now, we subtract Equation 1 from Equation 2:
\[ 10x - x = 7.7777\ldots - 0.7777\ldots \]
This simplifies to:
\[ 9x = 7 \]

### Step 6: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{7}{9} \]

### Conclusion
Thus, the simplest form of \( 0.\overline{7} \) is:
\[ \frac{7}{9} \]

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