Which of the following numbers is largest among all ?
`0.7, 0.bar7, 0.0bar7 0.bar07`

Options

A. `0.bar07`
B. `0.0bar7`
C. `0.7`
D. `0.bar7`

Answer

To determine which of the following numbers is the largest among `0.7`, `0.7̅`, `0.0̅7`, and `0.0̅07`, we will analyze each number step by step.

### Step 1: Understand the Numbers
1. **0.7**: This is a decimal number that is equal to 7 tenths.
2. **0.7̅**: This means that the digit 7 repeats indefinitely. Therefore, it can be expressed as:
\[ 0.7̅ = 0.77777... = \frac{7}{9} \text{ (using the formula for repeating decimals)} \]
3. **0.0̅7**: This means that the digit 7 repeats indefinitely after a leading zero. Therefore, it can be expressed as:
\[ 0.0̅7 = 0.077777... = 0.07 + \frac{7}{90} = 0.07 + 0.0777... = 0.07 + \frac{7}{90} \]
To simplify, we can convert \(0.07\) to a fraction:
\[ 0.07 = \frac{7}{100} \]
So,
\[ 0.0̅7 = \frac{7}{100} + \frac{7}{90} \]
To add these fractions, we need a common denominator (which is 900):
\[ 0.0̅7 = \frac{63}{900} + \frac{70}{900} = \frac{133}{900} \]
4. **0.0̅07**: This means that the digit 07 repeats indefinitely after a leading zero. Therefore, it can be expressed as:
\[ 0.0̅07 = 0.007777... = 0.007 + \frac{7}{900} = \frac{7}{1000} + \frac{7}{900} \]
Again, we convert \(0.007\) to a fraction:
\[ 0.007 = \frac{7}{1000} \]
So,
\[ 0.0̅07 = \frac{7}{1000} + \frac{7}{900} \]
The common denominator here is 9000:
\[ 0.0̅07 = \frac{63}{9000} + \frac{70}{9000} = \frac{133}{9000} \]

### Step 2: Compare the Values
Now we have:
- \(0.7 = \frac{7}{10} = \frac{6300}{9000}\)
- \(0.7̅ = \frac{7}{9} = \frac{7000}{9000}\)
- \(0.0̅7 = \frac{133}{900}\)
- \(0.0̅07 = \frac{133}{9000}\)

### Step 3: Determine the Largest
To compare:
- \(0.7 = \frac{6300}{9000}\)
- \(0.7̅ = \frac{7000}{9000}\)
- \(0.0̅7 = \frac{1330}{9000}\)
- \(0.0̅07 = \frac{133}{9000}\)

Clearly, \(0.7̅\) (which is \(\frac{7000}{9000}\)) is the largest, followed by \(0.7\) (which is \(\frac{6300}{9000}\)), then \(0.0̅7\) and \(0.0̅07\).

### Conclusion
The largest number among the given options is **0.7̅**.