Correct expression of `1.4bar(27)`. (the bar indicates repeating decimal)

Options

A. `(147)/(1000)`
B. `(157)/(110)`
C. `(1427)/(10000)`
D. `(157)/(111)`

Answer

To find the correct expression of \(1.4\overline{27}\) (where the bar indicates that 27 is the repeating part), we can follow these steps:

### Step 1: Define the variable
Let \( x = 1.427272727...\)

### Step 2: Multiply by 100
To eliminate the repeating decimal, we multiply both sides of the equation by 100:
\[ 100x = 142.727272727... \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 1.427272727...\)
2. \( 100x = 142.727272727...\)

### Step 4: Subtract the first equation from the second
Now, we subtract the first equation from the second:
\[ 100x - x = 142.727272727... - 1.427272727... \]
This simplifies to:
\[ 99x = 141.3 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{141.3}{99} \]

### Step 6: Convert to a fraction
To convert \(141.3\) to a fraction, we can express it as:
\[ 141.3 = \frac{1413}{10} \]
Thus,
\[ x = \frac{1413}{10 \times 99} = \frac{1413}{990} \]

### Step 7: Simplify the fraction
Now we simplify \(\frac{1413}{990}\):
- The greatest common divisor (GCD) of 1413 and 990 is 3.
- Dividing both the numerator and the denominator by 3 gives:
\[ x = \frac{471}{330} \]

### Step 8: Final simplification
We can check if \(471\) and \(330\) can be simplified further. The GCD of 471 and 330 is 3 again:
\[ x = \frac{471 \div 3}{330 \div 3} = \frac{157}{110} \]

Thus, the correct expression of \(1.4\overline{27}\) is:
\[ \frac{157}{110} \]