`(0.8 bar(3) divide 7.5)/(2.3 bar(21) - 0.0bar(98))` is equal to

Options

A. 0.6
B. 0.1
C. 0.06
D. 0.05

Answer

To solve the expression \((0.8\overline{3} \div 7.5) / (2.3\overline{21} - 0.0\overline{98})\), we will convert the repeating decimals into fractions and then simplify the expression step by step.

### Step 1: Convert \(0.8\overline{3}\) into a fraction
Let \(x = 0.8\overline{3}\).
Then, \(10x = 8.3\overline{3}\).
Subtracting the first equation from the second:
\[ 10x - x = 8.3\overline{3} - 0.8\overline{3} \]
This simplifies to:
\[ 9x = 8.3 - 0.8 \]
\[ 9x = 7.5 \]
Thus,
\[ x = \frac{7.5}{9} = \frac{75}{90} = \frac{5}{6} \]

### Step 2: Convert \(7.5\) into a fraction
\[ 7.5 = \frac{75}{10} = \frac{15}{2} \]

### Step 3: Calculate \(0.8\overline{3} \div 7.5\)
Now we compute:
\[ \frac{5/6}{15/2} = \frac{5}{6} \times \frac{2}{15} = \frac{10}{90} = \frac{1}{9} \]

### Step 4: Convert \(2.3\overline{21}\) into a fraction
Let \(y = 2.3\overline{21}\).
Then, \(100y = 321.\overline{21}\).
Subtracting the first equation from the second:
\[ 100y - y = 321.\overline{21} - 2.3\overline{21} \]
This simplifies to:
\[ 99y = 321 - 2.3 \]
Convert \(2.3\) into a fraction:
\[ 2.3 = \frac{23}{10} \]
Thus:
\[ 99y = 321 - \frac{23}{10} = \frac{3210 - 23}{10} = \frac{3187}{10} \]
So,
\[ y = \frac{3187}{990} \]

### Step 5: Convert \(0.0\overline{98}\) into a fraction
Let \(z = 0.0\overline{98}\).
Then, \(100z = 9.8\overline{98}\).
Subtracting the first equation from the second:
\[ 100z - z = 9.8\overline{98} - 0.0\overline{98} \]
This simplifies to:
\[ 99z = 9.8 \]
Convert \(9.8\) into a fraction:
\[ 9.8 = \frac{98}{10} = \frac{49}{5} \]
Thus:
\[ 99z = \frac{49}{5} \Rightarrow z = \frac{49}{495} \]

### Step 6: Calculate \(2.3\overline{21} - 0.0\overline{98}\)
Now we compute:
\[ \frac{3187}{990} - \frac{49}{495} \]
To subtract these fractions, we need a common denominator:
\[ \frac{3187}{990} - \frac{98}{990} = \frac{3187 - 98}{990} = \frac{3089}{990} \]

### Step 7: Calculate the final expression
Now we compute:
\[ \frac{1/9}{3089/990} = \frac{1}{9} \times \frac{990}{3089} = \frac{110}{3089} \]

### Step 8: Simplify the fraction
To convert this into decimal form:
\[ \frac{110}{3089} \approx 0.0356 \]

### Conclusion
The final result of the expression \((0.8\overline{3} \div 7.5) / (2.3\overline{21} - 0.0\overline{98})\) is approximately \(0.0356\).