The value of `0.bar(57)-0.4bar(32)+0.3bar5` is:

Options

A. `0.4bar(98)`
B. `0.bar(498)`
C. `0.4sqrt(94)`
D. `0.bar(494)`

Answer

To solve the problem \(0.\overline{57} - 0.\overline{32} + 0.\overline{35}\), we will first convert each repeating decimal into a fraction.

### Step 1: Convert \(0.\overline{57}\) to a fraction

Let \(x = 0.\overline{57}\).

Then, multiplying both sides by 100 (to shift the decimal point two places to the right):
\[ 100x = 57.\overline{57} \]

Now, subtract the original equation from this new equation:
\[ 100x - x = 57.\overline{57} - 0.\overline{57} \]
\[ 99x = 57 \]
\[ x = \frac{57}{99} \]

Now simplify \(\frac{57}{99}\):
\[ x = \frac{19}{33} \]

### Step 2: Convert \(0.\overline{32}\) to a fraction

Let \(y = 0.\overline{32}\).

Then, multiplying both sides by 100:
\[ 100y = 32.\overline{32} \]

Subtracting the original equation:
\[ 100y - y = 32.\overline{32} - 0.\overline{32} \]
\[ 99y = 32 \]
\[ y = \frac{32}{99} \]

### Step 3: Convert \(0.\overline{35}\) to a fraction

Let \(z = 0.\overline{35}\).

Then, multiplying both sides by 100:
\[ 100z = 35.\overline{35} \]

Subtracting the original equation:
\[ 100z - z = 35.\overline{35} - 0.\overline{35} \]
\[ 99z = 35 \]
\[ z = \frac{35}{99} \]

### Step 4: Substitute back into the original expression

Now we substitute these fractions back into the expression:
\[ 0.\overline{57} - 0.\overline{32} + 0.\overline{35} = \frac{19}{33} - \frac{32}{99} + \frac{35}{99} \]

### Step 5: Find a common denominator

The common denominator for \(33\) and \(99\) is \(99\). We need to convert \(\frac{19}{33}\) to have the same denominator:
\[ \frac{19}{33} = \frac{19 \times 3}{33 \times 3} = \frac{57}{99} \]

### Step 6: Combine the fractions

Now we can combine:
\[ \frac{57}{99} - \frac{32}{99} + \frac{35}{99} = \frac{57 - 32 + 35}{99} = \frac{60}{99} \]

### Step 7: Simplify the fraction

Now simplify \(\frac{60}{99}\):
\[ \frac{60}{99} = \frac{20}{33} \]

### Final Answer

Thus, the value of \(0.\overline{57} - 0.\overline{32} + 0.\overline{35}\) is:
\[ \frac{20}{33} \]

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