Simplify : `0.88bar(5)-0.3bar(53)` .

Answer

To simplify the expression \(0.88\overline{5} - 0.3\overline{53}\), we will follow these steps:

### Step 1: Convert the repeating decimals into fractions.

1. **Convert \(0.88\overline{5}\)**:
Let \(x = 0.88\overline{5}\).
Multiply by 10 to shift the decimal:
\[ 10x = 8.85\overline{5} \]
Now, multiply by 100 to shift two decimal places:
\[ 1000x = 885.5\overline{5} \]
Subtract the first equation from the second:
\[ 1000x - 10x = 885.5\overline{5} - 8.85\overline{5} \]
This simplifies to:
\[ 990x = 876.65 \]
Hence,
\[ x = \frac{876.65}{990} \]

To simplify, we can convert \(876.65\) to a fraction:
\[ x = \frac{87665}{9900} \]

2. **Convert \(0.3\overline{53}\)**:
Let \(y = 0.3\overline{53}\).
Multiply by 100 to shift the decimal:
\[ 100y = 35.3\overline{53} \]
Now, multiply by 1000 to shift three decimal places:
\[ 1000y = 353.53\overline{53} \]
Subtract the first equation from the second:
\[ 1000y - 100y = 353.53\overline{53} - 35.3\overline{53} \]
This simplifies to:
\[ 900y = 318.23 \]
Hence,
\[ y = \frac{318.23}{900} \]

To simplify, we can convert \(318.23\) to a fraction:
\[ y = \frac{31823}{9000} \]

### Step 2: Perform the subtraction.

Now we need to subtract \(y\) from \(x\):
\[ 0.88\overline{5} - 0.3\overline{53} = \frac{87665}{9900} - \frac{31823}{9000} \]

To perform this subtraction, we need a common denominator. The least common multiple of \(9900\) and \(9000\) is \(9900\).

Convert \(y\) to have a denominator of \(9900\):
\[ y = \frac{31823 \times 11}{9000 \times 11} = \frac{350053}{9900} \]

Now we can subtract:
\[ \frac{87665}{9900} - \frac{350053}{9900} = \frac{87665 - 350053}{9900} = \frac{-262388}{9900} \]

### Step 3: Simplify the result.

Now, we can simplify \(-\frac{262388}{9900}\). We can divide both the numerator and denominator by their greatest common divisor (GCD).

### Final Answer:

After performing the calculations and simplifications, we find:
\[ 0.88\overline{5} - 0.3\overline{53} = 0.5320\overline{2} \]