Solve the following : `19.3bar(68421) - 16.20bar(53)`

Answer

To solve the expression \( 19.3\overline{68421} - 16.2\overline{503} \), we will first convert the repeating decimals into fractions and then perform the subtraction.

### Step 1: Convert \( 19.3\overline{68421} \) into a fraction
Let \( x = 19.3\overline{68421} \).

To eliminate the repeating part, we can multiply \( x \) by \( 10^5 \) (since the repeating part has 5 digits):

\[ 100000x = 1930000 + 68421\overline{68421} \]

Now, we also express \( x \) in terms of its non-repeating part:

\[ 10x = 193 + 0.68421\overline{68421} \]

Next, we can express \( 0.68421\overline{68421} \) as a fraction. Let \( y = 0.68421\overline{68421} \):

\[ 100000y = 68421 + y \]
\[ 99999y = 68421 \implies y = \frac{68421}{99999} \]

Now substituting back into our equation for \( x \):

\[ x = 19.3 + \frac{68421}{99999} \]
\[ x = 19.3 + \frac{68421}{99999} = 19.3 + 0.68421 = 20.0 \]

### Step 2: Convert \( 16.2\overline{503} \) into a fraction
Let \( z = 16.2\overline{503} \).

Using a similar method, we multiply \( z \) by \( 1000 \) (since the repeating part has 3 digits):

\[ 1000z = 16200 + 503\overline{503} \]

Now, we express \( z \) in terms of its non-repeating part:

\[ 10z = 162 + 0.503\overline{503} \]

Let \( w = 0.503\overline{503} \):

\[ 1000w = 503 + w \]
\[ 999w = 503 \implies w = \frac{503}{999} \]

Substituting back into our equation for \( z \):

\[ z = 16.2 + \frac{503}{999} \]
\[ z = 16.2 + 0.503 = 16.703 \]

### Step 3: Perform the subtraction
Now we have:

\[ x - z = 20.0 - 16.703 \]

Calculating this gives:

\[ 20.0 - 16.703 = 3.297 \]

### Step 4: Convert the result back to a repeating decimal
The decimal \( 3.297 \) can be expressed as \( 3.297\overline{000} \) since there are no repeating digits after the decimal.

### Final Answer
Thus, the final answer is:

\[ 3.297\overline{000} \]