Evaluate `3.bar(2)-0.bar(16)`

Explanation

To evaluate \( 3.\overline{2} - 0.\overline{16} \), we first need to convert the repeating decimals into fractions.

### Step 1: Convert \( 3.\overline{2} \) to a fraction
Let \( x = 3.\overline{2} \).

To eliminate the repeating part, we can multiply by 10 (since the repeating part is one digit long):
\[ 10x = 32.\overline{2} \]

Now, we subtract the original equation from this new equation:
\[ 10x - x = 32.\overline{2} - 3.\overline{2} \]
\[ 9x = 32 - 3 \]
\[ 9x = 29 \]
\[ x = \frac{29}{9} \]

Thus, \( 3.\overline{2} = \frac{29}{9} \).

### Step 2: Convert \( 0.\overline{16} \) to a fraction
Let \( y = 0.\overline{16} \).

To eliminate the repeating part, we can multiply by 100 (since the repeating part is two digits long):
\[ 100y = 16.\overline{16} \]

Now, we subtract the original equation from this new equation:
\[ 100y - y = 16.\overline{16} - 0.\overline{16} \]
\[ 99y = 16 \]
\[ y = \frac{16}{99} \]

Thus, \( 0.\overline{16} = \frac{16}{99} \).

### Step 3: Perform the subtraction
Now we can subtract the two fractions:
\[ 3.\overline{2} - 0.\overline{16} = \frac{29}{9} - \frac{16}{99} \]

To subtract these fractions, we need a common denominator. The least common multiple of 9 and 99 is 99.

Convert \( \frac{29}{9} \) to have a denominator of 99:
\[ \frac{29}{9} = \frac{29 \times 11}{9 \times 11} = \frac{319}{99} \]

Now we can subtract:
\[ \frac{319}{99} - \frac{16}{99} = \frac{319 - 16}{99} = \frac{303}{99} \]

### Step 4: Simplify the fraction
Now we simplify \( \frac{303}{99} \):
\[ \frac{303 \div 3}{99 \div 3} = \frac{101}{33} \]

### Final Answer
Thus, the final answer is:
\[ 3.\overline{2} - 0.\overline{16} = \frac{101}{33} \]