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

Answer

To evaluate \(3.\overline{2} - 0.\overline{16}\), we can follow these steps:

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

1. **For \(3.\overline{2}\)**:
Let \(x = 3.\overline{2}\).
Then, \(x = 3.2222...\).

To eliminate the repeating part, multiply both sides by 10:
\[ 10x = 32.2222... \]
Now, subtract the first equation from this new equation:
\[ 10x - x = 32.2222... - 3.2222... \]
This simplifies to:
\[ 9x = 29 \]
Therefore,
\[ x = \frac{29}{9} \]

2. **For \(0.\overline{16}\)**:
Let \(y = 0.\overline{16}\).
Then, \(y = 0.161616...\).

Multiply both sides by 100:
\[ 100y = 16.161616... \]
Now, subtract the first equation from this new equation:
\[ 100y - y = 16.161616... - 0.161616... \]
This simplifies to:
\[ 99y = 16 \]
Therefore,
\[ y = \frac{16}{99} \]

### Step 2: Perform the subtraction.

Now we need to 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 perform the subtraction:
\[ \frac{319}{99} - \frac{16}{99} = \frac{319 - 16}{99} = \frac{303}{99} \]

### Step 3: Simplify the fraction.

Now we simplify \(\frac{303}{99}\):
\[ 303 \div 3 = 101 \quad \text{and} \quad 99 \div 3 = 33 \]
So,
\[ \frac{303}{99} = \frac{101}{33} \]

### Step 4: Convert back to decimal form.

To convert \(\frac{101}{33}\) back to decimal form, we can perform the division:
\[ 101 \div 33 = 3.060606... \]
This is equivalent to \(3.\overline{06}\).

### Final Answer:
Thus, the final answer is:
\[ 3.\overline{06} \]

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