Simplify : `8 (1)/(3)-4(5)/(9)+1/9-5(1)/(3)+2(1)/(3)-6(1)/(9)+1`

Options

A. `-4(2)/(9)`
B. `-2(2)/(9)`
C. `-2(1)/(3)`
D. `-4(2)/(3)`

Answer

To simplify the expression \( 8 \frac{1}{3} - 4 \frac{5}{9} + \frac{1}{9} - 5 \frac{1}{3} + 2 \frac{1}{3} - 6 \frac{1}{9} + 1 \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
- \( 8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{24 + 1}{3} = \frac{25}{3} \)
- \( 4 \frac{5}{9} = \frac{4 \times 9 + 5}{9} = \frac{36 + 5}{9} = \frac{41}{9} \)
- \( 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \)
- \( 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \)
- \( 6 \frac{1}{9} = \frac{6 \times 9 + 1}{9} = \frac{54 + 1}{9} = \frac{55}{9} \)
- \( 1 = \frac{1}{1} = \frac{9}{9} \)

Now the expression becomes:
\[ \frac{25}{3} - \frac{41}{9} + \frac{1}{9} - \frac{16}{3} + \frac{7}{3} - \frac{55}{9} + \frac{9}{9} \]

### Step 2: Find a Common Denominator
The common denominator for the fractions with denominators 3 and 9 is 9. We will convert all fractions to have a denominator of 9:
- \( \frac{25}{3} = \frac{25 \times 3}{3 \times 3} = \frac{75}{9} \)
- \( \frac{41}{9} \) remains \( \frac{41}{9} \)
- \( \frac{1}{9} \) remains \( \frac{1}{9} \)
- \( \frac{16}{3} = \frac{16 \times 3}{3 \times 3} = \frac{48}{9} \)
- \( \frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9} \)
- \( \frac{55}{9} \) remains \( \frac{55}{9} \)
- \( \frac{9}{9} \) remains \( \frac{9}{9} \)

Now the expression is:
\[ \frac{75}{9} - \frac{41}{9} + \frac{1}{9} - \frac{48}{9} + \frac{21}{9} - \frac{55}{9} + \frac{9}{9} \]

### Step 3: Combine the Fractions
Combine all the fractions over the common denominator:
\[ \frac{75 - 41 + 1 - 48 + 21 - 55 + 9}{9} \]
Calculating the numerator:
\[ 75 - 41 = 34 \]
\[ 34 + 1 = 35 \]
\[ 35 - 48 = -13 \]
\[ -13 + 21 = 8 \]
\[ 8 - 55 = -47 \]
\[ -47 + 9 = -38 \]

So, we have:
\[ \frac{-38}{9} \]

### Step 4: Convert to Mixed Number
To convert \( \frac{-38}{9} \) to a mixed number:
- Divide \( 38 \) by \( 9 \): \( 38 \div 9 = 4 \) remainder \( 2 \).
- So, \( \frac{-38}{9} = -4 \frac{2}{9} \).

### Final Answer
The simplified expression is:
\[ -4 \frac{2}{9} \]