Subtract the sum of `3(5)/(9) and 3(1)/(3)` from the sum of `5(5)/(6) and 4(1)/(9)`

Answer

To solve the problem of subtracting the sum of \(3\frac{5}{9}\) and \(3\frac{1}{3}\) from the sum of \(5\frac{5}{6}\) and \(4\frac{1}{9}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(3\frac{5}{9}\) to an improper fraction:
\[ 3\frac{5}{9} = \frac{3 \times 9 + 5}{9} = \frac{27 + 5}{9} = \frac{32}{9} \]

2. Convert \(3\frac{1}{3}\) to an improper fraction:
\[ 3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]

3. Convert \(5\frac{5}{6}\) to an improper fraction:
\[ 5\frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6} \]

4. Convert \(4\frac{1}{9}\) to an improper fraction:
\[ 4\frac{1}{9} = \frac{4 \times 9 + 1}{9} = \frac{36 + 1}{9} = \frac{37}{9} \]

### Step 2: Calculate the Sums
1. Calculate the sum of \(3\frac{5}{9}\) and \(3\frac{1}{3}\):
\[ \frac{32}{9} + \frac{10}{3} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9. Convert \(\frac{10}{3}\) to have a denominator of 9:
\[ \frac{10}{3} = \frac{10 \times 3}{3 \times 3} = \frac{30}{9} \]
Now, add:
\[ \frac{32}{9} + \frac{30}{9} = \frac{32 + 30}{9} = \frac{62}{9} \]

2. Calculate the sum of \(5\frac{5}{6}\) and \(4\frac{1}{9}\):
\[ \frac{35}{6} + \frac{37}{9} \]
The LCM of 6 and 9 is 18. Convert both fractions:
\[ \frac{35}{6} = \frac{35 \times 3}{6 \times 3} = \frac{105}{18} \]
\[ \frac{37}{9} = \frac{37 \times 2}{9 \times 2} = \frac{74}{18} \]
Now, add:
\[ \frac{105}{18} + \frac{74}{18} = \frac{105 + 74}{18} = \frac{179}{18} \]

### Step 3: Subtract the Sums
Now we need to subtract the sum of \(3\frac{5}{9}\) and \(3\frac{1}{3}\) from the sum of \(5\frac{5}{6}\) and \(4\frac{1}{9}\):
\[ \frac{179}{18} - \frac{62}{9} \]
Convert \(\frac{62}{9}\) to have a denominator of 18:
\[ \frac{62}{9} = \frac{62 \times 2}{9 \times 2} = \frac{124}{18} \]
Now, perform the subtraction:
\[ \frac{179}{18} - \frac{124}{18} = \frac{179 - 124}{18} = \frac{55}{18} \]

### Final Answer
The final result is:
\[ \frac{55}{18} \text{ or } 3\frac{1}{18} \]