Simplify : `(8 (3)/(5) + 7 (3)/(4) + 5 (2)/(3) - 4 (1)/(2))/(13 - 11 (9)/(10) + 10 (7)/(9) - 9 (17)/(20))"of" (2)/(11) "of 365"`

Options

A. `573 (3)/(11)`
B. `571 (7)/(11)`
C. `572 (3)/(11)`
D. `575 (4)/(11)`

Answer

To simplify the expression

\[ \frac{8 \frac{3}{5} + 7 \frac{3}{4} + 5 \frac{2}{3} - 4 \frac{1}{2}}{13 - 11 \frac{9}{10} + 10 \frac{7}{9} - 9 \frac{17}{20}} \text{ of } \frac{2}{11} \text{ of } 365, \]

we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Convert \(8 \frac{3}{5}\)**:
\[ 8 \frac{3}{5} = \frac{8 \times 5 + 3}{5} = \frac{40 + 3}{5} = \frac{43}{5} \]

2. **Convert \(7 \frac{3}{4}\)**:
\[ 7 \frac{3}{4} = \frac{7 \times 4 + 3}{4} = \frac{28 + 3}{4} = \frac{31}{4} \]

3. **Convert \(5 \frac{2}{3}\)**:
\[ 5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \]

4. **Convert \(4 \frac{1}{2}\)**:
\[ 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \]

5. **Convert \(11 \frac{9}{10}\)**:
\[ 11 \frac{9}{10} = \frac{11 \times 10 + 9}{10} = \frac{110 + 9}{10} = \frac{119}{10} \]

6. **Convert \(10 \frac{7}{9}\)**:
\[ 10 \frac{7}{9} = \frac{10 \times 9 + 7}{9} = \frac{90 + 7}{9} = \frac{97}{9} \]

7. **Convert \(9 \frac{17}{20}\)**:
\[ 9 \frac{17}{20} = \frac{9 \times 20 + 17}{20} = \frac{180 + 17}{20} = \frac{197}{20} \]

### Step 2: Substitute Improper Fractions into the Expression
Now substituting these values into the expression, we have:

\[ \frac{\frac{43}{5} + \frac{31}{4} + \frac{17}{3} - \frac{9}{2}}{13 - \frac{119}{10} + \frac{97}{9} - \frac{197}{20}} \]

### Step 3: Find a Common Denominator for the Numerator
The denominators are \(5, 4, 3, 2\). The least common multiple (LCM) is \(60\).

- Convert each fraction:
- \(\frac{43}{5} = \frac{43 \times 12}{60} = \frac{516}{60}\)
- \(\frac{31}{4} = \frac{31 \times 15}{60} = \frac{465}{60}\)
- \(\frac{17}{3} = \frac{17 \times 20}{60} = \frac{340}{60}\)
- \(\frac{9}{2} = \frac{9 \times 30}{60} = \frac{270}{60}\)

Now, the numerator becomes:
\[ \frac{516 + 465 + 340 - 270}{60} = \frac{1051}{60} \]

### Step 4: Find a Common Denominator for the Denominator
The denominators are \(10, 9, 20\). The LCM is \(180\).

- Convert each fraction:
- \(13 = \frac{13 \times 180}{180} = \frac{2340}{180}\)
- \(-\frac{119}{10} = -\frac{119 \times 18}{180} = -\frac{2142}{180}\)
- \(\frac{97}{9} = \frac{97 \times 20}{180} = \frac{1940}{180}\)
- \(-\frac{197}{20} = -\frac{197 \times 9}{180} = -\frac{1773}{180}\)

Now, the denominator becomes:
\[ \frac{2340 - 2142 + 1940 - 1773}{180} = \frac{365}{180} \]

### Step 5: Combine Numerator and Denominator
Now we can simplify the entire expression:
\[ \frac{\frac{1051}{60}}{\frac{365}{180}} = \frac{1051 \times 180}{60 \times 365} = \frac{189180}{21900} \]

### Step 6: Simplify Further
Now, we can reduce this fraction:
\[ \frac{189180 \div 60}{21900 \div 60} = \frac{3153}{365} \]

### Step 7: Multiply by \(\frac{2}{11} \text{ of } 365\)
Now we multiply by \(\frac{2}{11} \times 365\):
\[ \frac{3153}{365} \times \frac{2}{11} \times 365 = \frac{3153 \times 2}{11} = \frac{6306}{11} \]

### Step 8: Convert to Mixed Fraction
Now, dividing \(6306\) by \(11\):
\[ 6306 \div 11 = 573 \quad \text{remainder } 3 \]
So, we have:
\[ 573 \frac{3}{11} \]

### Final Answer
The simplified expression is:
\[ \boxed{573 \frac{3}{11}} \]