The value of `((7)/(8) div 5(1)/(4) xx 7(1)/(5) - (3)/(20) " of " (2)/(3)) div (1)/(2) + ((3)/(5) xx 7(1)/(2) + (2)/(3) div (8)/(15))` is equal to `7+k`, where k =

Options

A. `(3)/(5)`
B. `(1)/(5)`
C. `(1)/(20)`
D. `(19)/(20)`

Answer

To solve the expression `((7)/(8) div 5(1)/(4) xx 7(1)/(5) - (3)/(20) " of " (2)/(3)) div (1)/(2) + ((3)/(5) xx 7(1)/(2) + (2)/(3) div (8)/(15))`, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert the mixed numbers into improper fractions:
- \(5(1/4) = \frac{21}{4}\)
- \(7(1/5) = \frac{36}{5}\)
- \(7(1/2) = \frac{15}{2}\)

So the expression becomes:
\[ \left(\frac{7}{8} \div \frac{21}{4} \times \frac{36}{5} - \frac{3}{20} \text{ of } \frac{2}{3}\right) \div \frac{1}{2} + \left(\frac{3}{5} \times \frac{15}{2} + \frac{2}{3} \div \frac{8}{15}\right) \]

### Step 2: Solve the First Part
Calculate the first part:
1. **Division**: \( \frac{7}{8} \div \frac{21}{4} = \frac{7}{8} \times \frac{4}{21} = \frac{28}{168} = \frac{1}{6} \)
2. **Multiplication**: \( \frac{1}{6} \times \frac{36}{5} = \frac{36}{30} = \frac{6}{5} \)
3. **Of**: \( \frac{3}{20} \text{ of } \frac{2}{3} = \frac{3}{20} \times \frac{2}{3} = \frac{1}{10} \)
4. **Subtraction**: \( \frac{6}{5} - \frac{1}{10} = \frac{12}{10} - \frac{1}{10} = \frac{11}{10} \)

Now we have:
\[ \left(\frac{11}{10}\right) \div \frac{1}{2} = \frac{11}{10} \times 2 = \frac{22}{10} = \frac{11}{5} \]

### Step 3: Solve the Second Part
Calculate the second part:
1. **Multiplication**: \( \frac{3}{5} \times \frac{15}{2} = \frac{45}{10} = \frac{9}{2} \)
2. **Division**: \( \frac{2}{3} \div \frac{8}{15} = \frac{2}{3} \times \frac{15}{8} = \frac{30}{24} = \frac{5}{4} \)
3. **Addition**: \( \frac{9}{2} + \frac{5}{4} = \frac{18}{4} + \frac{5}{4} = \frac{23}{4} \)

### Step 4: Combine Both Parts
Now, combine both parts:
\[ \frac{11}{5} + \frac{23}{4} \]

To add these fractions, find a common denominator (which is 20):
1. Convert \( \frac{11}{5} = \frac{44}{20} \)
2. Convert \( \frac{23}{4} = \frac{115}{20} \)

Now add:
\[ \frac{44}{20} + \frac{115}{20} = \frac{159}{20} \]

### Step 5: Set Equal to \( 7 + k \)
We have:
\[ \frac{159}{20} = 7 + k \]
Convert 7 to a fraction with a denominator of 20:
\[ 7 = \frac{140}{20} \]

Now, set the equation:
\[ \frac{159}{20} = \frac{140}{20} + k \]

### Step 6: Solve for \( k \)
Subtract \( \frac{140}{20} \) from both sides:
\[ k = \frac{159}{20} - \frac{140}{20} = \frac{19}{20} \]

Thus, the value of \( k \) is:
\[ \boxed{\frac{19}{20}} \]