Solve the following and select the correct option.
(i) `-(-5(2)/(3))"of"6(3)/(8)-0.696xx10.00+3(5)/(4)+(-6.663)`
(ii) `4(3)/(5)-(-4(1)/(6))+(-3(7)/(6))+((-6))/(7)-(-1(4)/(5))+4(5)/(7)`

Options

A. (i) 26.752
(ii) `10(9)/(35)`
B. (i)26.767
(ii) `12(3)/(45)`
C. (i) 28.837
(ii) `10(9)/(35)`
D. (i)26.752
(ii) `6(19)/(35)`

Answer

To solve the given question, we will break it down into two parts as specified.

### Part (i)

The expression to solve is:
\[ -(-5\frac{2}{3}) \text{ of } 6\frac{3}{8} - 0.696 \times 10.00 + 3\frac{5}{4} + (-6.663) \]

**Step 1: Convert mixed numbers to improper fractions.**
- For \(-5\frac{2}{3}\):
\[ -5\frac{2}{3} = -\left(5 \times 3 + 2\right) / 3 = -\frac{17}{3} \]

- For \(6\frac{3}{8}\):
\[ 6\frac{3}{8} = \left(6 \times 8 + 3\right) / 8 = \frac{51}{8} \]

- For \(3\frac{5}{4}\):
\[ 3\frac{5}{4} = \left(3 \times 4 + 5\right) / 4 = \frac{17}{4} \]

**Step 2: Substitute back into the expression.**
\[ -(-\frac{17}{3}) \text{ of } \frac{51}{8} - 0.696 \times 10 + \frac{17}{4} - 6.663 \]

**Step 3: Calculate the multiplication (of means multiply).**
\[ \frac{17}{3} \times \frac{51}{8} = \frac{867}{24} \]

**Step 4: Calculate \(0.696 \times 10\).**
\[ 0.696 \times 10 = 6.96 \]

**Step 5: Substitute and simplify the expression.**
\[ \frac{867}{24} - 6.96 + \frac{17}{4} - 6.663 \]

**Step 6: Convert \(\frac{17}{4}\) to a common denominator with \(\frac{867}{24}\).**
\[ \frac{17}{4} = \frac{102}{24} \]
Now the expression becomes:
\[ \frac{867}{24} + \frac{102}{24} - 6.96 - 6.663 \]

**Step 7: Combine the fractions.**
\[ \frac{867 + 102}{24} = \frac{969}{24} \]

**Step 8: Calculate the decimal value of \(\frac{969}{24}\).**
\[ \frac{969}{24} = 40.375 \]

**Step 9: Combine the decimal values.**
\[ 40.375 - 6.96 - 6.663 = 40.375 - 13.623 = 26.752 \]

### Part (ii)

The expression to solve is:
\[ 4\frac{3}{5} - (-4\frac{1}{6}) + (-3\frac{7}{6}) + \left(-\frac{6}{7}\right) - (-1\frac{4}{5}) + 4\frac{5}{7} \]

**Step 1: Convert mixed numbers to improper fractions.**
- For \(4\frac{3}{5}\):
\[ 4\frac{3}{5} = \frac{23}{5} \]

- For \(-4\frac{1}{6}\):
\[ -4\frac{1}{6} = -\frac{25}{6} \]

- For \(-3\frac{7}{6}\):
\[ -3\frac{7}{6} = -\frac{25}{6} \]

- For \(-1\frac{4}{5}\):
\[ -1\frac{4}{5} = -\frac{9}{5} \]

- For \(4\frac{5}{7}\):
\[ 4\frac{5}{7} = \frac{33}{7} \]

**Step 2: Substitute back into the expression.**
\[ \frac{23}{5} + \frac{25}{6} - \frac{25}{6} - \frac{6}{7} + \frac{9}{5} + \frac{33}{7} \]

**Step 3: Combine like terms.**
The \(-\frac{25}{6}\) cancels out with \(+\frac{25}{6}\):
\[ \frac{23}{5} + \frac{9}{5} - \frac{6}{7} + \frac{33}{7} \]

**Step 4: Combine the fractions.**
\[ \frac{32}{5} - \frac{6}{7} + \frac{33}{7} \]

**Step 5: Find a common denominator (LCM of 5 and 7 is 35).**
\[ \frac{32 \times 7}{35} - \frac{6 \times 5}{35} + \frac{33 \times 5}{35} \]
This gives:
\[ \frac{224}{35} - \frac{30}{35} + \frac{165}{35} \]

**Step 6: Combine the fractions.**
\[ \frac{224 - 30 + 165}{35} = \frac{359}{35} \]

**Step 7: Convert to a mixed number.**
\[ 359 \div 35 = 10 \text{ remainder } 9 \Rightarrow 10\frac{9}{35} \]

### Final Answers
- Part (i): \(26.752\)
- Part (ii): \(10\frac{9}{35}\)