Evaluate :
(i) `(3)/(4)+(5)/(6)+(-1)/(4)+(-7)/(6)`
(ii) `(9)/(-10)+(4)/(15)+(-3)/(20)+(-3)/(10)+(8)/(15)+(9)/(-20)`

Answer

Let's solve the given problems step by step.

### Part (i): Evaluate \( \frac{3}{4} + \frac{5}{6} + \left(-\frac{1}{4}\right) + \left(-\frac{7}{6}\right) \)

**Step 1: Identify the fractions.**
We have the fractions: \( \frac{3}{4}, \frac{5}{6}, -\frac{1}{4}, -\frac{7}{6} \).

**Step 2: Find the LCM of the denominators.**
The denominators are 4 and 6. The LCM of 4 and 6 is 12.

**Step 3: Convert each fraction to have a common denominator of 12.**
- For \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]
- For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]
- For \( -\frac{1}{4} \):
\[ -\frac{1}{4} = -\frac{1 \times 3}{4 \times 3} = -\frac{3}{12} \]
- For \( -\frac{7}{6} \):
\[ -\frac{7}{6} = -\frac{7 \times 2}{6 \times 2} = -\frac{14}{12} \]

**Step 4: Rewrite the expression with the common denominator.**
Now we can rewrite the expression:
\[ \frac{9}{12} + \frac{10}{12} - \frac{3}{12} - \frac{14}{12} \]

**Step 5: Combine the fractions.**
Combine the numerators:
\[ \frac{9 + 10 - 3 - 14}{12} = \frac{2}{12} \]

**Step 6: Simplify the fraction.**
\[ \frac{2}{12} = \frac{1}{6} \]

### Answer for Part (i):
\[ \frac{1}{6} \]

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### Part (ii): Evaluate \( \frac{9}{-10} + \frac{4}{15} + \left(-\frac{3}{20}\right) + \left(-\frac{3}{10}\right) + \frac{8}{15} + \frac{9}{-20} \)

**Step 1: Identify the fractions.**
The fractions are \( \frac{9}{-10}, \frac{4}{15}, -\frac{3}{20}, -\frac{3}{10}, \frac{8}{15}, \frac{9}{-20} \).

**Step 2: Find the LCM of the denominators.**
The denominators are 10, 15, and 20. The LCM of 10, 15, and 20 is 60.

**Step 3: Convert each fraction to have a common denominator of 60.**
- For \( \frac{9}{-10} \):
\[ \frac{9}{-10} = \frac{9 \times 6}{-10 \times 6} = \frac{54}{-60} \]
- For \( \frac{4}{15} \):
\[ \frac{4}{15} = \frac{4 \times 4}{15 \times 4} = \frac{16}{60} \]
- For \( -\frac{3}{20} \):
\[ -\frac{3}{20} = -\frac{3 \times 3}{20 \times 3} = -\frac{9}{60} \]
- For \( -\frac{3}{10} \):
\[ -\frac{3}{10} = -\frac{3 \times 6}{10 \times 6} = -\frac{18}{60} \]
- For \( \frac{8}{15} \):
\[ \frac{8}{15} = \frac{8 \times 4}{15 \times 4} = \frac{32}{60} \]
- For \( \frac{9}{-20} \):
\[ \frac{9}{-20} = \frac{9 \times 3}{-20 \times 3} = \frac{27}{-60} \]

**Step 4: Rewrite the expression with the common denominator.**
Now we can rewrite the expression:
\[ \frac{54}{-60} + \frac{16}{60} - \frac{9}{60} - \frac{18}{60} + \frac{32}{60} + \frac{27}{-60} \]

**Step 5: Combine the fractions.**
Combine the numerators:
\[ \frac{54 - 9 - 18 + 16 + 32 - 27}{60} = \frac{48}{60} \]

**Step 6: Simplify the fraction.**
\[ \frac{48}{60} = \frac{4}{5} \]

### Answer for Part (ii):
\[ \frac{4}{5} \]

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