Simplify the following expressions :
(i) `(2/3+3/4) div (7/5-5/6) " (ii) "(2""1/3-1""1/2)" of " 3/5+1""2/5 div 2""1/3`.

Answer

Let's simplify the given expressions step by step.

### Expression (i):
\[ \frac{2}{3} + \frac{3}{4} \div \left(\frac{7}{5} - \frac{5}{6}\right) \]

**Step 1: Solve the subtraction in the denominator.**
- Find the LCM of 5 and 6, which is 30.
- Convert \(\frac{7}{5}\) and \(\frac{5}{6}\) to have a common denominator:
\[ \frac{7}{5} = \frac{7 \times 6}{5 \times 6} = \frac{42}{30} \]
\[ \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \]
- Now subtract:
\[ \frac{42}{30} - \frac{25}{30} = \frac{42 - 25}{30} = \frac{17}{30} \]

**Step 2: Rewrite the expression.**
\[ \frac{2}{3} + \frac{3}{4} \div \frac{17}{30} \]

**Step 3: Convert division to multiplication by the reciprocal.**
\[ \frac{2}{3} + \frac{3}{4} \times \frac{30}{17} \]

**Step 4: Solve the multiplication.**
- Calculate \(\frac{3}{4} \times \frac{30}{17}\):
\[ = \frac{3 \times 30}{4 \times 17} = \frac{90}{68} \]
- Simplify \(\frac{90}{68}\) by dividing both numerator and denominator by 2:
\[ = \frac{45}{34} \]

**Step 5: Now add \(\frac{2}{3}\) and \(\frac{45}{34}\).**
- Find the LCM of 3 and 34, which is 102.
- Convert both fractions:
\[ \frac{2}{3} = \frac{2 \times 34}{3 \times 34} = \frac{68}{102} \]
\[ \frac{45}{34} = \frac{45 \times 3}{34 \times 3} = \frac{135}{102} \]
- Now add:
\[ \frac{68}{102} + \frac{135}{102} = \frac{203}{102} \]

**Final Result for Expression (i):**
\[ \frac{203}{102} \text{ or } 2 \frac{1}{102} \]

### Expression (ii):
\[ (2 \frac{1}{3} - 1 \frac{1}{2}) \text{ of } \left( \frac{3}{5} + 1 \frac{2}{5} \div 2 \frac{1}{3} \right) \]

**Step 1: Convert mixed numbers to improper fractions.**
- \(2 \frac{1}{3} = \frac{7}{3}\)
- \(1 \frac{1}{2} = \frac{3}{2}\)
- \(1 \frac{2}{5} = \frac{7}{5}\)
- \(2 \frac{1}{3} = \frac{7}{3}\)

**Step 2: Rewrite the expression.**
\[ \left(\frac{7}{3} - \frac{3}{2}\right) \text{ of } \left(\frac{3}{5} + \frac{7}{5} \div \frac{7}{3}\right) \]

**Step 3: Solve the division in the second part.**
\[ \frac{7}{5} \div \frac{7}{3} = \frac{7}{5} \times \frac{3}{7} = \frac{3}{5} \]

**Step 4: Now add \(\frac{3}{5} + \frac{3}{5}\).**
\[ \frac{3}{5} + \frac{3}{5} = \frac{6}{5} \]

**Step 5: Now solve the subtraction in the first part.**
- Find LCM of 3 and 2, which is 6:
\[ \frac{7}{3} = \frac{14}{6}, \quad \frac{3}{2} = \frac{9}{6} \]
- Now subtract:
\[ \frac{14}{6} - \frac{9}{6} = \frac{5}{6} \]

**Step 6: Now multiply \(\frac{5}{6}\) by \(\frac{6}{5}\).**
\[ \frac{5}{6} \times \frac{6}{5} = 1 \]

**Final Result for Expression (ii):**
\[ 1 \]

### Summary of Results:
1. Expression (i): \(\frac{203}{102} \text{ or } 2 \frac{1}{102}\)
2. Expression (ii): \(1\)