Simplify :
(i) `(13/(8)xx(12)/(13))+((-4)/(9)xx(3)/(-2))`
(ii) `((16)/(15)xx(-25)/(8))+((-14)/(27)xx(6)/(7))`
(iii) `((6)/(55)xx(-22)/(9))-((26)/(125)xx(-10)/(39))`
(iv) `((-12)/(7)xx(-14)/(27))-((-8)/(-45)xx(9)/(16))`

Answer

Let's simplify each part step by step.

### (i) Simplify:
\[ \frac{13}{8} \times \frac{12}{13} + \frac{-4}{9} \times \frac{3}{-2} \]

**Step 1:** Simplify the first term:
\[ \frac{13}{8} \times \frac{12}{13} = \frac{12}{8} = \frac{3}{2} \]

**Step 2:** Simplify the second term:
\[ \frac{-4}{9} \times \frac{3}{-2} = \frac{4 \times 3}{9 \times 2} = \frac{12}{18} = \frac{2}{3} \]

**Step 3:** Add the two results:
\[ \frac{3}{2} + \frac{2}{3} \]

**Step 4:** Find a common denominator (which is 6):
\[ \frac{3}{2} = \frac{9}{6} \quad \text{and} \quad \frac{2}{3} = \frac{4}{6} \]
So,
\[ \frac{9}{6} + \frac{4}{6} = \frac{13}{6} \]

**Final Answer for (i):**
\[ \frac{13}{6} \]

### (ii) Simplify:
\[ \frac{16}{15} \times \frac{-25}{8} + \frac{-14}{27} \times \frac{6}{7} \]

**Step 1:** Simplify the first term:
\[ \frac{16}{15} \times \frac{-25}{8} = \frac{-400}{120} = \frac{-10}{3} \]

**Step 2:** Simplify the second term:
\[ \frac{-14}{27} \times \frac{6}{7} = \frac{-84}{189} = \frac{-4}{9} \]

**Step 3:** Add the two results:
\[ \frac{-10}{3} + \frac{-4}{9} \]

**Step 4:** Find a common denominator (which is 9):
\[ \frac{-10}{3} = \frac{-30}{9} \]
So,
\[ \frac{-30}{9} + \frac{-4}{9} = \frac{-34}{9} \]

**Final Answer for (ii):**
\[ \frac{-34}{9} \]

### (iii) Simplify:
\[ \frac{6}{55} \times \frac{-22}{9} - \frac{26}{125} \times \frac{-10}{39} \]

**Step 1:** Simplify the first term:
\[ \frac{6}{55} \times \frac{-22}{9} = \frac{-132}{495} = \frac{-4}{15} \]

**Step 2:** Simplify the second term:
\[ \frac{26}{125} \times \frac{-10}{39} = \frac{-260}{4875} = \frac{-52}{975} \]

**Step 3:** Subtract the two results:
\[ \frac{-4}{15} - \frac{-52}{975} \]

**Step 4:** Find a common denominator (which is 975):
\[ \frac{-4}{15} = \frac{-260}{975} \]
So,
\[ \frac{-260}{975} + \frac{52}{975} = \frac{-208}{975} \]

**Final Answer for (iii):**
\[ \frac{-208}{975} \]

### (iv) Simplify:
\[ \frac{-12}{7} \times \frac{-14}{27} - \frac{-8}{-45} \times \frac{9}{16} \]

**Step 1:** Simplify the first term:
\[ \frac{-12}{7} \times \frac{-14}{27} = \frac{168}{189} = \frac{8}{9} \]

**Step 2:** Simplify the second term:
\[ \frac{-8}{-45} \times \frac{9}{16} = \frac{72}{720} = \frac{1}{10} \]

**Step 3:** Subtract the two results:
\[ \frac{8}{9} - \frac{1}{10} \]

**Step 4:** Find a common denominator (which is 90):
\[ \frac{8}{9} = \frac{80}{90} \quad \text{and} \quad \frac{1}{10} = \frac{9}{90} \]
So,
\[ \frac{80}{90} - \frac{9}{90} = \frac{71}{90} \]

**Final Answer for (iv):**
\[ \frac{71}{90} \]