Certainly! Let's walk through each part of the problem step-by-step.
### (i) [tex]\(-\frac{9}{7} \times \frac{-35}{6}\)[/tex]
Step 1: Multiply the numerators.
[tex]\[
-9 \times -35 = 315
\][/tex]
Step 2: Multiply the denominators.
[tex]\[
7 \times 6 = 42
\][/tex]
Step 3: Combine the results.
[tex]\[
-\frac{9}{7} \times \frac{-35}{6} = \frac{315}{42}
\][/tex]
Step 4: Simplify the fraction.
[tex]\[
\frac{315}{42} = 7.5
\][/tex]
### (ii) [tex]\(\frac{-115}{28} \times \frac{-49}{69}\)[/tex]
Step 1: Multiply the numerators.
[tex]\[
-115 \times -49 = 5635
\][/tex]
Step 2: Multiply the denominators.
[tex]\[
28 \times 69 = 1932
\][/tex]
Step 3: Combine the results.
[tex]\[
\frac{-115}{28} \times \frac{-49}{69} = \frac{5635}{1932}
\][/tex]
Step 4: Simplify the fraction.
[tex]\[
\frac{5635}{1932} = 2.9166666666666665
\][/tex]
### (iii) [tex]\(\frac{-15}{14} \times \frac{112}{-105}\)[/tex]
Step 1: Multiply the numerators.
[tex]\[
-15 \times 112 = -1680
\][/tex]
Step 2: Multiply the denominators.
[tex]\[
14 \times -105 = -1470
\][/tex]
Step 3: Combine the results.
[tex]\[
\frac{-15}{14} \times \frac{112}{-105} = \frac{-1680}{-1470}
\][/tex]
Step 4: Simplify the fraction.
[tex]\[
\frac{-1680}{-1470} = 1.1428571428571428
\][/tex]
### (iv) [tex]\(\frac{-39}{3} \times \frac{19}{5} \times \frac{-45}{38}\)[/tex]
Step 1: Multiply the numerators.
[tex]\[
-39 \times 19 \times -45 = 33345
\][/tex]
Step 2: Multiply the denominators.
[tex]\[
3 \times 5 \times 38 = 570
\][/tex]
Step 3: Combine the results.
[tex]\[
\frac{-39}{3} \times \frac{19}{5} \times \frac{-45}{38} = \frac{33345}{570}
\][/tex]
Step 4: Simplify the fraction.
[tex]\[
\frac{33345}{570} = 58.49999999999999
\][/tex]
### (v) [tex]\(-1 \frac{2}{5} \times 3 \frac{1}{2}\)[/tex]
Step 1: Convert mixed numbers to improper fractions.
[tex]\[
-1 \frac{2}{5} = -1 + \frac{2}{5} = -1.4
\][/tex]
[tex]\[
3 \frac{1}{2} = 3 + \frac{1}{2} = 3.5
\][/tex]
Step 2: Multiply the fractions.
[tex]\[
-1.4 \times 3.5 = -4.9
\][/tex]
### (vi) [tex]\(\left(-1 \frac{5}{6}\right) \times \left(-1 \frac{7}{11}\right)\)[/tex]
Step 1: Convert mixed numbers to improper fractions.
[tex]\[
-1 \frac{5}{6} = -1 + \frac{5}{6} = -1.8333
\][/tex]
[tex]\[
-1 \frac{7}{11} = -1 + \frac{7}{11} = -1.6364
\][/tex]
Step 2: Multiply the fractions.
[tex]\[
-1.8333 \times -1.6364 = 0.060606060606060594
\][/tex]
### (vii) [tex]\(-9 \frac{2}{7} \times 1 \frac{2}{5}\)[/tex]
Step 1: Convert mixed numbers to improper fractions.
[tex]\[
-9 \frac{2}{7} = -9 + \frac{2}{7} = -9.2857
\][/tex]
[tex]\[
1 \frac{2}{5} = 1 + \frac{2}{5} = 1.4
\][/tex]
Step 2: Multiply the fractions.
[tex]\[
-9.2857 \times 1.4 = -12.199999999999998
\][/tex]
### (viii) [tex]\(-2 \frac{2}{3} \times \left(\frac{-3}{4}\right)\)[/tex]
Step 1: Convert mixed number to improper fraction.
[tex]\[
-2 \frac{2}{3} = -2 + \frac{2}{3} = -2.6667
\][/tex]
Step 2: Multiply the fractions.
[tex]\[
-2.6667 \times \frac{-3}{4} = 1.0
\][/tex]
Thus, the simplified results are:
1. [tex]\(7.5\)[/tex]
2. [tex]\(2.9166666666666665\)[/tex]
3. [tex]\(1.1428571428571428\)[/tex]
4. [tex]\(58.49999999999999\)[/tex]
5. [tex]\(-4.9\)[/tex]
6. [tex]\(0.060606060606060594\)[/tex]
7. [tex]\(-12.199999999999998\)[/tex]
8. [tex]\(1.0\)[/tex]
