Answer :
Sure, let's solve each part step-by-step.
### Part i.
We start with the multiplication of two mixed numbers:
[tex]\[ \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] \][/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ -2 \frac{5}{8} = -\left(2 + \frac{5}{8}\right) = -\left(\frac{16}{8} + \frac{5}{8}\right) = -\frac{21}{8} \][/tex]
[tex]\[ 1 \frac{3}{7} = 1 + \frac{3}{7} = \frac{7}{7} + \frac{3}{7} = \frac{10}{7} \][/tex]
Now, multiply the improper fractions:
[tex]\[ -\frac{21}{8} \times \frac{10}{7} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ \left(-21 \times 10\right) / \left(8 \times 7\right) = -210 / 56 \][/tex]
Simplify the fraction:
[tex]\[ -210 / 56 = -\frac{15}{4} = -3.75 \][/tex]
So, the result is:
[tex]\[ \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] = -3.75 \][/tex]
### Part ii.
Next, we have the division of a mixed number by a fraction:
[tex]\[ 3 \frac{1}{7} \div \frac{11}{-2} \][/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 3 \frac{1}{7} = 3 + \frac{1}{7} = \frac{21}{7} + \frac{1}{7} = \frac{22}{7} \][/tex]
Since dividing by a fraction is the same as multiplying by its reciprocal, we have:
[tex]\[ \frac{22}{7} \div \frac{11}{-2} = \frac{22}{7} \times -\frac{2}{11} \][/tex]
Now, multiply the fractions:
[tex]\[ \frac{22 \times -2}{7 \times 11} = \frac{-44}{77} \][/tex]
Simplify the fraction:
[tex]\[ \frac{-44}{77} = -\frac{4}{7} \approx -0.5714285714285714 \][/tex]
So, the result is:
[tex]\[ 3 \frac{1}{7} \div \frac{11}{-2} \approx -0.5714285714285714 \][/tex]
### Part iii.
Lastly, we have a combination of multiplication and division with fractions:
[tex]\[ \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] \][/tex]
First, multiply the fractions:
[tex]\[ \frac{-3}{7} \times \frac{-2}{3} = \frac{(-3) \times (-2)}{7 \times 3} = \frac{6}{21} \][/tex]
Then, simplify the fraction:
[tex]\[ \frac{6}{21} = \frac{2}{7} \][/tex]
Next, divide by the fraction:
[tex]\[ \left[\frac{2}{7}\right] \div \left[\frac{16}{-21}\right] = \frac{2}{7} \times \left(-\frac{21}{16}\right) \][/tex]
Now, multiply the fractions:
[tex]\[ \frac{2 \times -21}{7 \times 16} = \frac{-42}{112} \][/tex]
Simplify the fraction:
[tex]\[ \frac{-42}{112} = -\frac{3}{8} = -0.375 \][/tex]
So, the result is:
[tex]\[ \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] = -0.375 \][/tex]
### Conclusion
To summarize, the values are:
[tex]\[ i. \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] = -3.75 \][/tex]
[tex]\[ ii. 3 \frac{1}{7} \div \frac{11}{-2} \approx -0.5714285714285714 \][/tex]
[tex]\[ iii. \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] = -0.375 \][/tex]
### Part i.
We start with the multiplication of two mixed numbers:
[tex]\[ \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] \][/tex]
First, convert the mixed numbers to improper fractions:
[tex]\[ -2 \frac{5}{8} = -\left(2 + \frac{5}{8}\right) = -\left(\frac{16}{8} + \frac{5}{8}\right) = -\frac{21}{8} \][/tex]
[tex]\[ 1 \frac{3}{7} = 1 + \frac{3}{7} = \frac{7}{7} + \frac{3}{7} = \frac{10}{7} \][/tex]
Now, multiply the improper fractions:
[tex]\[ -\frac{21}{8} \times \frac{10}{7} \][/tex]
Multiplying the numerators together and the denominators together:
[tex]\[ \left(-21 \times 10\right) / \left(8 \times 7\right) = -210 / 56 \][/tex]
Simplify the fraction:
[tex]\[ -210 / 56 = -\frac{15}{4} = -3.75 \][/tex]
So, the result is:
[tex]\[ \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] = -3.75 \][/tex]
### Part ii.
Next, we have the division of a mixed number by a fraction:
[tex]\[ 3 \frac{1}{7} \div \frac{11}{-2} \][/tex]
First, convert the mixed number to an improper fraction:
[tex]\[ 3 \frac{1}{7} = 3 + \frac{1}{7} = \frac{21}{7} + \frac{1}{7} = \frac{22}{7} \][/tex]
Since dividing by a fraction is the same as multiplying by its reciprocal, we have:
[tex]\[ \frac{22}{7} \div \frac{11}{-2} = \frac{22}{7} \times -\frac{2}{11} \][/tex]
Now, multiply the fractions:
[tex]\[ \frac{22 \times -2}{7 \times 11} = \frac{-44}{77} \][/tex]
Simplify the fraction:
[tex]\[ \frac{-44}{77} = -\frac{4}{7} \approx -0.5714285714285714 \][/tex]
So, the result is:
[tex]\[ 3 \frac{1}{7} \div \frac{11}{-2} \approx -0.5714285714285714 \][/tex]
### Part iii.
Lastly, we have a combination of multiplication and division with fractions:
[tex]\[ \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] \][/tex]
First, multiply the fractions:
[tex]\[ \frac{-3}{7} \times \frac{-2}{3} = \frac{(-3) \times (-2)}{7 \times 3} = \frac{6}{21} \][/tex]
Then, simplify the fraction:
[tex]\[ \frac{6}{21} = \frac{2}{7} \][/tex]
Next, divide by the fraction:
[tex]\[ \left[\frac{2}{7}\right] \div \left[\frac{16}{-21}\right] = \frac{2}{7} \times \left(-\frac{21}{16}\right) \][/tex]
Now, multiply the fractions:
[tex]\[ \frac{2 \times -21}{7 \times 16} = \frac{-42}{112} \][/tex]
Simplify the fraction:
[tex]\[ \frac{-42}{112} = -\frac{3}{8} = -0.375 \][/tex]
So, the result is:
[tex]\[ \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] = -0.375 \][/tex]
### Conclusion
To summarize, the values are:
[tex]\[ i. \left[-2 \frac{5}{8}\right] \times \left[1 \frac{3}{7}\right] = -3.75 \][/tex]
[tex]\[ ii. 3 \frac{1}{7} \div \frac{11}{-2} \approx -0.5714285714285714 \][/tex]
[tex]\[ iii. \left[\frac{-3}{7} \times \frac{-2}{3}\right] \div \left[\frac{16}{-21}\right] = -0.375 \][/tex]
