Sure! Let's go through each of the problems step by step.
### (i) [tex]\(\frac{3}{5} \div \frac{1}{2}\)[/tex]
When dividing fractions, we multiply by the reciprocal of the divisor.
So, [tex]\(\frac{3}{5} \div \frac{1}{2}\)[/tex] is the same as [tex]\(\frac{3}{5} \times \frac{2}{1}\)[/tex].
[tex]\[ \frac{3}{5} \times \frac{2}{1} = \frac{3 \times 2}{5 \times 1} = \frac{6}{5} = 1.2 \][/tex]
Thus, [tex]\(\frac{3}{5} \div \frac{1}{2} = 1.2\)[/tex].
### (ii) [tex]\(\frac{1}{2} \div \frac{3}{5}\)[/tex]
Again, we multiply by the reciprocal of the divisor.
So, [tex]\(\frac{1}{2} \div \frac{3}{5}\)[/tex] is the same as [tex]\(\frac{1}{2} \times \frac{5}{3}\)[/tex].
[tex]\[ \frac{1}{2} \times \frac{5}{3} = \frac{1 \times 5}{2 \times 3} = \frac{5}{6} \][/tex]
[tex]\(\frac{5}{6} \approx 0.8333\)[/tex]
Thus, [tex]\(\frac{1}{2} \div \frac{3}{5} \approx 0.8333\)[/tex].
### (iii) [tex]\(2 \frac{1}{2} \div \frac{3}{5}\)[/tex]
First, convert the mixed number [tex]\(2 \frac{1}{2}\)[/tex] to an improper fraction.
[tex]\[ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \][/tex]
Now, divide by multiplying by the reciprocal of the divisor:
[tex]\[ \frac{5}{2} \div \frac{3}{5} = \frac{5}{2} \times \frac{5}{3} \][/tex]
[tex]\[ \frac{5}{2} \times \frac{5}{3} = \frac{5 \times 5}{2 \times 3} = \frac{25}{6} \][/tex]
[tex]\(\frac{25}{6} \approx 4.1667\)[/tex]
Thus, [tex]\(2 \frac{1}{2} \div \frac{3}{5} \approx 4.1667\)[/tex].
### (iv) [tex]\(5 \frac{1}{6} \div \frac{9}{2}\)[/tex]
First, convert the mixed number [tex]\(5 \frac{1}{6}\)[/tex] to an improper fraction.
[tex]\[ 5 \frac{1}{6} = 5 + \frac{1}{6} = \frac{30}{6} + \frac{1}{6} = \frac{31}{6} \][/tex]
Now, divide by multiplying by the reciprocal of the divisor:
[tex]\[ \frac{31}{6} \div \frac{9}{2} = \frac{31}{6} \times \frac{2}{9} \][/tex]
[tex]\[ \frac{31}{6} \times \frac{2}{9} = \frac{31 \times 2}{6 \times 9} = \frac{62}{54} \][/tex]
Simplify [tex]\(\frac{62}{54}\)[/tex]:
[tex]\(\frac{62}{54} = \frac{31}{27} \)[/tex]
[tex]\(\frac{31}{27} \approx 1.1481\)[/tex]
Thus, [tex]\(5 \frac{1}{6} \div \frac{9}{2} \approx 1.1481\)[/tex].
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To sum up:
(i) [tex]\(\frac{3}{5} \div \frac{1}{2} = 1.2\)[/tex]
(ii) [tex]\(\frac{1}{2} \div \frac{3}{5} \approx 0.8333\)[/tex]
(iii) [tex]\(2 \frac{1}{2} \div \frac{3}{5} \approx 4.1667\)[/tex]
(iv) [tex]\(5 \frac{1}{6} \div \frac{9}{2} \approx 1.1481\)[/tex]
