Answer :
Let's solve each part of the given mathematical problems step by step.
### Part (a) [tex]\( 18 \div \frac{1}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{1}{3}\)[/tex] is 3.
3. Thus, [tex]\(18 \div \frac{1}{3} = 18 \cdot 3 = 54\)[/tex].
Answer: 54
### Part (b) [tex]\( 16 \div \frac{1}{4} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{1}{4}\)[/tex] is 4.
3. Thus, [tex]\(16 \div \frac{1}{4} = 16 \cdot 4 = 64\)[/tex].
Answer: 64
### Part (c) [tex]\( 32 \div 1 \frac{7}{9} \)[/tex]
1. Convert the mixed number [tex]\(1 \frac{7}{9}\)[/tex] to an improper fraction:
[tex]\(1 \frac{7}{9} = \frac{16}{9}\)[/tex].
2. Dividing by [tex]\(\frac{16}{9}\)[/tex] is equivalent to multiplying by its reciprocal, [tex]\(\frac{9}{16}\)[/tex].
3. Thus, [tex]\(32 \div \frac{16}{9} = 32 \cdot \frac{9}{16} = 2 \cdot 9 = 18\)[/tex].
Answer: 18
### Part (I) [tex]\( 64 \div \frac{8}{5} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{8}{5}\)[/tex] is [tex]\(\frac{5}{8}\)[/tex].
3. Thus, [tex]\(64 \div \frac{8}{5} = 64 \cdot \frac{5}{8} = 8 \cdot 5 = 40\)[/tex].
Answer: 40
### Part (i) [tex]\( 75 \div \frac{2}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Thus, [tex]\(75 \div \frac{2}{3} = 75 \cdot \frac{3}{2} = \frac{75 \cdot 3}{2} = \frac{225}{2} = 112.5\)[/tex].
Answer: 112.5
### Part (j) [tex]\( 18 \frac{1}{6} \div \frac{7}{12} \)[/tex]
1. Convert the mixed number [tex]\(18 \frac{1}{6}\)[/tex] to an improper fraction:
[tex]\(18 \frac{1}{6} = \frac{109}{6}\)[/tex].
2. Dividing by [tex]\(\frac{7}{12}\)[/tex] is equivalent to multiplying by its reciprocal, [tex]\(\frac{12}{7}\)[/tex].
3. Thus, [tex]\(\frac{109}{6} \div \frac{7}{12} = \frac{109}{6} \cdot \frac{12}{7} = \frac{109 \cdot 12}{6 \cdot 7} = \frac{1308}{42} = 31.142857142857142\)[/tex].
Answer: 31.142857142857142
### Part (m) [tex]\( \frac{25}{6} \div 10 \)[/tex]
1. Dividing by a whole number is equivalent to multiplying by its reciprocal.
2. The reciprocal of 10 is [tex]\(\frac{1}{10}\)[/tex].
3. Thus, [tex]\(\frac{25}{6} \div 10 = \frac{25}{6} \cdot \frac{1}{10} = \frac{25 \cdot 1}{6 \cdot 10} = \frac{25}{60} = \frac{5}{12} = 0.4166666666666667\)[/tex].
Answer: 0.4166666666666667
### Part (ii) [tex]\( \frac{4}{9} \div \frac{2}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Thus, [tex]\(\frac{4}{9} \div \frac{2}{3} = \frac{4}{9} \cdot \frac{3}{2} = \frac{4 \cdot 3}{9 \cdot 2} = \frac{12}{18} = \frac{2}{3} = 0.6666666666666666\)[/tex].
Answer: 0.6666666666666666
Each part was solved by either converting the division of fractions to multiplication by the reciprocal or converting mixed numbers to improper fractions as needed.
### Part (a) [tex]\( 18 \div \frac{1}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{1}{3}\)[/tex] is 3.
3. Thus, [tex]\(18 \div \frac{1}{3} = 18 \cdot 3 = 54\)[/tex].
Answer: 54
### Part (b) [tex]\( 16 \div \frac{1}{4} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{1}{4}\)[/tex] is 4.
3. Thus, [tex]\(16 \div \frac{1}{4} = 16 \cdot 4 = 64\)[/tex].
Answer: 64
### Part (c) [tex]\( 32 \div 1 \frac{7}{9} \)[/tex]
1. Convert the mixed number [tex]\(1 \frac{7}{9}\)[/tex] to an improper fraction:
[tex]\(1 \frac{7}{9} = \frac{16}{9}\)[/tex].
2. Dividing by [tex]\(\frac{16}{9}\)[/tex] is equivalent to multiplying by its reciprocal, [tex]\(\frac{9}{16}\)[/tex].
3. Thus, [tex]\(32 \div \frac{16}{9} = 32 \cdot \frac{9}{16} = 2 \cdot 9 = 18\)[/tex].
Answer: 18
### Part (I) [tex]\( 64 \div \frac{8}{5} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{8}{5}\)[/tex] is [tex]\(\frac{5}{8}\)[/tex].
3. Thus, [tex]\(64 \div \frac{8}{5} = 64 \cdot \frac{5}{8} = 8 \cdot 5 = 40\)[/tex].
Answer: 40
### Part (i) [tex]\( 75 \div \frac{2}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Thus, [tex]\(75 \div \frac{2}{3} = 75 \cdot \frac{3}{2} = \frac{75 \cdot 3}{2} = \frac{225}{2} = 112.5\)[/tex].
Answer: 112.5
### Part (j) [tex]\( 18 \frac{1}{6} \div \frac{7}{12} \)[/tex]
1. Convert the mixed number [tex]\(18 \frac{1}{6}\)[/tex] to an improper fraction:
[tex]\(18 \frac{1}{6} = \frac{109}{6}\)[/tex].
2. Dividing by [tex]\(\frac{7}{12}\)[/tex] is equivalent to multiplying by its reciprocal, [tex]\(\frac{12}{7}\)[/tex].
3. Thus, [tex]\(\frac{109}{6} \div \frac{7}{12} = \frac{109}{6} \cdot \frac{12}{7} = \frac{109 \cdot 12}{6 \cdot 7} = \frac{1308}{42} = 31.142857142857142\)[/tex].
Answer: 31.142857142857142
### Part (m) [tex]\( \frac{25}{6} \div 10 \)[/tex]
1. Dividing by a whole number is equivalent to multiplying by its reciprocal.
2. The reciprocal of 10 is [tex]\(\frac{1}{10}\)[/tex].
3. Thus, [tex]\(\frac{25}{6} \div 10 = \frac{25}{6} \cdot \frac{1}{10} = \frac{25 \cdot 1}{6 \cdot 10} = \frac{25}{60} = \frac{5}{12} = 0.4166666666666667\)[/tex].
Answer: 0.4166666666666667
### Part (ii) [tex]\( \frac{4}{9} \div \frac{2}{3} \)[/tex]
1. Dividing by a fraction is equivalent to multiplying by its reciprocal.
2. The reciprocal of [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Thus, [tex]\(\frac{4}{9} \div \frac{2}{3} = \frac{4}{9} \cdot \frac{3}{2} = \frac{4 \cdot 3}{9 \cdot 2} = \frac{12}{18} = \frac{2}{3} = 0.6666666666666666\)[/tex].
Answer: 0.6666666666666666
Each part was solved by either converting the division of fractions to multiplication by the reciprocal or converting mixed numbers to improper fractions as needed.
