Sure! Let's solve each of these division problems step by step.
### (a) [tex]\[7 \div \frac{3}{5}\][/tex]
1. When dividing by a fraction, we multiply by its reciprocal.
2. [tex]\[7 \div \frac{3}{5} = 7 \times \frac{5}{3} = \frac{7 \times 5}{3} = \frac{35}{3} \approx 11.666666666666668\][/tex]
### (b) [tex]\[3 \frac{1}{2} \div \frac{1}{21}\][/tex]
1. Convert the mixed number to an improper fraction.
2. [tex]\[3 \frac{1}{2} = \frac{7}{2}\][/tex]
3. When dividing by a fraction, we multiply by its reciprocal.
4. [tex]\[\frac{7}{2} \div \frac{1}{21} = \frac{7}{2} \times 21 = \frac{7 \times 21}{2} = \frac{147}{2} = 73.5\][/tex]
### (c) [tex]\[\frac{5}{7} \div \frac{10}{21}\][/tex]
1. When dividing by a fraction, we multiply by its reciprocal.
2. [tex]\[\frac{5}{7} \div \frac{10}{21} = \frac{5}{7} \times \frac{21}{10} = \frac{5 \times 21}{7 \times 10} = \frac{105}{70} = \frac{3}{2} = 1.5\][/tex]
### (d) [tex]\[5 \frac{5}{6} \div \frac{5}{6}\][/tex]
1. Convert the mixed number to an improper fraction.
2. [tex]\[5 \frac{5}{6} = \frac{35}{6}\][/tex]
3. When dividing by a fraction, we multiply by its reciprocal.
4. [tex]\[\frac{35}{6} \div \frac{5}{6} = \frac{35}{6} \times \frac{6}{5} = \frac{35 \times 6}{6 \times 5} = \frac{210}{30} = 7\][/tex]
### (e) [tex]\[3 \div \frac{3}{5}\][/tex]
1. When dividing by a fraction, we multiply by its reciprocal.
2. [tex]\[3 \div \frac{3}{5} = 3 \times \frac{5}{3} = \frac{3 \times 5}{3} = 5\][/tex]
### (f) [tex]\[6 \frac{1}{2} \div 1 \frac{3}{8}\][/tex]
1. Convert the mixed numbers to improper fractions.
2. [tex]\[6 \frac{1}{2} = \frac{13}{2}\][/tex]
3. [tex]\[1 \frac{3}{8} = \frac{11}{8}\][/tex]
4. When dividing by a fraction, we multiply by its reciprocal.
5. [tex]\[\frac{13}{2} \div \frac{11}{8} = \frac{13}{2} \times \frac{8}{11} = \frac{13 \times 8}{2 \times 11} = \frac{104}{22} \approx 4.7272727272727275\][/tex]
### (g) [tex]\[83 \div 83\][/tex]
1. Dividing a number by itself gives us [tex]\(1\)[/tex].
2. [tex]\[83 \div 83 = 1\][/tex]
These detailed solutions match the numerical results provided.
