Answer :
Let's solve the given problems step-by-step.
### Problem d: [tex]\(\frac{12}{5} \times \frac{10}{64} \div \frac{1}{9}\)[/tex]
1. Simplify [tex]\(\frac{12}{5} \times \frac{10}{64}\)[/tex]:
- [tex]\(\frac{12}{5}\)[/tex] can be left as it is.
- Reduce [tex]\(\frac{10}{64}\)[/tex]. The greatest common divisor (GCD) of 10 and 64 is 2.
[tex]\[ \frac{10}{64} = \frac{10 \div 2}{64 \div 2} = \frac{5}{32} \][/tex]
- Now, multiply the fractions:
[tex]\[ \frac{12}{5} \times \frac{5}{32} = \frac{12 \times 5}{5 \times 32} = \frac{60}{160} = \frac{3}{8} \][/tex]
2. Divide by [tex]\(\frac{1}{9}\)[/tex]:
- Division by a fraction [tex]\(\frac{a}{b}\)[/tex] is the same as multiplication by its reciprocal [tex]\(\frac{b}{a}\)[/tex].
- So, [tex]\(\frac{3}{8} \div \frac{1}{9} = \frac{3}{8} \times \frac{9}{1} = \frac{3 \times 9}{8 \times 1} = \frac{27}{8}\)[/tex]
- Simplify [tex]\(\frac{27}{8} = 3.375\)[/tex]
Hence, the result for part d is [tex]\( \frac{27}{8} \)[/tex] or 3.375.
### Problem b: [tex]\(1 \frac{2}{3}\left\{\left(\overline{\frac{7}{8}+\frac{3}{4}} \times \frac{2}{3}\right) \div 1 \frac{3}{4}+\frac{5}{8}\right.\)[/tex]
1. Simplify [tex]\(1 \frac{2}{3}\)[/tex]:
- Convert the mixed number to an improper fraction:
[tex]\[ 1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \][/tex]
2. Simplify [tex]\(\frac{7}{8} + \frac{3}{4}\)[/tex]:
- Convert [tex]\(\frac{3}{4}\)[/tex] to a fraction with a common denominator of 8:
[tex]\[ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \][/tex]
- Now, add:
[tex]\[ \frac{7}{8} + \frac{6}{8} = \frac{13}{8} \][/tex]
3. Multiply [tex]\(\frac{13}{8} \times \frac{2}{3}\)[/tex]:
- Multiply the numerators and the denominators:
[tex]\[ \frac{13}{8} \times \frac{2}{3} = \frac{13 \times 2}{8 \times 3} = \frac{26}{24} = \frac{13}{12} \approx 1.083 \][/tex]
4. Divide [tex]\(\frac{13}{12} \div 1 \frac{3}{4}\)[/tex]:
- Convert [tex]\(1 \frac{3}{4}\)[/tex] to an improper fraction:
[tex]\[ 1 \frac{3}{4} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \][/tex]
- Division by a fraction is multiplication by its reciprocal:
[tex]\[ \frac{13}{12} \div \frac{7}{4} = \frac{13}{12} \times \frac{4}{7} = \frac{13 \times 4}{12 \times 7} = \frac{52}{84} = \frac{13}{21} \approx 0.619 \][/tex]
5. Add [tex]\(\frac{13}{21}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex]:
- Find a common denominator for [tex]\(\frac{13}{21}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex], which is 168:
[tex]\[ \frac{13}{21} = \frac{13 \times 8}{21 \times 8} = \frac{104}{168} \][/tex]
[tex]\[ \frac{5}{8} = \frac{5 \times 21}{8 \times 21} = \frac{105}{168} \][/tex]
- Add the two fractions:
[tex]\[ \frac{104}{168} + \frac{105}{168} = \frac{209}{168} \approx 1.244 \][/tex]
6. Multiply [tex]\(1 \frac{2}{3}\)[/tex] by the sum:
- Convert [tex]\(1 \frac{2}{3}\)[/tex] back to [tex]\(\frac{5}{3}\)[/tex] and multiply it by [tex]\(\frac{209}{168}\)[/tex]:
[tex]\[ \frac{5}{3} \times \frac{209}{168} = \frac{5 \times 209}{3 \times 168} = \frac{1045}{504} \approx 2.073 \][/tex]
Hence, the result for part b is [tex]\( \frac{1045}{504} \)[/tex] or approximately 2.073.
### Problem d: [tex]\(\frac{12}{5} \times \frac{10}{64} \div \frac{1}{9}\)[/tex]
1. Simplify [tex]\(\frac{12}{5} \times \frac{10}{64}\)[/tex]:
- [tex]\(\frac{12}{5}\)[/tex] can be left as it is.
- Reduce [tex]\(\frac{10}{64}\)[/tex]. The greatest common divisor (GCD) of 10 and 64 is 2.
[tex]\[ \frac{10}{64} = \frac{10 \div 2}{64 \div 2} = \frac{5}{32} \][/tex]
- Now, multiply the fractions:
[tex]\[ \frac{12}{5} \times \frac{5}{32} = \frac{12 \times 5}{5 \times 32} = \frac{60}{160} = \frac{3}{8} \][/tex]
2. Divide by [tex]\(\frac{1}{9}\)[/tex]:
- Division by a fraction [tex]\(\frac{a}{b}\)[/tex] is the same as multiplication by its reciprocal [tex]\(\frac{b}{a}\)[/tex].
- So, [tex]\(\frac{3}{8} \div \frac{1}{9} = \frac{3}{8} \times \frac{9}{1} = \frac{3 \times 9}{8 \times 1} = \frac{27}{8}\)[/tex]
- Simplify [tex]\(\frac{27}{8} = 3.375\)[/tex]
Hence, the result for part d is [tex]\( \frac{27}{8} \)[/tex] or 3.375.
### Problem b: [tex]\(1 \frac{2}{3}\left\{\left(\overline{\frac{7}{8}+\frac{3}{4}} \times \frac{2}{3}\right) \div 1 \frac{3}{4}+\frac{5}{8}\right.\)[/tex]
1. Simplify [tex]\(1 \frac{2}{3}\)[/tex]:
- Convert the mixed number to an improper fraction:
[tex]\[ 1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \][/tex]
2. Simplify [tex]\(\frac{7}{8} + \frac{3}{4}\)[/tex]:
- Convert [tex]\(\frac{3}{4}\)[/tex] to a fraction with a common denominator of 8:
[tex]\[ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \][/tex]
- Now, add:
[tex]\[ \frac{7}{8} + \frac{6}{8} = \frac{13}{8} \][/tex]
3. Multiply [tex]\(\frac{13}{8} \times \frac{2}{3}\)[/tex]:
- Multiply the numerators and the denominators:
[tex]\[ \frac{13}{8} \times \frac{2}{3} = \frac{13 \times 2}{8 \times 3} = \frac{26}{24} = \frac{13}{12} \approx 1.083 \][/tex]
4. Divide [tex]\(\frac{13}{12} \div 1 \frac{3}{4}\)[/tex]:
- Convert [tex]\(1 \frac{3}{4}\)[/tex] to an improper fraction:
[tex]\[ 1 \frac{3}{4} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \][/tex]
- Division by a fraction is multiplication by its reciprocal:
[tex]\[ \frac{13}{12} \div \frac{7}{4} = \frac{13}{12} \times \frac{4}{7} = \frac{13 \times 4}{12 \times 7} = \frac{52}{84} = \frac{13}{21} \approx 0.619 \][/tex]
5. Add [tex]\(\frac{13}{21}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex]:
- Find a common denominator for [tex]\(\frac{13}{21}\)[/tex] and [tex]\(\frac{5}{8}\)[/tex], which is 168:
[tex]\[ \frac{13}{21} = \frac{13 \times 8}{21 \times 8} = \frac{104}{168} \][/tex]
[tex]\[ \frac{5}{8} = \frac{5 \times 21}{8 \times 21} = \frac{105}{168} \][/tex]
- Add the two fractions:
[tex]\[ \frac{104}{168} + \frac{105}{168} = \frac{209}{168} \approx 1.244 \][/tex]
6. Multiply [tex]\(1 \frac{2}{3}\)[/tex] by the sum:
- Convert [tex]\(1 \frac{2}{3}\)[/tex] back to [tex]\(\frac{5}{3}\)[/tex] and multiply it by [tex]\(\frac{209}{168}\)[/tex]:
[tex]\[ \frac{5}{3} \times \frac{209}{168} = \frac{5 \times 209}{3 \times 168} = \frac{1045}{504} \approx 2.073 \][/tex]
Hence, the result for part b is [tex]\( \frac{1045}{504} \)[/tex] or approximately 2.073.
