Answer :
Sure, let's go through each problem step-by-step:
1. Problem: [tex]\(0.25 \times \frac{5}{9}\)[/tex]
Solution: To multiply, multiply 0.25 by the numerator and then divide by the denominator:
[tex]\[
0.25 \times \frac{5}{9} = \frac{0.25 \times 5}{9} = \frac{1.25}{9} \approx 0.138888\dots
\][/tex]
2. Problem: [tex]\(\left(0.\overline{3} \times \frac{6}{11}\right) \times \left(\frac{8}{9} \times \frac{11}{6}\right)\)[/tex]
Solution: Convert [tex]\(0.\overline{3}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]. The other fractions simplify as follows:
[tex]\[
\frac{1}{3} \times \frac{6}{11} = \frac{6}{33} = \frac{2}{11}
\][/tex]
[tex]\[
\frac{8}{9} \times \frac{11}{6} = \frac{88}{54} = \frac{44}{27}
\][/tex]
[tex]\[
\frac{2}{11} \times \frac{44}{27} = \frac{88}{297} = \frac{8}{27}
\][/tex]
3. Problem: [tex]\(0.\overline{22} \times \frac{3}{8}\)[/tex]
Solution: Convert [tex]\(0.\overline{22}\)[/tex] to [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[
\frac{2}{9} \times \frac{3}{8} = \frac{6}{72} = \frac{1}{12}
\][/tex]
4. Problem: [tex]\(2.4 \times \frac{9}{10}\)[/tex]
Solution: Multiply directly:
[tex]\[
2.4 \times \frac{9}{10} = \frac{21.6}{10} = 2.16
\][/tex]
5. Problem: [tex]\(1.7 \times \frac{18}{20}\)[/tex]
Solution: Simplify the fraction first:
[tex]\[
\frac{18}{20} = \frac{9}{10}
\][/tex]
Then multiply:
[tex]\[
1.7 \times \frac{9}{10} = \frac{15.3}{10} = 1.53
\][/tex]
6. Problem: [tex]\(\frac{0.\overline{2}}{0.\overline{3}}\)[/tex]
Solution: Convert and divide:
[tex]\[
\frac{2}{9} \div \frac{1}{3} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3}
\][/tex]
7. Problem: [tex]\(\frac{0.\overline{42}}{0.\overline{18}}\)[/tex]
Solution: Convert to fractions:
[tex]\[
\frac{42}{99} \div \frac{18}{99} = \frac{42}{99} \times \frac{99}{18} = \frac{42}{18} = \frac{7}{3}
\][/tex]
8. Problem: [tex]\(\frac{2.\overline{7}}{0.\overline{5}}\)[/tex]
Solution: Convert repeating decimals:
[tex]\[
\frac{25}{9} \div \frac{5}{9} = \frac{25}{9} \times \frac{9}{5} = 5
\][/tex]
9. Problem: [tex]\(\frac{0.\overline{21}}{0.\overline{3}}\)[/tex]
Solution: Convert to fractions:
[tex]\[
\frac{21}{99} \div \frac{1}{3} = \frac{21}{99} \times \frac{3}{1} = \frac{63}{99} = \frac{7}{11}
\][/tex]
10. Problem: [tex]\(\frac{0.5}{0.5}\)[/tex]
Solution: Division of equal numbers:
[tex]\[
\frac{0.5}{0.5} = 1
\][/tex]
I hope this step-by-step guide helps you understand how to solve similar problems involving repeating decimals and fractions!
1. Problem: [tex]\(0.25 \times \frac{5}{9}\)[/tex]
Solution: To multiply, multiply 0.25 by the numerator and then divide by the denominator:
[tex]\[
0.25 \times \frac{5}{9} = \frac{0.25 \times 5}{9} = \frac{1.25}{9} \approx 0.138888\dots
\][/tex]
2. Problem: [tex]\(\left(0.\overline{3} \times \frac{6}{11}\right) \times \left(\frac{8}{9} \times \frac{11}{6}\right)\)[/tex]
Solution: Convert [tex]\(0.\overline{3}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]. The other fractions simplify as follows:
[tex]\[
\frac{1}{3} \times \frac{6}{11} = \frac{6}{33} = \frac{2}{11}
\][/tex]
[tex]\[
\frac{8}{9} \times \frac{11}{6} = \frac{88}{54} = \frac{44}{27}
\][/tex]
[tex]\[
\frac{2}{11} \times \frac{44}{27} = \frac{88}{297} = \frac{8}{27}
\][/tex]
3. Problem: [tex]\(0.\overline{22} \times \frac{3}{8}\)[/tex]
Solution: Convert [tex]\(0.\overline{22}\)[/tex] to [tex]\(\frac{2}{9}\)[/tex]:
[tex]\[
\frac{2}{9} \times \frac{3}{8} = \frac{6}{72} = \frac{1}{12}
\][/tex]
4. Problem: [tex]\(2.4 \times \frac{9}{10}\)[/tex]
Solution: Multiply directly:
[tex]\[
2.4 \times \frac{9}{10} = \frac{21.6}{10} = 2.16
\][/tex]
5. Problem: [tex]\(1.7 \times \frac{18}{20}\)[/tex]
Solution: Simplify the fraction first:
[tex]\[
\frac{18}{20} = \frac{9}{10}
\][/tex]
Then multiply:
[tex]\[
1.7 \times \frac{9}{10} = \frac{15.3}{10} = 1.53
\][/tex]
6. Problem: [tex]\(\frac{0.\overline{2}}{0.\overline{3}}\)[/tex]
Solution: Convert and divide:
[tex]\[
\frac{2}{9} \div \frac{1}{3} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3}
\][/tex]
7. Problem: [tex]\(\frac{0.\overline{42}}{0.\overline{18}}\)[/tex]
Solution: Convert to fractions:
[tex]\[
\frac{42}{99} \div \frac{18}{99} = \frac{42}{99} \times \frac{99}{18} = \frac{42}{18} = \frac{7}{3}
\][/tex]
8. Problem: [tex]\(\frac{2.\overline{7}}{0.\overline{5}}\)[/tex]
Solution: Convert repeating decimals:
[tex]\[
\frac{25}{9} \div \frac{5}{9} = \frac{25}{9} \times \frac{9}{5} = 5
\][/tex]
9. Problem: [tex]\(\frac{0.\overline{21}}{0.\overline{3}}\)[/tex]
Solution: Convert to fractions:
[tex]\[
\frac{21}{99} \div \frac{1}{3} = \frac{21}{99} \times \frac{3}{1} = \frac{63}{99} = \frac{7}{11}
\][/tex]
10. Problem: [tex]\(\frac{0.5}{0.5}\)[/tex]
Solution: Division of equal numbers:
[tex]\[
\frac{0.5}{0.5} = 1
\][/tex]
I hope this step-by-step guide helps you understand how to solve similar problems involving repeating decimals and fractions!
