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Answer :
Certainly! Let's tackle each problem step-by-step, simplifying the expressions involving repeating decimals.
### 1. [tex]\( 0.\overline{25} \left(\frac{9}{5}\right) \)[/tex]
[tex]\[ 0.\overline{25} \][/tex] is a repeating decimal, which can be expressed as the fraction [tex]\(\frac{25}{99}\)[/tex].
Now, multiply this fraction by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{25}{99} \times \frac{9}{5} = \frac{225}{495} = \frac{15}{33} = \frac{5}{11} \approx 0.4545
\][/tex]
### 2. [tex]\( 0.\overline{3} \left(\frac{6}{11}\right) \)[/tex]
[tex]\[ 0.\overline{3} \][/tex] is equal to the fraction [tex]\(\frac{1}{3}\)[/tex].
Multiply [tex]\(\frac{1}{3}\)[/tex] by [tex]\(\frac{6}{11}\)[/tex]:
[tex]\[
\frac{1}{3} \times \frac{6}{11} = \frac{6}{33} = \frac{2}{11} \approx 0.1818
\][/tex]
### 3. [tex]\( 0.\overline{22} \left(\frac{3}{8}\right) \)[/tex]
[tex]\[ 0.\overline{22} \][/tex] is equivalent to the fraction [tex]\(\frac{2}{9}\)[/tex].
Multiply [tex]\(\frac{2}{9}\)[/tex] by [tex]\(\frac{3}{8}\)[/tex]:
[tex]\[
\frac{2}{9} \times \frac{3}{8} = \frac{6}{72} = \frac{1}{12} \approx 0.0833
\][/tex]
### 4. [tex]\( 2.\overline{4} \left(\frac{9}{10}\right) \)[/tex]
[tex]\[ 2.\overline{4} \][/tex] equals [tex]\(\frac{22}{9}\)[/tex] (since the repeating part is 4 over 9).
Multiply [tex]\(2.\overline{4}\)[/tex] by [tex]\(\frac{9}{10}\)[/tex]:
[tex]\[
\frac{22}{9} \times \frac{9}{10} = \frac{198}{90} = \frac{11}{5} = 2.2
\][/tex]
### 5. [tex]\( 1.7 \left(\frac{18}{20}\right) \)[/tex]
1.7 is already a decimal number.
Multiply this by [tex]\(\frac{18}{20}\)[/tex] (which simplifies to [tex]\(\frac{9}{10}\)[/tex]):
[tex]\[
1.7 \times \frac{9}{10} = 1.53
\][/tex]
### 6. [tex]\(\left(\frac{0.\overline{2}}{0.\overline{3}}\right)\)[/tex]
[tex]\[ 0.\overline{2} \][/tex] is [tex]\(\frac{2}{9}\)[/tex] and [tex]\[ 0.\overline{3} \][/tex] is [tex]\(\frac{1}{3}\)[/tex].
Divide [tex]\(\frac{2}{9}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{2}{9} \div \frac{1}{3} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3} \approx 0.6667
\][/tex]
### 7. [tex]\(\left(\frac{0.\overline{42}}{0.\overline{18}}\right)\)[/tex]
[tex]\[ 0.\overline{42} \][/tex] is [tex]\(\frac{14}{33}\)[/tex] and [tex]\[ 0.\overline{18} \][/tex] is [tex]\(\frac{2}{11}\)[/tex].
Divide [tex]\(\frac{14}{33}\)[/tex] by [tex]\(\frac{2}{11}\)[/tex]:
[tex]\[
\frac{14}{33} \div \frac{2}{11} = \frac{14}{33} \times \frac{11}{2} = \frac{154}{66} = \frac{77}{33} \approx 2.333
\][/tex]
### 8. [tex]\(\left(\frac{2.\overline{7}}{0.\overline{5}}\right)\)[/tex]
[tex]\[ 2.\overline{7} \][/tex] equals [tex]\(\frac{8}{3}\)[/tex] and [tex]\[ 0.\overline{5} \][/tex] is [tex]\(\frac{5}{9}\)[/tex].
Divide [tex]\(\frac{8}{3}\)[/tex] by [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[
\frac{8}{3} \div \frac{5}{9} = \frac{8}{3} \times \frac{9}{5} = \frac{72}{15} = \frac{24}{5} = 4.8
\][/tex]
### 9. [tex]\(\left(\frac{0.\overline{21}}{0.\overline{3}}\right)\)[/tex]
[tex]\[ 0.\overline{21} \][/tex] equals [tex]\(\frac{7}{33}\)[/tex] and [tex]\[ 0.\overline{3} \][/tex] is [tex]\(\frac{1}{3}\)[/tex].
Divide [tex]\(\frac{7}{33}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{7}{33} \div \frac{1}{3} = \frac{7}{33} \times \frac{3}{1} = \frac{21}{33} = \frac{7}{11} \approx 0.636
\][/tex]
### 10. [tex]\(\left(\frac{0.5}{0.5}\right)\)[/tex]
Both numbers are the same:
[tex]\[
\frac{0.5}{0.5} = 1.0
\][/tex]
I hope this helps! If you have any questions or need further clarification on any steps, feel free to ask!
### 1. [tex]\( 0.\overline{25} \left(\frac{9}{5}\right) \)[/tex]
[tex]\[ 0.\overline{25} \][/tex] is a repeating decimal, which can be expressed as the fraction [tex]\(\frac{25}{99}\)[/tex].
Now, multiply this fraction by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{25}{99} \times \frac{9}{5} = \frac{225}{495} = \frac{15}{33} = \frac{5}{11} \approx 0.4545
\][/tex]
### 2. [tex]\( 0.\overline{3} \left(\frac{6}{11}\right) \)[/tex]
[tex]\[ 0.\overline{3} \][/tex] is equal to the fraction [tex]\(\frac{1}{3}\)[/tex].
Multiply [tex]\(\frac{1}{3}\)[/tex] by [tex]\(\frac{6}{11}\)[/tex]:
[tex]\[
\frac{1}{3} \times \frac{6}{11} = \frac{6}{33} = \frac{2}{11} \approx 0.1818
\][/tex]
### 3. [tex]\( 0.\overline{22} \left(\frac{3}{8}\right) \)[/tex]
[tex]\[ 0.\overline{22} \][/tex] is equivalent to the fraction [tex]\(\frac{2}{9}\)[/tex].
Multiply [tex]\(\frac{2}{9}\)[/tex] by [tex]\(\frac{3}{8}\)[/tex]:
[tex]\[
\frac{2}{9} \times \frac{3}{8} = \frac{6}{72} = \frac{1}{12} \approx 0.0833
\][/tex]
### 4. [tex]\( 2.\overline{4} \left(\frac{9}{10}\right) \)[/tex]
[tex]\[ 2.\overline{4} \][/tex] equals [tex]\(\frac{22}{9}\)[/tex] (since the repeating part is 4 over 9).
Multiply [tex]\(2.\overline{4}\)[/tex] by [tex]\(\frac{9}{10}\)[/tex]:
[tex]\[
\frac{22}{9} \times \frac{9}{10} = \frac{198}{90} = \frac{11}{5} = 2.2
\][/tex]
### 5. [tex]\( 1.7 \left(\frac{18}{20}\right) \)[/tex]
1.7 is already a decimal number.
Multiply this by [tex]\(\frac{18}{20}\)[/tex] (which simplifies to [tex]\(\frac{9}{10}\)[/tex]):
[tex]\[
1.7 \times \frac{9}{10} = 1.53
\][/tex]
### 6. [tex]\(\left(\frac{0.\overline{2}}{0.\overline{3}}\right)\)[/tex]
[tex]\[ 0.\overline{2} \][/tex] is [tex]\(\frac{2}{9}\)[/tex] and [tex]\[ 0.\overline{3} \][/tex] is [tex]\(\frac{1}{3}\)[/tex].
Divide [tex]\(\frac{2}{9}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{2}{9} \div \frac{1}{3} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3} \approx 0.6667
\][/tex]
### 7. [tex]\(\left(\frac{0.\overline{42}}{0.\overline{18}}\right)\)[/tex]
[tex]\[ 0.\overline{42} \][/tex] is [tex]\(\frac{14}{33}\)[/tex] and [tex]\[ 0.\overline{18} \][/tex] is [tex]\(\frac{2}{11}\)[/tex].
Divide [tex]\(\frac{14}{33}\)[/tex] by [tex]\(\frac{2}{11}\)[/tex]:
[tex]\[
\frac{14}{33} \div \frac{2}{11} = \frac{14}{33} \times \frac{11}{2} = \frac{154}{66} = \frac{77}{33} \approx 2.333
\][/tex]
### 8. [tex]\(\left(\frac{2.\overline{7}}{0.\overline{5}}\right)\)[/tex]
[tex]\[ 2.\overline{7} \][/tex] equals [tex]\(\frac{8}{3}\)[/tex] and [tex]\[ 0.\overline{5} \][/tex] is [tex]\(\frac{5}{9}\)[/tex].
Divide [tex]\(\frac{8}{3}\)[/tex] by [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[
\frac{8}{3} \div \frac{5}{9} = \frac{8}{3} \times \frac{9}{5} = \frac{72}{15} = \frac{24}{5} = 4.8
\][/tex]
### 9. [tex]\(\left(\frac{0.\overline{21}}{0.\overline{3}}\right)\)[/tex]
[tex]\[ 0.\overline{21} \][/tex] equals [tex]\(\frac{7}{33}\)[/tex] and [tex]\[ 0.\overline{3} \][/tex] is [tex]\(\frac{1}{3}\)[/tex].
Divide [tex]\(\frac{7}{33}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{7}{33} \div \frac{1}{3} = \frac{7}{33} \times \frac{3}{1} = \frac{21}{33} = \frac{7}{11} \approx 0.636
\][/tex]
### 10. [tex]\(\left(\frac{0.5}{0.5}\right)\)[/tex]
Both numbers are the same:
[tex]\[
\frac{0.5}{0.5} = 1.0
\][/tex]
I hope this helps! If you have any questions or need further clarification on any steps, feel free to ask!
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