Express the following rational numbers in the form (p)/(q),p,q are integers, q!=0. (i) 6. bar(46) (ii) 0. bar(136) (iii) 3. bar(146) (iv) -5. bar(12)

Set Variable x: To convert a repeating decimal to a fraction, we let the repeating decimal be equal to a variable, say x. For example, if we have 6.bar(46), we can write x = 6.464646...Multiply by Power of 10: Next, we multiply x by a power of 10 that matches the length of the repeating pattern to shift the decimal point to the right, just before the pattern starts repeating again. For 6.bar(46), the repeating pattern is 46, which is two digits long, so we multiply by 10^2 (which is 100). We get 100x = 646.464646...Subtract to Eliminate Decimals: Now, we subtract the original x from this new equation to get rid of the repeating decimals. So, 100x - x = 646.464646... - 6.464646... This gives us 99x = 640.Solve for x: We then solve for x by dividing both sides of the equation by 99. So, x = 640/99. This fraction can be simplified if necessary.Convert to Fraction: For 6.bar(46), the simplified fraction is x = 640/99, which cannot be simplified further. So, the fraction form of 6.bar(46) is 640/99.Repeat for Other Decimals: We repeat the process for 0.bar(136). Let x = 0.136136136...Multiply x by 10^3 (which is 1000) because the repeating pattern 136 is three digits long. We get 1000x = 136.136136...Subtract the original x from this new equation: 1000x - x = 136.136136... - 0.136136... This gives us 999x = 136.Solve for x by dividing both sides by 999. So, x = 136/999. This fraction cannot be simplified further. So, the fraction form of 0.bar(136) is 136/999.For 3.bar(146), let x = 3.146146146...Multiply x by 10^3 (which is 1000) because the repeating pattern 146 is three digits long. We get 1000x = 3146.146146...Subtract the original x from this new equation: 1000x - x = 3146.146146... - 3.146146... This gives us 999x = 3143.Solve for x by dividing both sides by 999. So, x = 3143/999. This fraction can be simplified by dividing both numerator and denominator by 9. So, x = 349/111.For -5.bar(12), let x = -5.121212...Multiply x by 10^2 (which is 100) because the repeating pattern 12 is two digits long. We get 100x = -512.121212...Subtract the original x from this new equation: 100x - x = -512.121212... + 5.121212... This gives us 99x = -507.Solve for x by dividing both sides by 99. So, x = -507/99. This fraction can be simplified by dividing both numerator and denominator by 3. So, x = -169/33.We have now converted all the given repeating decimals to fractions. The final answers are: (i) 6.bar(46) = 640/99 (ii) 0.bar(136) = 136/999 (iii) 3.bar(146) = 349/111 (iv) -5.bar(12) = -169/33

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Express the following rational numbers in the form
(p)/(q),p,q are integers,
q!=0.
(i)
6. bar(46)
(ii)
0. bar(136)
(iii)
3. bar(146)
(iv)
-5. bar(12)

Express the following rational numbers in the form $\frac{p}{q}, p, q$ are integers, $q \neq 0$ . $\newline$ (i) $6 . \overline{46}$ $\newline$ (ii) $0 . \overline{136}$ $\newline$ (iii) $3 . \overline{146}$ $\newline$ (iv) $-5 . \overline{12}$

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Q. Express the following rational numbers in the form

\frac{p}{q}, p, q

are integers,

q \neq 0

\newline

(i)

6 . \overline{46}

\newline

(ii)

0 . \overline{136}

\newline

(iii)

3 . \overline{146}

\newline

(iv)

-5 . \overline{12}

Set Variable $x$ : To convert a repeating decimal to a fraction, we let the repeating decimal be equal to a variable, say $x$ . For example, if we have $6.\overline{46}$ , we can write $x = 6.464646\ldots$
Multiply by Power of $10$ : Next, we multiply $x$ by a power of $10$ that matches the length of the repeating pattern to shift the decimal point to the right, just before the pattern starts repeating again. For $6.\overline{46}$ , the repeating pattern is $46$ , which is two digits long, so we multiply by $10^2$ (which is $100$ ). We get $100x = 646.464646\ldots$
Subtract to Eliminate Decimals: Now, we subtract the original $x$ from this new equation to get rid of the repeating decimals. So, $100x - x = 646.464646... - 6.464646...$ This gives us $99x = 640$ .
Solve for x: We then solve for x by dividing both sides of the equation by $99$ . So, $x = \frac{640}{99}$ . This fraction can be simplified if necessary.
Convert to Fraction: For $6.\overline{46}$ , the simplified fraction is $x = \frac{640}{99}$ , which cannot be simplified further. So, the fraction form of $6.\overline{46}$ is $\frac{640}{99}$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ . For $10^3$ $0$ , let $10^3$ $1$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ . For $10^3$ $0$ , let $10^3$ $1$ Multiply $x$ by $10^3$ $3$ (which is $10^3$ $4$ ) because the repeating pattern $10^3$ $5$ is two digits long. We get $10^3$ $6$
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ . For $10^3$ $0$ , let $10^3$ $1$ Multiply $x$ by $10^3$ $3$ (which is $10^3$ $4$ ) because the repeating pattern $10^3$ $5$ is two digits long. We get $10^3$ $6$ Subtract the original $x$ from this new equation: $10^3$ $8$ This gives us $10^3$ $9$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ . For $10^3$ $0$ , let $10^3$ $1$ Multiply $x$ by $10^3$ $3$ (which is $10^3$ $4$ ) because the repeating pattern $10^3$ $5$ is two digits long. We get $10^3$ $6$ Subtract the original $x$ from this new equation: $10^3$ $8$ This gives us $10^3$ $9$ . Solve for $x$ by dividing both sides by $1000$ $1$ . So, $1000$ $2$ . This fraction can be simplified by dividing both numerator and denominator by $1000$ $3$ . So, $1000$ $4$ .
Repeat for Other Decimals: We repeat the process for $0.\overline{136}$ . Let $x = 0.136136136\ldots$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $136$ is three digits long. We get $1000x = 136.136136\ldots$ Subtract the original $x$ from this new equation: $1000x - x = 136.136136\ldots - 0.136136\ldots$ This gives us $999x = 136$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x = 0.136136136\ldots$ $2$ . This fraction cannot be simplified further. So, the fraction form of $0.\overline{136}$ is $x = 0.136136136\ldots$ $4$ . For $x = 0.136136136\ldots$ $5$ , let $x = 0.136136136\ldots$ $6$ Multiply $x$ by $10^3$ (which is $1000$ ) because the repeating pattern $x$ $0$ is three digits long. We get $x$ $1$ Subtract the original $x$ from this new equation: $x$ $3$ This gives us $x$ $4$ . Solve for $x$ by dividing both sides by $x = 0.136136136\ldots$ $1$ . So, $x$ $7$ . This fraction can be simplified by dividing both numerator and denominator by $x$ $8$ . So, $x$ $9$ . For $10^3$ $0$ , let $10^3$ $1$ Multiply $x$ by $10^3$ $3$ (which is $10^3$ $4$ ) because the repeating pattern $10^3$ $5$ is two digits long. We get $10^3$ $6$ Subtract the original $x$ from this new equation: $10^3$ $8$ This gives us $10^3$ $9$ . Solve for $x$ by dividing both sides by $1000$ $1$ . So, $1000$ $2$ . This fraction can be simplified by dividing both numerator and denominator by $1000$ $3$ . So, $1000$ $4$ . We have now converted all the given repeating decimals to fractions. The final answers are: (i) $1000$ $5$ (ii) $1000$ $6$ (iii) $1000$ $7$ (iv) $1000$ $8$

Express the following rational numbers in the form $\frac{p}{q}, p, q$ are integers, $q \neq 0$ . $\newline$ (i) $6 . \overline{46}$ $\newline$ (ii) $0 . \overline{136}$ $\newline$ (iii) $3 . \overline{146}$ $\newline$ (iv) $-5 . \overline{12}$

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