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Prove algebraically that the recurring decimal 0.681 can be written as [tex]\(\frac{15}{22}\)[/tex].
To convert the recurring decimal [tex]\(0.681818 \ldots\)[/tex] (where [tex]\(18\)[/tex] repeats) into a fraction, follow these steps:
1. Let [tex]\( x \)[/tex] represent the recurring decimal: [tex]\[
x = 0.681818 \ldots
\][/tex]
2. Express the repeating decimal part by shifting the decimal point: Since the repeating part has two digits ("18"), multiply [tex]\( x \)[/tex] by 100 to shift the decimal point two places: [tex]\[
100x = 68.181818 \ldots
\][/tex]
3. Form a second equation with [tex]\( x \)[/tex]: Now, we have two expressions for the repeating decimal: [tex]\[
x = 0.681818 \ldots
\][/tex] and [tex]\[
100x = 68.181818 \ldots
\][/tex]
4. Subtract the first equation from the second equation to eliminate the repeating part: [tex]\[
100x - x = 68.181818 \ldots - 0.681818 \ldots
\][/tex] [tex]\[
99x = 67.5
\][/tex]
5. Solve for [tex]\( x \)[/tex]: [tex]\[
x = \frac{67.5}{99}
\][/tex]
6. Convert [tex]\(67.5\)[/tex] into a fraction: Since [tex]\(67.5\)[/tex] can be written as [tex]\(\frac{675}{10}\)[/tex]: [tex]\[
x = \frac{\frac{675}{10}}{99}
\][/tex]
7. Simplify the complex fraction: Simplify the fraction by multiplying the numerator and the denominator by 10 to eliminate the decimal point: [tex]\[
x = \frac{675}{990}
\][/tex]
8. Simplify [tex]\(\frac{675}{990}\)[/tex] by finding the greatest common divisor (GCD): To simplify this fraction, find the greatest common divisor of 675 and 990. The GCD of 675 and 990 is 45.
Divide both the numerator and the denominator by their GCD: [tex]\[
\frac{675 \div 45}{990 \div 45} = \frac{15}{22}
\][/tex]
Thus, the recurring decimal [tex]\(0.681818 \ldots\)[/tex] can be written as the fraction: [tex]\[
\frac{15}{22}
\][/tex]
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