To express the repeating decimal [tex]\( 6.927927927\ldots \)[/tex] as a rational number in the form [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] have no common factors, follow these steps:
1. Define the repeating decimal:
Let [tex]\( x = 6.927927927\ldots \)[/tex].
2. Identify the repeating block:
The repeating block is [tex]\(927\)[/tex] which has 3 digits.
3. Form an equation by shifting the decimal point to the right:
Multiply [tex]\(x\)[/tex] by [tex]\(1000\)[/tex] (since the repeating block is 3 digits long, we multiply by [tex]\(10^3\)[/tex]):
[tex]\[ 1000x = 6927.927927927\ldots \][/tex]
4. Subtract the original equation from the new one:
Subtract [tex]\(x = 6.927927927\ldots\)[/tex]:
[tex]\[
1000x - x = 6927.927927927\ldots - 6.927927927\ldots
\][/tex]
[tex]\[ 999x = 6921 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Isolate [tex]\(x\)[/tex] by dividing both sides by 999:
[tex]\[ x = \frac{6921}{999} \][/tex]
6. Simplify the fraction:
To simplify [tex]\(\frac{6921}{999}\)[/tex], find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 6921 and 999 is 9.
Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{6921}{999} = \frac{6921 \div 9}{999 \div 9} = \frac{769}{111}
\][/tex]
Thus, the repeating decimal [tex]\( 6.927927927\ldots \)[/tex] can be expressed as the rational number:
[tex]\[
\frac{p}{q} = \frac{769}{111}
\][/tex]
Where [tex]\(p = 769\)[/tex] and [tex]\(q = 111\)[/tex].
[tex]\[
\begin{array}{l}
p=769 \text { and } \\
q=111
\end{array}
\][/tex]
