To express the repeating decimal [tex]\(6.603603603 \ldots\)[/tex] as a rational number, we follow these steps:
1. Define the repeating decimal: Let [tex]\(x = 6.603603603 \cdots\)[/tex].
2. Isolate the repeating part: Notice that the repeating part is [tex]\(603\)[/tex]. We can use an equation to isolate the repeating part.
- Multiply [tex]\(x\)[/tex] by [tex]\(1000\)[/tex] because the repeating block has three digits (shifting the decimal three places to the right):
[tex]\[
1000x = 6603.603603603 \ldots
\][/tex]
3. Subtract to eliminate the repeating part: Subtract the original [tex]\(x\)[/tex] from this new equation:
[tex]\[
1000x - x = 6603.603603603 \ldots - 6.603603603 \cdots
\][/tex]
This simplifies to:
[tex]\[
999x = 6597
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{6597}{999}
\][/tex]
5. Simplify the fraction: To simplify [tex]\(\frac{6597}{999}\)[/tex], we need to find the greatest common divisor (GCD) of 6597 and 999.
After determining the GCD, we divide both the numerator and the denominator by this GCD to get the simplest form of the fraction.
6. Express the fraction in simplest form:
The GCD of 6597 and 999 is 9. Therefore:
[tex]\[
p = \frac{6597}{9} = 733
\][/tex]
[tex]\[
q = \frac{999}{9} = 111
\][/tex]
Hence, the repeating decimal [tex]\(6.603603603 \ldots\)[/tex] can be expressed as the rational number [tex]\(\frac{733}{111}\)[/tex].
So, the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are:
[tex]\[
\begin{array}{l}
p = 733 \text{ and} \\
q = 111
\end{array}
\][/tex]
