To express the repeating decimal \(0.1543715437\ldots \) as a fraction in \( \frac{p}{q} \) form, follow these steps:
1. Identify the repeating decimal: The given repeating decimal is \( 0.1543715437\ldots \).
2. Set up an equation for the repeating decimal:
Let \( x = 0.1543715437\ldots \).
3. Express the repeating decimal as a fraction: The fractional representation of \( x \) must account for its repeating nature.
4. Find a common expansion form: We need to stop the repeating part. Note that the decimal repeats after 10 digits in this case. This helps in moving the decimal point to convert it into a fraction.
5. Create an equation to solve for \( x \):
By multiplying both sides of \( x = 0.1543715437\ldots \) by \( 10^{10} \) (since the period of the repeating decimal is 10 digits), we get:
[tex]\[
10^{10} \times x = 1543715437.1543715437\ldots
\][/tex]
6. Subtract the original equation from this new equation:
[tex]\[
10^{10}x - x = 1543715437.1543715437\ldots - 0.1543715437\ldots
\][/tex]
Simplify the subtraction:
[tex]\[
9999999999x = 1543715437
\][/tex]
7. Solve for \( x \):
[tex]\[
x = \frac{1543715437}{9999999999}
\][/tex]
8. Simplify the fraction:
Divide the numerator and the denominator by their greatest common divisor (GCD). In this case:
[tex]\[
\frac{1543715437}{9999999999} = \frac{1543715437}{10000000000}
\][/tex]
Since 9999999999 is assumed to be a prime factor here without extra detail needed to correct input assumptions on how precision was approximate.
Thus, the fraction form of \( 0.1543715437\ldots \) is:
[tex]\[
\frac{1543715437}{10000000000}
\][/tex]
So, the repeating decimal \( 0.1543715437\ldots \) can be expressed as the fraction:
[tex]\[
\frac{1543715437}{10000000000}
\][/tex]
This fraction is already in its simplest form.
