Certainly! Let's express the mixed recurring decimal [tex]\( 11.\overline{142} \)[/tex] in the form of a fraction [tex]\(\frac{p}{q}\)[/tex], and then find the value of [tex]\( p - q \)[/tex].
### Step-by-Step Solution:
1. Letting the decimal be [tex]\( x \)[/tex]:
Let [tex]\( x = 11.\overline{142} \)[/tex]. This means [tex]\( x \)[/tex] represents the repeating decimal [tex]\( 11.142142142\ldots \)[/tex].
2. Shifting the decimal point:
To eliminate the repeating part, we multiply [tex]\( x \)[/tex] by 1000 (since the repeating sequence has 3 digits: 142):
[tex]\[
1000x = 11412.142142142\ldots
\][/tex]
3. Subtracting the original [tex]\( x \)[/tex] from this equation:
Now, subtract the original [tex]\( x \)[/tex] from the above equation to eliminate the repeating decimal:
[tex]\[
1000x - x = 11412.142142142\ldots - 11.142142142\ldots
\][/tex]
Simplifying this, we get:
[tex]\[
999x = 11401
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
Divide both sides of the equation by 999:
[tex]\[
x = \frac{11401}{999}
\][/tex]
5. Simplifying the fraction:
In this particular case, [tex]\(\frac{11401}{999}\)[/tex] is already in its simplest form as there are no common factors other than 1 between 11401 and 999.
6. Identifying [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
From the fraction [tex]\(\frac{11401}{999}\)[/tex], we have:
[tex]\[
p = 11401, \quad q = 999
\][/tex]
7. Finding [tex]\( p - q \)[/tex]:
Now, calculate [tex]\( p - q \)[/tex]:
[tex]\[
p - q = 11401 - 999 = 10402
\][/tex]
### Conclusion:
- The mixed recurring decimal [tex]\( 11.\overline{142} \)[/tex] can be expressed as the fraction [tex]\(\frac{11401}{999}\)[/tex].
- The value of [tex]\( p - q \)[/tex] is [tex]\( 10402 \)[/tex].
