To express [tex]\(4.0 \overline{35}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers, we need to convert the repeating decimal to a fraction. Let's go through the steps in detail:
1. Define the repeating decimal:
Let [tex]\( x = 4.0 \overline{35} \)[/tex]. This means:
[tex]\[
x = 4.03535353535\ldots
\][/tex]
2. Separate the non-repeating part from the repeating part:
The non-repeating part is 4.0, and the repeating part is [tex]\(0.\overline{35}\)[/tex].
Let [tex]\( y = 0.\overline{35} \)[/tex], so that:
[tex]\[
x = 4 + y
\][/tex]
3. Express [tex]\(y = 0.\overline{35}\)[/tex] as a fraction:
- Let [tex]\( y = 0.\overline{35} \)[/tex].
- To eliminate the repeating part, multiply [tex]\( y \)[/tex] by 100 (since the repeating part is two digits long):
[tex]\[
100y = 35.35353535\ldots
\][/tex]
- Now, we set up an equation with the original [tex]\( y \)[/tex]:
[tex]\[
100y = 35.35353535\ldots
\][/tex]
[tex]\[
y = 0.35353535\ldots
\][/tex]
- Subtract the second equation from the first to eliminate the repeating part:
[tex]\[
100y - y = 35.35353535\ldots - 0.35353535\ldots
\][/tex]
[tex]\[
99y = 35
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{35}{99}
\][/tex]
4. Combine the integer part and the fraction:
Recall that:
[tex]\[
x = 4 + y = 4 + \frac{35}{99}
\][/tex]
5. Combine the terms into a single fraction:
First, make the integer 4 into a fraction with the same denominator:
[tex]\[
4 = \frac{4 \cdot 99}{99} = \frac{396}{99}
\][/tex]
Then, add the fractions:
[tex]\[
x = \frac{396}{99} + \frac{35}{99} = \frac{396 + 35}{99} = \frac{431}{99}
\][/tex]
6. Simplify the fraction (if possible):
Since 431 and 99 are both prime relative to each other (they have no common factors other than 1), the fraction is already in its simplest form.
Hence, the repeating decimal [tex]\(4.0 \overline{35}\)[/tex] expressed in the form [tex]\(\frac{p}{q}\)[/tex] is:
[tex]\[
\frac{431}{99}
\][/tex]
