To express the rational number [tex]\(\frac{-5}{6}\)[/tex] with a given numerator, we need to find the corresponding denominator for each case. The general approach is to set up a proportion that ensures the new fraction is equivalent to the original fraction.
### Part (a)
Given numerator: [tex]\(35\)[/tex]
Let's set up the proportion to find the denominator:
[tex]\[
\frac{-5}{6} = \frac{35}{d_a}
\][/tex]
We solve for [tex]\(d_a\)[/tex] by cross-multiplying:
[tex]\[
-5 \times d_a = 35 \times 6
\][/tex]
So,
[tex]\[
-5d_a = 210
\][/tex]
Next, solve for [tex]\(d_a\)[/tex]:
[tex]\[
d_a = \frac{210}{-5} = -42
\][/tex]
Therefore, the rational number [tex]\(\frac{-5}{6}\)[/tex] with numerator 35 is [tex]\(\frac{35}{-42}\)[/tex].
### Part (b)
Given numerator: [tex]\(-110\)[/tex]
Let's set up the proportion to find the denominator for this new numerator:
[tex]\[
\frac{-5}{6} = \frac{-110}{d_b}
\][/tex]
We solve for [tex]\(d_b\)[/tex] by cross-multiplying:
[tex]\[
-5 \times d_b = -110 \times 6
\][/tex]
So,
[tex]\[
-5d_b = -660
\][/tex]
Next, solve for [tex]\(d_b\)[/tex]:
[tex]\[
d_b = \frac{-660}{-5} = 132
\][/tex]
Therefore, the rational number [tex]\(\frac{-5}{6}\)[/tex] with numerator [tex]\(-110\)[/tex] is [tex]\(\frac{-110}{132}\)[/tex].
### Summary
- For numerator 35, the fraction is [tex]\(\frac{35}{-42}\)[/tex].
- For numerator [tex]\(-110\)[/tex], the fraction is [tex]\(\frac{-110}{132}\)[/tex].
