Certainly! To express the rational number [tex]\(\frac{6}{-11}\)[/tex] with a new numerator of 30, we need to find the corresponding denominator.
Step-by-step, here is what we do:
1. Identify the original rational number:
The given rational number is [tex]\(\frac{6}{-11}\)[/tex].
2. Set up the new fraction with the required numerator:
We need to express this fraction so that the numerator becomes 30. Let the new fraction be [tex]\(\frac{30}{x}\)[/tex], where [tex]\(x\)[/tex] is the unknown denominator we need to find.
3. Use the property of equivalent fractions:
Two fractions are equivalent if their cross products are equal. This means:
[tex]\[
\frac{6}{-11} = \frac{30}{x}
\][/tex]
4. Set up the equation based on the equivalence of the fractions:
[tex]\[
6 \times x = 30 \times (-11)
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
6x = -330
\][/tex]
To isolate [tex]\(x\)[/tex], divide both sides by 6:
[tex]\[
x = \frac{-330}{6}
\][/tex]
Simplify the right-hand side:
[tex]\[
x = -55
\][/tex]
6. Write the new rational number:
With the new numerator and denominator, the fraction becomes:
[tex]\[
\frac{30}{-55}
\][/tex]
Therefore, the rational number [tex]\(\frac{6}{-11}\)[/tex] expressed with a numerator of 30 is [tex]\(\frac{30}{-55}\)[/tex].
