Alright, let’s go through the problem step-by-step to express the fraction [tex]\(\frac{4}{17}\)[/tex] with [tex]\(-64\)[/tex] as its numerator.
1. Understand the Problem:
We have an initial fraction [tex]\(\frac{4}{17}\)[/tex]. We need to find a new fraction with the same value but with a new numerator [tex]\(-64\)[/tex]. Let's denote the unknown denominator of this new fraction as [tex]\(x\)[/tex]. So, we need to find [tex]\(x\)[/tex] such that:
[tex]\[
\frac{4}{17} = \frac{-64}{x}
\][/tex]
2. Set Up the Proportion:
To keep the two fractions equivalent, we set up the equation:
[tex]\[
\frac{4}{17} = \frac{-64}{x}
\][/tex]
3. Cross Multiply:
To solve for [tex]\(x\)[/tex], we cross multiply. Cross multiplying involves multiplying the numerator of each fraction by the denominator of the other fraction:
[tex]\[
4x = -64 \times 17
\][/tex]
4. Calculate the Right Side of the Equation:
Next, we calculate the right-hand side:
[tex]\[
-64 \times 17 = -1088
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Now, we have the equation:
[tex]\[
4x = -1088
\][/tex]
We need to solve for [tex]\(x\)[/tex] by isolating it. So, we divide both sides of the equation by 4:
[tex]\[
x = \frac{-1088}{4}
\][/tex]
This simplifies to:
[tex]\[
x = -272
\][/tex]
So, the new fraction with numerator [tex]\(-64\)[/tex] that is equivalent to [tex]\(\frac{4}{17}\)[/tex] has a denominator of [tex]\(-272\)[/tex]. Therefore, the fraction [tex]\(\frac{4}{17}\)[/tex] expressed with [tex]\(-64\)[/tex] as its numerator is:
[tex]\[
\frac{-64}{-272}
\][/tex]
This form can be simplified back to [tex]\(\frac{4}{17}\)[/tex] if needed, indicating that our new fraction [tex]\(\frac{-64}{-272} \equiv \frac{4}{17}\)[/tex] is indeed correct.
