To express the rational number [tex]\(\frac{-15}{16}\)[/tex] in different forms with specified numerators or denominators, we need to perform equivalent fraction operations. Here’s the step-by-step process for each case:
### (i) New numerator = -30
We want to find the new denominator when the numerator is [tex]\(-30\)[/tex].
Given the original fraction [tex]\(\frac{-15}{16}\)[/tex]:
[tex]\[
\frac{-15}{16} = \frac{-30}{x}
\][/tex]
To find the new denominator [tex]\(x\)[/tex], we solve:
[tex]\[
-15 \cdot x = -30 \cdot 16
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-30 \cdot 16}{-15} = \frac{480}{15} = 32
\][/tex]
So, the equivalent fraction is [tex]\(\frac{-30}{32}\)[/tex].
### (ii) New numerator = 75
We want to find the new denominator when the numerator is [tex]\(75\)[/tex].
Given the original fraction [tex]\(\frac{-15}{16}\)[/tex]:
[tex]\[
\frac{-15}{16} = \frac{75}{x}
\][/tex]
To find the new denominator [tex]\(x\)[/tex], we solve:
[tex]\[
-15 \cdot x = 75 \cdot 16
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{75 \cdot 16}{-15} = \frac{1200}{-15} = -80
\][/tex]
So, the equivalent fraction is [tex]\(\frac{75}{-80}\)[/tex].
### (iii) New denominator = 48
We want to find the new numerator when the denominator is [tex]\(48\)[/tex].
Given the original fraction [tex]\(\frac{-15}{16}\)[/tex]:
[tex]\[
\frac{-15}{16} = \frac{y}{48}
\][/tex]
To find the new numerator [tex]\(y\)[/tex], we solve:
[tex]\[
-15 \cdot 48 = y \cdot 16
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-15 \cdot 48}{16} = \frac{-720}{16} = -45
\][/tex]
So, the equivalent fraction is [tex]\(\frac{-45}{48}\)[/tex].
### (iv) New denominator = -96
We want to find the new numerator when the denominator is [tex]\(-96\)[/tex].
Given the original fraction [tex]\(\frac{-15}{16}\)[/tex]:
[tex]\[
\frac{-15}{16} = \frac{y}{-96}
\][/tex]
To find the new numerator [tex]\(y\)[/tex], we solve:
[tex]\[
-15 \cdot -96 = y \cdot 16
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-15 \cdot -96}{16} = \frac{1440}{16} = 90
\][/tex]
So, the equivalent fraction is [tex]\(\frac{90}{-96}\)[/tex].
Therefore, the rational number [tex]\(\frac{-15}{16}\)[/tex] expressed as specified is:
1. [tex]\(\frac{-30}{32}\)[/tex]
2. [tex]\(\frac{75}{-80}\)[/tex]
3. [tex]\(\frac{-45}{48}\)[/tex]
4. [tex]\(\frac{90}{-96}\)[/tex]
