To find the number by which [tex]\(-2 \frac{1}{3}\)[/tex] should be multiplied to get [tex]\(-8 \frac{3}{4}\)[/tex], let’s solve this step-by-step.
1. Convert the mixed numbers to improper fractions:
- For [tex]\(-2 \frac{1}{3}\)[/tex]:
[tex]\[
-2 \frac{1}{3} = -2 - \frac{1}{3}
= -\left(2 + \frac{1}{3}\right)
= -\left(\frac{6}{3} + \frac{1}{3}\right)
= -\frac{7}{3}
\][/tex]
- For [tex]\(-8 \frac{3}{4}\)[/tex]:
[tex]\[
-8 \frac{3}{4} = -8 - \frac{3}{4}
= -\left(8 + \frac{3}{4}\right)
= -\left(\frac{32}{4} + \frac{3}{4}\right)
= -\frac{35}{4}
\][/tex]
2. Set up the equation:
Let [tex]\( x \)[/tex] be the number by which [tex]\(-2 \frac{1}{3}\)[/tex] is to be multiplied. Thus:
[tex]\[
-\frac{7}{3} \times x = -\frac{35}{4}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides by [tex]\(-\frac{7}{3}\)[/tex]:
[tex]\[
x = \frac{-\frac{35}{4}}{-\frac{7}{3}}
= \frac{35}{4} \times \frac{3}{7}
= \frac{35 \times 3}{4 \times 7}
= \frac{105}{28}
\][/tex]
Simplify [tex]\(\frac{105}{28}\)[/tex]:
[tex]\[
x = \frac{105 \div 7}{28 \div 7}
= \frac{15}{4}
= 3.75
\][/tex]
Now that we know the multiplier is [tex]\(3.75\)[/tex], let’s match it with the given options:
(a) [tex]\(-4 \frac{3}{4}\)[/tex] = [tex]\(-4.75\)[/tex]
(b) [tex]\(3 \frac{3}{4}\)[/tex] = [tex]\(3.75\)[/tex]
(c) [tex]\(-3 \frac{3}{4}\)[/tex] = [tex]\(-3.75\)[/tex]
(d) [tex]\(4 \frac{3}{4}\)[/tex] = [tex]\(4.75\)[/tex]
The closest match is option (b) [tex]\(3 \frac{3}{4}\)[/tex], since [tex]\(3 \frac{3}{4} = 3.75\)[/tex].
Therefore, the number by which [tex]\(-2 \frac{1}{3}\)[/tex] should be multiplied to get [tex]\(-8 \frac{3}{4}\)[/tex] as a product is:
[tex]\( \boxed{3 \frac{3}{4}} \)[/tex]
