To determine the number by which [tex]\(-4 \frac{9}{8}\)[/tex] should be divided to obtain [tex]\(-3 \frac{1}{2}\)[/tex], follow these steps:
1. Convert the mixed numbers to improper fractions.
For the dividend [tex]\(-4 \frac{9}{8}\)[/tex]:
[tex]\[
-4 \frac{9}{8} = -\left(4 + \frac{9}{8}\right)
\][/tex]
Convert [tex]\(4\)[/tex] to a fraction with the same denominator:
[tex]\[
-\left(\frac{32}{8} + \frac{9}{8}\right) = -\left(\frac{32 + 9}{8}\right) = -\frac{41}{8}
\][/tex]
For the divisor result [tex]\(-3 \frac{1}{2}\)[/tex]:
[tex]\[
-3 \frac{1}{2} = -\left(3 + \frac{1}{2}\right)
\][/tex]
Convert [tex]\(3\)[/tex] to a fraction with the same denominator:
[tex]\[
-\left(\frac{6}{2} + \frac{1}{2}\right) = -\left(\frac{6 + 1}{2}\right) = -\frac{7}{2}
\][/tex]
2. Divide the dividend by the divisor result.
We have:
[tex]\[
\frac{-\frac{41}{8}}{-\frac{7}{2}}
\][/tex]
Division of fractions is carried out by multiplying the first fraction by the reciprocal of the second fraction. Thus:
[tex]\[
\frac{41}{8} \times \frac{2}{7} = \frac{41 \cdot 2}{8 \cdot 7} = \frac{82}{56}
\][/tex]
Simplify [tex]\(\frac{82}{56}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{82 \div 2}{56 \div 2} = \frac{41}{28}
\][/tex]
3. Convert the improper fraction to a decimal (if necessary).
[tex]\[
\frac{41}{28} \approx 1.15
\][/tex]
Therefore, [tex]\(-4 \frac{9}{8}\)[/tex] should be divided by approximately [tex]\( 1.15 \)[/tex] to obtain [tex]\(-3 \frac{1}{2}\)[/tex].
