Certainly! Let's solve the problem step-by-step.
Given the two mixed numbers:
[tex]\[ \left(-6 \frac{5}{12}\right) \quad \text{and} \quad \left(2 \frac{1}{16}\right) \][/tex]
First, we convert the mixed numbers to improper fractions.
1. Convert [tex]\(-6 \frac{5}{12}\)[/tex] to an improper fraction:
[tex]\[
-6 \frac{5}{12} = -\left(6 + \frac{5}{12}\right) = -\left(\frac{6 \times 12}{12} + \frac{5}{12}\right) = -\left(\frac{72}{12} + \frac{5}{12}\right) = -\left(\frac{72 + 5}{12}\right) = -\frac{77}{12}
\][/tex]
2. Convert [tex]\(2 \frac{1}{16}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{1}{16} = \left(2 + \frac{1}{16}\right) = \left(\frac{2 \times 16}{16} + \frac{1}{16}\right) = \left(\frac{32}{16} + \frac{1}{16}\right) = \left(\frac{32 + 1}{16}\right) = \frac{33}{16}
\][/tex]
Now, we perform the division of the two improper fractions:
[tex]\[
\left(-\frac{77}{12}\right) \div \left(\frac{33}{16}\right)
\][/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
-\frac{77}{12} \times \frac{16}{33}
\][/tex]
Next, we multiply the numerators and the denominators:
[tex]\[
-\frac{77 \times 16}{12 \times 33}
\][/tex]
Multiplying the numbers:
[tex]\[
-\frac{1232}{396}
\][/tex]
Now let's reduce the fraction to its simplest form. To do this, we find the greatest common divisor (GCD) of 1232 and 396 and divide both the numerator and denominator by the GCD.
The GCD of [tex]\(1232\)[/tex] and [tex]\(396\)[/tex] is [tex]\(44\)[/tex].
Divide both the numerator and denominator by [tex]\(44\)[/tex]:
[tex]\[
-\frac{1232 \div 44}{396 \div 44} = -\frac{28}{9}
\][/tex]
Thus, the result of the division [tex]\(\left(-6 \frac{5}{12}\right) \div \left(2 \frac{1}{16}\right)\)[/tex] in improper fraction form is:
[tex]\[
-\frac{28}{9}
\][/tex]
So, the two mixed numbers divided and reduced to their simplest form is:
[tex]\[
-\frac{28}{9}
\][/tex]
