Answer :
To check if Zachary's work is correct, let's go through each step of his process:
Step 1:
Zachary begins with the division of fractions:
[tex]\[
\frac{3}{4} \div \frac{5}{6}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, the correct operation should be:
[tex]\[
\frac{3}{4} \times \frac{6}{5}
\][/tex]
Zachary mistakenly said:
[tex]\[
\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}
\][/tex]
Here, Zachary made an error in Step 1 by not finding and using the reciprocal of [tex]\(\frac{5}{6}\)[/tex] correctly.
Step 2:
If we perform the correct multiplication from Step 1:
[tex]\[
\frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}
\][/tex]
This fraction simplifies to:
[tex]\[
\frac{18}{20} = \frac{9}{10}
\][/tex]
Zachary wrote:
[tex]\[
=\frac{20}{18} = \frac{10}{9}
\][/tex]
Here, Zachary made another error: he calculated the multiplication result incorrectly, which resulted in the wrong simplified fraction.
Conclusion:
Zachary made errors in both Step 1 and Step 2. The correct final result of [tex]\(\frac{3}{4} \div \frac{5}{6}\)[/tex] should be [tex]\(\frac{9}{10}\)[/tex].
Step 1:
Zachary begins with the division of fractions:
[tex]\[
\frac{3}{4} \div \frac{5}{6}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, the correct operation should be:
[tex]\[
\frac{3}{4} \times \frac{6}{5}
\][/tex]
Zachary mistakenly said:
[tex]\[
\frac{3}{4} \div \frac{5}{6} = \frac{4}{3} \times \frac{5}{6}
\][/tex]
Here, Zachary made an error in Step 1 by not finding and using the reciprocal of [tex]\(\frac{5}{6}\)[/tex] correctly.
Step 2:
If we perform the correct multiplication from Step 1:
[tex]\[
\frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}
\][/tex]
This fraction simplifies to:
[tex]\[
\frac{18}{20} = \frac{9}{10}
\][/tex]
Zachary wrote:
[tex]\[
=\frac{20}{18} = \frac{10}{9}
\][/tex]
Here, Zachary made another error: he calculated the multiplication result incorrectly, which resulted in the wrong simplified fraction.
Conclusion:
Zachary made errors in both Step 1 and Step 2. The correct final result of [tex]\(\frac{3}{4} \div \frac{5}{6}\)[/tex] should be [tex]\(\frac{9}{10}\)[/tex].
