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Is Zachary's work correct? If not, where did he make an error?

- Zachary's work is correct.
- Zachary made an error in Step 1.
- Zachary made an error in Step 2.
- Zachary made an error in Step 3.

[Step 1]
[tex]\[\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5}\][/tex]

[Step 2]
[tex]\[= \frac{18}{20}\][/tex]

[Step 3]
[tex]\[= \frac{9}{10}\][/tex]

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].

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