Sure, let's go through the process step-by-step to find the correct equivalent multiplication expression.
We are given the division of two fractions:
[tex]\[
\frac{7}{8} \div \frac{5}{6}
\][/tex]
In general, dividing by a fraction is the same as multiplying by its reciprocal. Therefore, to convert the division problem into a multiplication problem, we need to take the reciprocal of the second fraction ([tex]\(\frac{5}{6}\)[/tex]).
The reciprocal of [tex]\(\frac{5}{6}\)[/tex] is:
[tex]\[
\frac{6}{5}
\][/tex]
Now, replace the division with multiplication by the reciprocal:
[tex]\[
\frac{7}{8} \div \frac{5}{6} \implies \frac{7}{8} \cdot \frac{6}{5}
\][/tex]
So, the equivalent multiplication expression is:
[tex]\[
\frac{7}{8} \cdot \frac{6}{5}
\][/tex]
Let's validate this by checking the options provided:
1. [tex]\( \frac{7}{8} \cdot \frac{6}{5} \)[/tex]
2. [tex]\( \frac{8}{7} \cdot \frac{6}{5} \)[/tex]
3. [tex]\( \frac{7}{8} \cdot \frac{5}{6} \)[/tex]
4. [tex]\( \frac{8}{7} \cdot \frac{5}{6} \)[/tex]
The correct equivalent multiplication expression is:
[tex]\[
\frac{7}{8} \cdot \frac{6}{5}
\][/tex]
Which corresponds to the first option in the list.
So, the answer is:
[tex]\[
\boxed{\frac{7}{8} \cdot \frac{6}{5}}
\][/tex]
