To solve the problem [tex]\(\frac{2}{3} \times \frac{3}{4} \div \frac{5}{8}\)[/tex], we will break it down into step-by-step operations and simplify it to its lowest terms.
### Step 1: Multiplication of Fractions
First, let's multiply the fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex].
[tex]\[
\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12}
\][/tex]
### Step 2: Simplify the Result
Now, simplify the fraction [tex]\(\frac{6}{12}\)[/tex].
[tex]\[
\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}
\][/tex]
So, [tex]\(\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}\)[/tex].
### Step 3: Division of Fractions
Next, we need to divide [tex]\(\frac{1}{2}\)[/tex] by [tex]\(\frac{5}{8}\)[/tex].
To divide by a fraction, we multiply by its reciprocal. The reciprocal of [tex]\(\frac{5}{8}\)[/tex] is [tex]\(\frac{8}{5}\)[/tex].
[tex]\[
\frac{1}{2} \div \frac{5}{8} = \frac{1}{2} \times \frac{8}{5} = \frac{1 \times 8}{2 \times 5} = \frac{8}{10}
\][/tex]
### Step 4: Simplify the Division Result
Now, simplify the fraction [tex]\(\frac{8}{10}\)[/tex].
[tex]\[
\frac{8}{10} = \frac{8 \div 2}{10 \div 2} = \frac{4}{5}
\][/tex]
Therefore, [tex]\(\frac{2}{3} \times \frac{3}{4} \div \frac{5}{8} = \frac{4}{5}\)[/tex].
### Final Reduction to Lowest Terms
Since [tex]\(\frac{4}{5}\)[/tex] is already in its lowest terms, this is our final answer.
So, the simplified result is:
[tex]\[
\frac{4}{5}
\][/tex]
