To solve the expression [tex]\(\frac{2}{3} \div 5 \frac{3}{4}\)[/tex], let's go through it step-by-step:
1. Convert the Mixed Number to an Improper Fraction:
First, convert the mixed number [tex]\(5 \frac{3}{4}\)[/tex] to an improper fraction.
[tex]\[
5 \frac{3}{4} = \frac{5 \cdot 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}
\][/tex]
2. Reciprocal of the Fraction:
To divide by a fraction, we multiply by its reciprocal. The reciprocal of [tex]\(\frac{23}{4}\)[/tex] is [tex]\(\frac{4}{23}\)[/tex].
3. Rewrite the Division as a Multiplication:
The given expression [tex]\(\frac{2}{3} \div \frac{23}{4}\)[/tex] becomes:
[tex]\[
\frac{2}{3} \cdot \frac{4}{23}
\][/tex]
4. Multiply the Fractions:
Multiply the numerators and the denominators:
[tex]\[
\frac{2 \cdot 4}{3 \cdot 23} = \frac{8}{69}
\][/tex]
So, the correct operation simplifies to [tex]\((2/3) \cdot (4/23)\)[/tex], which corresponds to [tex]\( \frac{8}{69} \)[/tex].
Now, let's check the given options to see which one matches:
- Option (A): [tex]\( \frac{3}{2} \cdot \frac{23}{3} \)[/tex]
This doesn't match our calculation.
- Option (B): [tex]\( \frac{3}{2} \cdot \frac{2}{4} \)[/tex]
This doesn't match our calculation.
- Option (C): [tex]\( \frac{2}{3} \cdot \frac{3}{23} \)[/tex]
This doesn't match our calculation.
- Option (D): [tex]\( \frac{2}{3} \cdot \frac{3}{4} \)[/tex]
This doesn't match our calculation.
Since none of the options exactly match [tex]\(\frac{2}{3} \cdot \frac{4}{23}\)[/tex], the correct operation we derived is:
[tex]\((2/3) \cdot (4/23)\)[/tex].
Thus, the answer is not listed among the provided options.
