Certainly! To find the quotient of [tex]\(\frac{5}{31}\)[/tex] divided by [tex]\(\frac{15}{23}\)[/tex], you need to follow several steps, which include inverting the second fraction and multiplying the numerators and denominators. Let's break this down clearly:
1. Represent the Problem: You need to divide [tex]\(\frac{5}{31}\)[/tex] by [tex]\(\frac{15}{23}\)[/tex].
2. Inversion and Multiplication: When dividing by a fraction, you multiply by its reciprocal. Thus:
[tex]\[
\frac{5}{31} \div \frac{15}{23} = \frac{5}{31} \times \frac{23}{15}
\][/tex]
3. Multiply the Numerators and Denominators: Now, we perform the multiplication:
[tex]\[
\text{Numerator: } 5 \times 23 = 115
\][/tex]
[tex]\[
\text{Denominator: } 31 \times 15 = 465
\][/tex]
So, after multiplication, we get:
[tex]\[
\frac{115}{465}
\][/tex]
4. Reduction to Lowest Terms: To simplify [tex]\(\frac{115}{465}\)[/tex] to its lowest form, we need to find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD of 115 and 465 turns out to be 5.
Dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{115 \div 5}{465 \div 5} = \frac{23}{93}
\][/tex]
5. Conclusion: The fraction [tex]\(\frac{23}{93}\)[/tex] is in its lowest terms. Comparing this with the given choices, the correct answer corresponds to:
[tex]\[
\boxed{C. \frac{23}{93}}
\][/tex]
Therefore, the best answer is:
C. [tex]\(\frac{23}{93}\)[/tex].
