To find the quotient of [tex]\(\frac{5}{31}\)[/tex] divided by [tex]\(\frac{15}{23}\)[/tex], let's follow these steps:
1. Understand the division of fractions: Dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction.
2. Reciprocal of the second fraction: The reciprocal of [tex]\(\frac{15}{23}\)[/tex] is [tex]\(\frac{23}{15}\)[/tex].
3. Multiply the first fraction by the reciprocal of the second fraction:
[tex]\[
\frac{5}{31} \div \frac{15}{23} = \frac{5}{31} \times \frac{23}{15}
\][/tex]
4. Multiply the numerators: Multiply the numerators of the fractions:
[tex]\[
5 \times 23 = 115
\][/tex]
5. Multiply the denominators: Multiply the denominators of the fractions:
[tex]\[
31 \times 15 = 465
\][/tex]
6. Form the resulting fraction: Place the resulting numerator over the resulting denominator:
[tex]\[
\frac{115}{465}
\][/tex]
7. Simplify the fraction: Simplify [tex]\(\frac{115}{465}\)[/tex] to its lowest terms. We can find the greatest common divisor (GCD) of 115 and 465, which is 5. Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{115 \div 5}{465 \div 5} = \frac{23}{93}
\][/tex]
So, the quotient of [tex]\(\frac{5}{31}\)[/tex] divided by [tex]\(\frac{15}{23}\)[/tex], reduced to its lowest terms, is:
[tex]\[
\frac{23}{93}
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{23}{93}} \][/tex]
From the options given:
A. [tex]\( \frac{115}{465} \)[/tex]
B. [tex]\( \frac{93}{23}, \text{ or } \frac{41}{23} \)[/tex]
C. [tex]\( \frac{75}{373} \)[/tex]
D. [tex]\( \frac{23}{93} \)[/tex]
The correct choice is:
[tex]\[ \boxed{D} \][/tex]
