To find the quotient of the fractions [tex]\( \frac{5}{31} \)[/tex] divided by [tex]\( \frac{15}{23} \)[/tex], we follow these steps:
1. Understand Division of Fractions: Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore,
[tex]\[
\frac{5}{31} \div \frac{15}{23} = \frac{5}{31} \times \frac{23}{15}
\][/tex]
2. Perform the Multiplication: Multiply the numerators together and the denominators together.
[tex]\[
\frac{5 \times 23}{31 \times 15} = \frac{115}{465}
\][/tex]
3. Reduce the Fraction: To simplify [tex]\(\frac{115}{465}\)[/tex], we need to find the greatest common divisor (GCD) of 115 and 465.
4. Calculate the GCD: The GCD of 115 and 465 is 5.
5. Divide Both Numerator and Denominator by the GCD:
[tex]\[
\frac{115 \div 5}{465 \div 5} = \frac{23}{93}
\][/tex]
Thus, the quotient of [tex]\(\frac{5}{31}\)[/tex] divided by [tex]\(\frac{15}{23}\)[/tex] when reduced to its lowest terms is [tex]\(\frac{23}{93}\)[/tex]. Therefore, the correct option is:
D) [tex]\(\frac{23}{93}\)[/tex]
