To solve the problem of finding the quotient of two fractions, [tex]\(\frac{5}{31}\)[/tex] divided by [tex]\(\frac{15}{23}\)[/tex], follow these steps:
1. Division of Fractions:
Dividing by a fraction is the same as multiplying by its reciprocal. So, we need to multiply [tex]\(\frac{5}{31}\)[/tex] by the reciprocal of [tex]\(\frac{15}{23}\)[/tex], which is [tex]\(\frac{23}{15}\)[/tex].
2. Perform the Multiplication:
Multiply the numerators together and the denominators together:
[tex]\[
\text{Numerator: } 5 \times 23 = 115
\][/tex]
[tex]\[
\text{Denominator: } 31 \times 15 = 465
\][/tex]
So, the product is:
[tex]\[
\frac{5}{31} \div \frac{15}{23} = \frac{5 \times 23}{31 \times 15} = \frac{115}{465}
\][/tex]
3. Reduce the Fraction to the Lowest Terms:
Find the greatest common divisor (GCD) of 115 and 465. The GCD of 115 and 465 is 5. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{115 \div 5}{465 \div 5} = \frac{23}{93}
\][/tex]
Now, the fraction [tex]\(\frac{23}{93}\)[/tex] is in its simplest form as it cannot be reduced further.
Therefore, the solution to the given problem is:
[tex]\[
\boxed{\frac{23}{93}}
\][/tex]
So, the correct answer is:
C. [tex]\(\frac{23}{93}\)[/tex]
