What is the quotient?

[tex]\[
\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}
\][/tex]

A. [tex]\(\frac{1}{2y}\)[/tex]

B. [tex]\(\frac{3y + 2}{6y^2}\)[/tex]

C. [tex]\(\frac{1}{y}\)[/tex]

D. [tex]\(\frac{2(3y + 2)}{3}\)[/tex]



Answer :

Sure, let's solve the problem step-by-step to find the correct quotient.

Given the expression:
[tex]\[ \frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2} \][/tex]

First, recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite the expression as:
[tex]\[ \frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y} \][/tex]

Now, multiply the numerators and the denominators:
[tex]\[ \frac{(3y + 2) \times (3y + 2)}{3y \times (6y^2 + 4y)} \][/tex]

Next, simplify the numerator:
[tex]\[ (3y + 2) \times (3y + 2) = (3y + 2)^2 \][/tex]

So, this becomes:
[tex]\[ \frac{(3y + 2)^2}{3y \times (6y^2 + 4y)} \][/tex]

Move to the denominator now; factor out a common factor in the term [tex]\(6y^2 + 4y\)[/tex]:
[tex]\[ 6y^2 + 4y = 2y(3y + 2) \][/tex]

Replacing this in the expression:
[tex]\[ \frac{(3y + 2)^2}{3y \times 2y (3y + 2)} \][/tex]

Cancel the common factor of [tex]\(3y + 2\)[/tex] from the numerator and denominator:
[tex]\[ \frac{(3y + 2)}{3y \times 2y} = \frac{(3y + 2)}{6y^2} \][/tex]

Thus, the quotient simplifies to:
[tex]\[ \frac{3y + 2}{6y^2} \][/tex]

Therefore, the answer is:
[tex]\[ \boxed{\frac{3 y + 2}{6 y^2}} \][/tex]