Question

what is the quotient?\\(\frac{3y + 2}{3y} div \frac{6y^2 + 4y}{3y + 2}\\)\\(\frac{2(3y + 2)}{3}\\) \\(\frac{1}{2y}\\) \\(\frac{1}{y}\\) \\(\frac{3y + 2}{6y^2}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide by a fraction, multiply by its reciprocal. So, $\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}$ becomes $\frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}$.

Step2: Factor the denominator

Factor $6y^2 + 4y$. We can factor out a $2y$: $6y^2 + 4y = 2y(3y + 2)$.

Step3: Multiply the fractions

Now we have $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$. Multiply the numerators and denominators: $\frac{(3y + 2)(3y + 2)}{3y \times 2y(3y + 2)}$.

Step4: Simplify the fraction

Cancel out one $(3y + 2)$ from the numerator and denominator: $\frac{3y + 2}{6y^2}$. Wait, no, wait. Wait, let's do it again. Wait, numerator is $(3y + 2)(3y + 2)$, denominator is $3y \times 2y(3y + 2)$. So we can cancel one $(3y + 2)$: $\frac{3y + 2}{3y \times 2y}$. Wait, no, that's not right. Wait, original factoring: $6y^2 + 4y = 2y(3y + 2)$. So the denominator is $3y \times 2y(3y + 2)$, numerator is $(3y + 2)(3y + 2)$. So cancel one $(3y + 2)$: $\frac{3y + 2}{3y \times 2y}$. Wait, no, that's $\frac{3y + 2}{6y^2}$? Wait, no, wait, maybe I made a mistake. Wait, let's start over.

Wait, the reciprocal of $\frac{6y^2 + 4y}{3y + 2}$ is $\frac{3y + 2}{6y^2 + 4y}$. So the expression is $\frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}$. Then factor $6y^2 + 4y$ as $2y(3y + 2)$. So now we have $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$. Now, multiply numerators: $(3y + 2)(3y + 2)$, denominators: $3y \times 2y(3y + 2)$. Now, cancel one $(3y + 2)$ from numerator and denominator: $\frac{3y + 2}{3y \times 2y}$. Wait, no, that's $\frac{3y + 2}{6y^2}$? No, wait, no, the numerator after canceling is $(3y + 2)$, denominator is $3y \times 2y = 6y^2$? Wait, no, that can't be. Wait, maybe I messed up the reciprocal. Wait, the original problem is $\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}$. So reciprocal is $\frac{3y + 2}{6y^2 + 4y}$. So multiplying: $\frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}$. Then factor $6y^2 + 4y = 2y(3y + 2)$. So now it's $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$. Now, multiply numerators: $(3y + 2)(3y + 2)$, denominators: $3y \times 2y(3y + 2)$. Now, cancel one $(3y + 2)$: $\frac{3y + 2}{3y \times 2y} = \frac{3y + 2}{6y^2}$? Wait, but that's one of the options? Wait, no, the options are $\frac{2(3y + 2)}{3}$, $\frac{1}{2y}$, $\frac{1}{y}$, $\frac{3y + 2}{6y^2}$. Wait, but maybe I made a mistake. Wait, let's try again.

Wait, maybe the factoring is wrong. Wait, $6y^2 + 4y = 2y(3y + 2)$. So the denominator is $2y(3y + 2)$, so the reciprocal is $\frac{3y + 2}{2y(3y + 2)}$. Then multiplying by $\frac{3y + 2}{3y}$: $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$. Now, numerator: $(3y + 2)(3y + 2)$, denominator: $3y \times 2y(3y + 2)$. Cancel one $(3y + 2)$: $\frac{3y + 2}{3y \times 2y} = \frac{3y + 2}{6y^2}$. Wait, but that's option D? Wait, but let's check the options again. The options are:

1. $\frac{2(3y + 2)}{3}$

2. $\frac{1}{2y}$

3. $\frac{1}{y}$

4. $\frac{3y + 2}{6y^2}$

Wait, but maybe I made a mistake. Wait, let's do the division again. Wait, $\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2} = \frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}$. Factor $6y^2 + 4y = 2y(3y + 2)$. So now we have $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$. Now, multiply numerators: $(3y + 2)(3y + 2)$, denominators: $3y \times 2y(3y + 2)$. Now, cancel one $(3y + 2)$: $\frac{3y + 2}{3y \times 2y} = \frac{3y + 2}{6y^2}$. Wait, but that's option D. But let's check with another approach. Let's substitute a value for y. Let's take y = 1. Then original expression: $\frac{3(1) + 2}{3(1)} \div \frac{6(1)^2 + 4(1)}{3(1) + 2} = \frac{5}{3} \div \frac{10}{5} = \frac{5}{3} \div 2 = \frac{5}{6}$. Now check option D: $\frac{3(1) + 2}{6(1)^2} = \frac{5}{6}$. Which matches. Now check option B: $\frac{1}{2(1)} = \frac{1}{2}$, which doesn't match. Option C: $\frac{1}{1} = 1$, no. Option A: $\frac{2(5)}{3} = \frac{10}{3}$, no. So the correct answer is $\frac{3y + 2}{6y^2}$, which is the fourth option. Wait, but in the options, the fourth is $\frac{3y + 2}{6y^2}$. So that's the answer.

Wait, but let's do the simplification again. So:

$\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}$

= $\frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}$

Factor denominator of the second fraction: $6y^2 + 4y = 2y(3y + 2)$

So:

= $\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}$

Multiply numerators and denominators:

= $\frac{(3y + 2)(3y + 2)}{3y \times 2y(3y + 2)}$

Cancel one $(3y + 2)$ from numerator and denominator:

= $\frac{3y + 2}{6y^2}$

Yes, that's correct. So the quotient is $\frac{3y + 2}{6y^2}$, which is the fourth option (the one at the bottom left).

Answer:

$\frac{3y + 2}{6y^2}$ (the option with $\frac{3y + 2}{6y^2}$)