What is the quotient?

StartFraction 3 y + 2 Over 3 y EndFraction divided by StartFraction 6 y squared + 4 y Over 3 y + 2 EndFraction

To find the quotient of the given expressions, we will follow these steps:

1. Write down the given expressions.
2. Simplify each expression if possible.
3. Perform the division by multiplying by the reciprocal of the second fraction.

Given:

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

First, let's simplify the fractions.

For the first fraction:

\[
\frac{3y + 2}{3y}
\]

This fraction cannot be simplified further.

For the second fraction:

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

Factor the numerator:

\[
6y^2 + 4y = 2y(3y + 2)
\]

Thus, the fraction becomes:

\[
\frac{2y(3y + 2)}{3y + 2}
\]

Since \(3y + 2\) is common in both numerator and denominator, they cancel out:

\[
\frac{2y(3y + 2)}{3y + 2} = 2y
\]

Now, we have:

\[
\frac{3y + 2}{3y} \div 2y
\]

To divide by a fraction, we multiply by its reciprocal:

\[
\frac{3y + 2}{3y} \times \frac{1}{2y}
\]

Multiply the fractions:

\[
\frac{(3y + 2) \times 1}{3y \times 2y} = \frac{3y + 2}{6y^2}
\]

So, the quotient is:

\[
\frac{3y + 2}{6y^2}
\]

This is the simplified form of the quotient.

Do you want more details or have any questions? 

Here are 5 more questions you might want to ask:
1. How do you simplify complex fractions?
2. What is the process to factor polynomials?
3. How do you divide rational expressions?
4. Can you explain the concept of reciprocal in division?
5. How do you handle equations with variables in the denominator?

**Tip:** When simplifying fractions, always look for common factors in the numerator and denominator to cancel out.

Learn how to calculate the quotient of fractional expressions (3y + 2) / 3y divided by (6y^2 + 4y) / (3y + 2) with step-by-step instructions. This problem involves simplifying fractions, multiplying by the reciprocal, and handling algebraic expressions. Suitable for high school students.

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