Question

which expression is equivalent to \\(\frac{3x}{x+1}\\) divided by \\(x + 1\\)?
\\(\frac{x+1}{3x} \cdot \frac{x+1}{1}\\) \\(\frac{x+1}{1} \div \frac{3x}{x+1}\\) \\(\frac{3x}{x+1} \div \frac{1}{x+1}\\) \\(\frac{3x}{x+1} \cdot \frac{1}{x+1}\\)

Explanation:

Step1: Recall division of fractions rule

To divide by a fraction, multiply by its reciprocal. So, dividing by \(x + 1\) (which is \(\frac{x + 1}{1}\)) means multiplying by \(\frac{1}{x + 1}\).

Step2: Apply the rule to the given expression

The given expression is \(\frac{3x}{x + 1}\div(x + 1)\). Rewriting \(x + 1\) as \(\frac{x + 1}{1}\), we use the division rule: \(\frac{3x}{x + 1}\div\frac{x + 1}{1}=\frac{3x}{x + 1}\cdot\frac{1}{x + 1}\).

Step3: Analyze each option

• Option 1: \(\frac{x + 1}{3x}\cdot\frac{x + 1}{1}\) is not equivalent as it's the reciprocal of the correct multiplication.
• Option 2: \(\frac{x + 1}{1}\div\frac{3x}{x + 1}\) is a different operation, not the equivalent of the given division.
• Option 3: \(\frac{3x}{x + 1}\div\frac{1}{x + 1}\) is dividing by a different fraction, not equivalent to the given expression.
• Option 4: \(\frac{3x}{x + 1}\cdot\frac{1}{x + 1}\) matches the result from Step 2.

Answer:

The fourth option (the right - most one) \(\boldsymbol{\frac{3x}{x + 1}\cdot\frac{1}{x + 1}}\)