- The division of a fraction \(\frac{A}{B}\) by an expression \(C\) follows the rule: \(\frac{A}{B} \div C = \frac{A}{B} \cdot \frac{1}{C}\).
- Applying this to the problem: \(\frac{3x}{x+1} \div (x+1) = \frac{3x}{x+1} \cdot \frac{1}{x+1}\).
- This matches the first option provided in the problem description.
- The correct equivalent expression is \(\boxed{\frac{3x}{x+1} \cdot \frac{1}{x+1}}\).
Explanation
- Analyzing the Division Operation
To determine which expression is equivalent to \(\frac{3x}{x+1}\) divided by \(x+1\), we start by analyzing the operation. We have the expression \(\frac{3x}{x+1}\) and we are instructed to divide it by \((x+1)\). In algebra, dividing a fraction by a value (or another expression) \(C\) is equivalent to multiplying that fraction by the reciprocal of \(C\). That is, \(\frac{A}{B} \div C = \frac{A}{B} \times \frac{1}{C}\).
- Applying the Rule of Division by Reciprocals
Now, we apply the division rule to our specific expression. Let \(A = 3x\), \(B = x+1\), and \(C = x+1\). The operation is \(\frac{3x}{x+1} \div (x+1)\). To express the divisor \(x+1\) as a fraction, we can write it as \(\frac{x+1}{1}\). Dividing by \(\frac{x+1}{1}\) is the same as multiplying by its reciprocal, which is \(\frac{1}{x+1}\). Therefore, the expression becomes \(\frac{3x}{x+1} \times \frac{1}{x+1}\).
- Identifying the Correct Expression
Comparing our result, \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\), with the provided options, we see that it matches the first choice exactly. Therefore, this is the correct equivalent expression. The operation is equivalent to multiplying the original fraction by \(\frac{1}{x+1}\), which simplifies to \(\frac{3x}{(x+1)^2}\), but the required task is simply to identify the equivalent multiplication form.
Examples
In real life, division by a value is akin to partitioning a total quantity into smaller equal parts. For example, if you have a certain amount of material represented by \(\frac{3x}{x+1}\) and you need to split it equally among \(x+1\) people, you are essentially multiplying that amount by the fraction \(\frac{1}{x+1}\) for each share. Understanding how to handle these algebraic fractions helps in managing resources or proportions efficiently in engineering, finance, or even simple task planning.