What is the product?
$\frac{4 n}{4 n-4} \cdot \frac{n-1}{n+1}$
A. $\frac{4 n}{n+1}$
B. $\frac{n}{n+1}$
C. $\frac{1}{n+1}$
D. $\frac{4}{n+1}$
Answer (1)
Factor out 4 from the denominator: 4 ( n − 1 ) 4 n ⋅ n + 1 n − 1 .
Cancel the common factor of 4: n − 1 n ⋅ n + 1 n − 1 .
Cancel the common factor of n − 1 : n + 1 n .
The product is n + 1 n .
Explanation
Problem Analysis We are given the expression 4 n − 4 4 n ⋅ n + 1 n − 1 and asked to simplify it. Our goal is to find the product and choose the correct answer from the list of options.
Factoring First, we factor out 4 from the denominator of the first fraction: 4 n − 4 4 n = 4 ( n − 1 ) 4 n Now the expression becomes: 4 ( n − 1 ) 4 n ⋅ n + 1 n − 1
Simplifying Next, we cancel the common factor of 4 in the numerator and the denominator of the first fraction: 4 ( n − 1 ) 4 n ⋅ n + 1 n − 1 = n − 1 n ⋅ n + 1 n − 1
Further Simplification Now, we cancel the common factor of n − 1 in the numerator and the denominator: n − 1 n ⋅ n + 1 n − 1 = n + 1 n
Final Answer Therefore, the simplified expression is n + 1 n . Comparing this to the given options, we see that it matches the second option.
Examples
In real-world applications, simplifying rational expressions like this can be useful in various fields such as physics and engineering. For example, when calculating the flow rate of fluids through pipes, the expressions often involve rational functions. Simplifying these expressions can make the calculations easier and more efficient. Also, in electrical engineering, when analyzing circuits, simplifying complex impedance expressions often involves simplifying rational expressions.
