1Cademy - Example 7.27: Simplifying $$\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}$$ by Writing it as Division

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Procedure to Simplify a Complex Rational Expression by Writing it as Division

Example

Example 7.27: Simplifying $\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}$ by Writing it as Division

To simplify the complex rational expression $\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}$ by writing it as division, perform the following steps:

Step 1. Simplify the numerator and denominator. Find common denominators for the numerator and the denominator separately. In the numerator, the common denominator is $n+5$ . Rewrite $n$ as $\frac{n(n+5)}{n+5}$ . Subtracting gives $\frac{n(n+5)-4n}{n+5}$ . Simplifying the numerator of this expression yields $\frac{n^2+5n-4n}{n+5} = \frac{n^2+n}{n+5}$ . In the denominator, the common denominators are $n+5$ and $n-5$ . The common denominator is $(n+5)(n-5)$ . Multiply $\frac{1}{n+5}$ by $\frac{n-5}{n-5}$ and $\frac{1}{n-5}$ by $\frac{n+5}{n+5}$ . Adding them gives $\frac{1(n-5)+1(n+5)}{(n+5)(n-5)} = \frac{n-5+n+5}{(n+5)(n-5)} = \frac{2n}{(n+5)(n-5)}$ . The complex rational expression is now: $\frac{\frac{n^2+n}{n+5}}{\frac{2n}{(n+5)(n-5)}}$ .

Step 2. Rewrite as fraction division. Replace the main fraction bar with a division sign: $\frac{n^2+n}{n+5} \div \frac{2n}{(n+5)(n-5)}$

Step 3. Multiply the first fraction by the reciprocal of the second. $\frac{n^2+n}{n+5} \cdot \frac{(n+5)(n-5)}{2n}$

Step 4. Factor any expressions if possible and remove common factors. Factor the numerator of the first fraction: $n^2+n = n(n+1)$ . $\frac{n(n+1)}{n+5} \cdot \frac{(n+5)(n-5)}{2n} = \frac{n(n+1)(n+5)(n-5)}{2n(n+5)}$ Divide out the common factors of $n$ and $(n+5)$ . The simplified result is: $\frac{(n+1)(n-5)}{2}$

Updated 2026-04-30

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OpenStax Intermediate Algebra

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