Question
Simplify. Express your answer as a single fraction in simplest form. \( \frac{10}{u+5}+\frac{5}{2 u+10} \)
Ask by Nguyen Bowen. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(\frac{25}{2(u+5)}\)
Solution
Let’s simplify the expression step by step.
1. Start with the given expression:
\[
\frac{10}{u+5}+\frac{5}{2u+10}
\]
2. Notice that the denominator in the second fraction can be factored:
\[
2u+10 = 2(u+5)
\]
So, the expression becomes:
\[
\frac{10}{u+5}+\frac{5}{2(u+5)}
\]
3. The common denominator between the two fractions is \(2(u+5)\). To rewrite the first fraction with this common denominator, multiply the numerator and denominator of the first term by 2:
\[
\frac{10}{u+5}=\frac{10\cdot 2}{2(u+5)}=\frac{20}{2(u+5)}
\]
4. Now, add the two fractions:
\[
\frac{20}{2(u+5)}+\frac{5}{2(u+5)}=\frac{20+5}{2(u+5)}=\frac{25}{2(u+5)}
\]
Thus, the simplified expression is:
\[
\frac{25}{2(u+5)}
\]
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Simplify this solution Extra Insights
To simplify the expression \( \frac{10}{u+5}+\frac{5}{2u+10} \), we first notice that \( 2u + 10 \) can be factored as \( 2(u + 5) \). Thus, we rewrite the second fraction as: \[ \frac{5}{2(u + 5)} \] Now, we will find a common denominator for the two fractions. The common denominator is \( 2(u + 5) \). We can then rewrite the first fraction with the common denominator: \[ \frac{10}{u + 5} = \frac{10 \cdot 2}{(u + 5) \cdot 2} = \frac{20}{2(u + 5)} \] Now, we can combine the two fractions: \[ \frac{20}{2(u + 5)} + \frac{5}{2(u + 5)} = \frac{20 + 5}{2(u + 5)} = \frac{25}{2(u + 5)} \] Thus, the simplified form of the expression is: \[ \frac{25}{2(u + 5)} \]
