Add the rational expressions.
$\frac{5 m}{2 n}+\frac{5 m}{2 n} = $ (Simplify your answer.)
Answer (1)
Add the numerators since the denominators are the same: 5 m + 5 m = 10 m .
Write the sum as a single fraction: 2 n 10 m .
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
The simplified sum is n 5 m .
Explanation
Understanding the Problem We are given two rational expressions, 2 n 5 m and 2 n 5 m , which we need to add.
Adding the Numerators Since the denominators of both expressions are the same, we can add the numerators directly.
Combining the Fractions Adding the numerators, we have 5 m + 5 m = 10 m . So, the sum of the two rational expressions is 2 n 10 m .
Simplifying the Fraction Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, 2 n 10 m = 2 ÷ 2 10 ÷ 2 ⋅ n m = n 5 m .
Final Answer Therefore, the simplified sum of the given rational expressions is n 5 m .
Examples
Rational expressions are useful in many real-world applications, such as calculating rates, proportions, and ratios. For example, if you are mixing a solution that requires a certain ratio of ingredients, rational expressions can help you determine the correct amounts to use. Suppose you are making a cleaning solution that requires 5 parts water for every 1 part of concentrate. This ratio can be expressed as the rational expression 1 5 . If you want to make a larger batch of the solution, you can use this rational expression to calculate the amount of water and concentrate needed. Understanding how to add and simplify rational expressions is essential for solving these types of problems.
