Perform the indicated operation. Simplify if possible.
[tex]\frac{12 a}{b}+\frac{5 b}{3}[/tex]
[tex]\frac{12 a}{b}+\frac{5 b}{3}=[/tex] $\square$ (Simplify your answer.)
Answer (1)
Find a common denominator for the two fractions, which is 3 b .
Rewrite each fraction with the common denominator: b 12 a = 3 b 36 a and 3 5 b = 3 b 5 b 2 .
Add the fractions: 3 b 36 a + 3 b 5 b 2 = 3 b 36 a + 5 b 2 .
The simplified expression is 3 b 36 a + 5 b 2 .
Explanation
Understanding the problem We are asked to add two fractions: b 12 a and 3 5 b . To do this, we need to find a common denominator.
Finding the common denominator The least common denominator (LCD) of b and 3 is 3 b . We will rewrite each fraction with this common denominator.
Rewriting the first fraction To rewrite b 12 a with a denominator of 3 b , we multiply both the numerator and the denominator by 3 : b 12 a × 3 3 = 3 b 36 a
Rewriting the second fraction To rewrite 3 5 b with a denominator of 3 b , we multiply both the numerator and the denominator by b : 3 5 b × b b = 3 b 5 b 2
Adding the fractions Now we can add the two fractions: 3 b 36 a + 3 b 5 b 2 = 3 b 36 a + 5 b 2
Final Answer The expression 3 b 36 a + 5 b 2 is simplified as much as possible since there are no common factors in the numerator and denominator. Therefore, the final answer is 3 b 36 a + 5 b 2 .
Examples
Fractions are a fundamental concept in mathematics and have numerous real-world applications. For instance, when baking, you might need to combine fractions of ingredients, like 2 1 cup of flour and 4 1 cup of sugar. Adding these fractions helps you determine the total amount of dry ingredients needed. Similarly, in construction, combining different lengths of materials often involves adding fractions to ensure accurate measurements and proper assembly. Understanding how to add fractions is essential for precise calculations in various practical scenarios.
