Express $\frac{2}{6 x}+\frac{2}{8}$ as a single fraction in its lowest terms.
Answer (1)
Simplify the fractions: 6 x 2 = 3 x 1 and 8 2 = 4 1 .
Find a common denominator: The common denominator for 3 x 1 and 4 1 is 12 x .
Rewrite the fractions with the common denominator: 3 x 1 = 12 x 4 and 4 1 = 12 x 3 x .
Add the fractions: 12 x 4 + 12 x 3 x = 12 x 4 + 3 x .
The final answer is 12 x 3 x + 4 .
Explanation
Problem Analysis We are given the expression 6 x 2 + 8 2 and we want to express it as a single fraction in its lowest terms.
Simplifying Fractions First, let's simplify each fraction separately. We have 6 x 2 = 3 x 1 and 8 2 = 4 1 . So the expression becomes 3 x 1 + 4 1 .
Finding Common Denominator To add these two fractions, we need to find a common denominator. The least common multiple of 3 x and 4 is 12 x . Therefore, we will rewrite each fraction with the common denominator 12 x .
Rewriting Fractions We have 3 x 1 = 3 x × 4 1 × 4 = 12 x 4 and 4 1 = 4 × 3 x 1 × 3 x = 12 x 3 x .
Adding Fractions Now we can add the two fractions: 12 x 4 + 12 x 3 x = 12 x 4 + 3 x .
Final Answer The expression is now a single fraction. We need to check if the fraction can be simplified further. The numerator is 4 + 3 x and the denominator is 12 x . There are no common factors between 4 + 3 x and 12 x , so the fraction is in its lowest terms. Therefore, the final answer is 12 x 3 x + 4 .
Examples
Fractions are a fundamental concept in mathematics and are used in various real-life situations. For example, when you are cooking and need to combine different ingredients in specific proportions, you are using fractions. If a recipe calls for 3 1 cup of flour and 4 1 cup of sugar, you need to find a common denominator to add these fractions together, just like we did in the problem. This ensures that you get the correct ratio of ingredients in your recipe, leading to a delicious outcome. Understanding how to manipulate and combine fractions is essential for accurate measurements and successful cooking!
