Evaluate. Write your answer as a fraction. [tex]$\frac{4}{15}-\frac{4}{5}+\frac{1}{2}$[/tex]
Answer (1)
Find the least common denominator (LCD) of the fractions, which is 30.
Convert each fraction to an equivalent fraction with the LCD: 15 4 = 30 8 , 5 4 = 30 24 , 2 1 = 30 15 .
Combine the fractions: 30 8 − 30 24 + 30 15 = 30 8 − 24 + 15 .
Simplify the numerator to get the final answer: 30 − 1 = − 30 1 .
The final answer is − 30 1 .
Explanation
Understanding the problem We are asked to evaluate the expression 15 4 − 5 4 + 2 1 and write the answer as a fraction.
Finding a common denominator To evaluate this expression, we need to find a common denominator for the three fractions. The denominators are 15, 5, and 2. The least common multiple (LCM) of these three numbers is 30.
Rewriting the fractions Now we rewrite each fraction with the common denominator of 30:
15 4 = 15 × 2 4 × 2 = 30 8
5 4 = 5 × 6 4 × 6 = 30 24
2 1 = 2 × 15 1 × 15 = 30 15
Substituting the equivalent fractions Now we substitute these equivalent fractions back into the original expression:
30 8 − 30 24 + 30 15
Combining the fractions Combine the fractions:
30 8 − 24 + 15
Simplifying the numerator Simplify the numerator:
8 − 24 + 15 = − 16 + 15 = − 1
Final result So the final result is:
30 − 1 = − 30 1
Examples
Fractions are used in everyday life, such as when calculating proportions in recipes, determining discounts while shopping, or understanding probabilities in games. For example, if a recipe calls for 2 1 cup of flour and you only want to make half the recipe, you need to calculate 2 1 of 2 1 , which is 4 1 cup. Understanding how to add, subtract, multiply, and divide fractions is essential for accurate measurements and calculations in various real-world scenarios.
