Simplify. [tex]$\frac{8}{12 x}+\frac{7}{15 x}$[/tex] [tex]$\frac{[?]}{x}$[/tex]
Answer (1)
Find a common denominator for the fractions, which is 60 x .
Rewrite the fractions with the common denominator: 12 x 8 = 60 x 40 and 15 x 7 = 60 x 28 .
Add the fractions: 60 x 40 + 60 x 28 = 60 x 68 .
Simplify the fraction: 60 x 68 = 15 x 17 = x 17/15 . The final answer is 15 17 .
Explanation
Understanding the Problem We are given the expression 12 x 8 + 15 x 7 and asked to simplify it into the form x [ ?] . To do this, we need to find a common denominator and add the two fractions.
Finding the Common Denominator The denominators are 12 x and 15 x . The least common multiple of 12 and 15 is 60. Therefore, the least common denominator is 60 x .
Rewriting the First Fraction Rewrite the fractions with the common denominator 60 x . Multiply the numerator and denominator of the first fraction by 5: 12 x 8 = 12 x × 5 8 × 5 = 60 x 40 .
Rewriting the Second Fraction Multiply the numerator and denominator of the second fraction by 4: 15 x 7 = 15 x × 4 7 × 4 = 60 x 28 .
Adding the Fractions Add the two fractions: 60 x 40 + 60 x 28 = 60 x 40 + 28 = 60 x 68 .
Simplifying the Fraction Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 4: 60 x 68 = 60 x ÷ 4 68 ÷ 4 = 15 x 17 .
Final Answer Rewrite the simplified fraction in the form x [ ?] : 15 x 17 = x 17/15 . Therefore, the simplified expression is x 17/15 .
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. Simplifying these fractions helps you accurately measure and combine the ingredients to achieve the desired recipe outcome. Similarly, in construction, combining fractional lengths of materials is essential for precise measurements and project completion.
