Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's simplify the expression [tex]\(\frac{2}{3} + \frac{-3}{5} + \frac{1}{6} + \frac{-8}{15}\)[/tex].
Step 1: Identify the fractions:
[tex]\[ \frac{2}{3}, \quad \frac{-3}{5}, \quad \frac{1}{6}, \quad \frac{-8}{15} \][/tex]
Step 2: Determine the least common denominator (LCD) for all the fractions.
The denominators are [tex]\(3\)[/tex], [tex]\(5\)[/tex], [tex]\(6\)[/tex], and [tex]\(15\)[/tex].
- The prime factorization of [tex]\(3\)[/tex] is [tex]\(3\)[/tex].
- The prime factorization of [tex]\(5\)[/tex] is [tex]\(5\)[/tex].
- The prime factorization of [tex]\(6\)[/tex] is [tex]\(2 \times 3\)[/tex].
- The prime factorization of [tex]\(15\)[/tex] is [tex]\(3 \times 5\)[/tex].
The LCD is the smallest number that each of these numbers can divide into, which includes each prime factor the greatest number of times it occurs in any factorization:
[tex]\[ \text{LCD} = 2 \times 3 \times 5 = 30 \][/tex]
Step 3: Convert each fraction to an equivalent fraction with the LCD as the denominator:
[tex]\[ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \][/tex]
[tex]\[ \frac{-3}{5} = \frac{-3 \times 6}{5 \times 6} = \frac{-18}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \][/tex]
[tex]\[ \frac{-8}{15} = \frac{-8 \times 2}{15 \times 2} = \frac{-16}{30} \][/tex]
Step 4: Add the fractions:
[tex]\[ = \frac{20}{30} + \frac{-18}{30} + \frac{5}{30} + \frac{-16}{30} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ = \frac{20 + (-18) + 5 + (-16)}{30} \][/tex]
[tex]\[ = \frac{20 - 18 + 5 - 16}{30} \][/tex]
[tex]\[ = \frac{-9}{30} \][/tex]
Step 5: Simplify the final fraction:
The greatest common divisor (GCD) of [tex]\(-9\)[/tex] and [tex]\(30\)[/tex] is [tex]\(3\)[/tex]. Divide the numerator and the denominator by their GCD:
[tex]\[ = \frac{-9 \div 3}{30 \div 3} \][/tex]
[tex]\[ = \frac{-3}{10} \][/tex]
Thus, the simplified sum of the fractions [tex]\(\frac{2}{3} + \frac{-3}{5} + \frac{1}{6} + \frac{-8}{15}\)[/tex] is:
[tex]\[ -0.3 \][/tex]
So, the simplified result of the expression [tex]\(\frac{2}{3} + \frac{-3}{5} + \frac{1}{6} + \frac{-8}{15}\)[/tex] is [tex]\(-0.3\)[/tex].
Step 1: Identify the fractions:
[tex]\[ \frac{2}{3}, \quad \frac{-3}{5}, \quad \frac{1}{6}, \quad \frac{-8}{15} \][/tex]
Step 2: Determine the least common denominator (LCD) for all the fractions.
The denominators are [tex]\(3\)[/tex], [tex]\(5\)[/tex], [tex]\(6\)[/tex], and [tex]\(15\)[/tex].
- The prime factorization of [tex]\(3\)[/tex] is [tex]\(3\)[/tex].
- The prime factorization of [tex]\(5\)[/tex] is [tex]\(5\)[/tex].
- The prime factorization of [tex]\(6\)[/tex] is [tex]\(2 \times 3\)[/tex].
- The prime factorization of [tex]\(15\)[/tex] is [tex]\(3 \times 5\)[/tex].
The LCD is the smallest number that each of these numbers can divide into, which includes each prime factor the greatest number of times it occurs in any factorization:
[tex]\[ \text{LCD} = 2 \times 3 \times 5 = 30 \][/tex]
Step 3: Convert each fraction to an equivalent fraction with the LCD as the denominator:
[tex]\[ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \][/tex]
[tex]\[ \frac{-3}{5} = \frac{-3 \times 6}{5 \times 6} = \frac{-18}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \][/tex]
[tex]\[ \frac{-8}{15} = \frac{-8 \times 2}{15 \times 2} = \frac{-16}{30} \][/tex]
Step 4: Add the fractions:
[tex]\[ = \frac{20}{30} + \frac{-18}{30} + \frac{5}{30} + \frac{-16}{30} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ = \frac{20 + (-18) + 5 + (-16)}{30} \][/tex]
[tex]\[ = \frac{20 - 18 + 5 - 16}{30} \][/tex]
[tex]\[ = \frac{-9}{30} \][/tex]
Step 5: Simplify the final fraction:
The greatest common divisor (GCD) of [tex]\(-9\)[/tex] and [tex]\(30\)[/tex] is [tex]\(3\)[/tex]. Divide the numerator and the denominator by their GCD:
[tex]\[ = \frac{-9 \div 3}{30 \div 3} \][/tex]
[tex]\[ = \frac{-3}{10} \][/tex]
Thus, the simplified sum of the fractions [tex]\(\frac{2}{3} + \frac{-3}{5} + \frac{1}{6} + \frac{-8}{15}\)[/tex] is:
[tex]\[ -0.3 \][/tex]
So, the simplified result of the expression [tex]\(\frac{2}{3} + \frac{-3}{5} + \frac{1}{6} + \frac{-8}{15}\)[/tex] is [tex]\(-0.3\)[/tex].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
