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].
