Sure, let's solve the expression step by step:
1. Simplify the first product:
[tex]\[
\frac{2}{3} \times \frac{7}{6}
\][/tex]
To multiply two fractions, you multiply the numerators together and the denominators together:
[tex]\[
\frac{2 \times 7}{3 \times 6} = \frac{14}{18}
\][/tex]
Now, simplify [tex]\(\frac{14}{18}\)[/tex] by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 2:
[tex]\[
\frac{14 \div 2}{18 \div 2} = \frac{7}{9}
\][/tex]
So, the simplified result of the first product is [tex]\(\frac{7}{9}\)[/tex].
2. Simplify the second product:
[tex]\[
\frac{-4}{15} \times \frac{-5}{-8}
\][/tex]
First, we note that [tex]\(\frac{-5}{-8}\)[/tex] simplifies to [tex]\(\frac{5}{8}\)[/tex] because a negative divided by a negative is a positive. Now, multiply the two fractions:
[tex]\[
\frac{-4}{15} \times \frac{5}{8} = \frac{-4 \times 5}{15 \times 8} = \frac{-20}{120}
\][/tex]
Simplify [tex]\(\frac{-20}{120}\)[/tex] by finding the GCD, which is 20:
[tex]\[
\frac{-20 \div 20}{120 \div 20} = \frac{-1}{6}
\][/tex]
So, the simplified result of the second product is [tex]\(\frac{-1}{6}\)[/tex].
3. Add the two simplified results:
[tex]\[
\frac{7}{9} + \frac{-1}{6}
\][/tex]
To add these fractions, they need a common denominator. The least common multiple (LCM) of 9 and 6 is 18. Convert each fraction:
[tex]\[
\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}
\][/tex]
[tex]\[
\frac{-1}{6} = \frac{-1 \times 3}{6 \times 3} = \frac{-3}{18}
\][/tex]
Now, add the two fractions with a common denominator:
[tex]\[
\frac{14}{18} + \frac{-3}{18} = \frac{14 - 3}{18} = \frac{11}{18}
\][/tex]
4. Convert the final result to a decimal:
[tex]\[
\frac{11}{18} \approx 0.6111 \quad (\text{We typically round to 4 decimal places})
\][/tex]
So, the individual products are:
- [tex]\(\frac{2}{3} \times \frac{7}{6} \approx 0.7778\)[/tex]
- [tex]\(\frac{-4}{15} \times \frac{-5}{-8} \approx -0.1667\)[/tex]
And the final result of the expression [tex]\(\left(\frac{2}{3} \times \frac{7}{6} \right) + \left(\frac{-4}{15} \times \frac{-5}{-8}\right)\)[/tex] is approximately:
[tex]\[
0.6111
\][/tex]
