To find the value of [tex]\(\frac{3}{5}+\left(-\frac{1}{2}\right)-\frac{2}{3} \div \frac{8}{9}\)[/tex], we will follow these steps:
1. Simplify the division:
We need to divide [tex]\(\frac{2}{3}\)[/tex] by [tex]\(\frac{8}{9}\)[/tex]. Dividing by a fraction is the same as multiplying by its reciprocal. Thus:
[tex]\[
\frac{2}{3} \div \frac{8}{9} = \frac{2}{3} \times \frac{9}{8}
\][/tex]
To multiply two fractions, we multiply the numerators and the denominators:
[tex]\[
\frac{2 \times 9}{3 \times 8} = \frac{18}{24}
\][/tex]
Simplify [tex]\(\frac{18}{24}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 6:
[tex]\[
\frac{18 \div 6}{24 \div 6} = \frac{3}{4}
\][/tex]
2. Perform the remaining operations:
We now substitute [tex]\(\frac{3}{4}\)[/tex] for [tex]\(\frac{2}{3} \div \frac{8}{9}\)[/tex] in the original expression:
[tex]\[
\frac{3}{5} + \left(-\frac{1}{2}\right) - \frac{3}{4}
\][/tex]
3. Find a common denominator:
To add and subtract these fractions, they need a common denominator. The denominators are 5, 2, and 4. The least common multiple (LCM) of these numbers is 20. Convert each fraction:
[tex]\[
\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}
\][/tex]
[tex]\[
-\frac{1}{2} = -\frac{1 \times 10}{2 \times 10} = -\frac{10}{20}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}
\][/tex]
4. Combine the fractions:
Now that we have a common denominator, we can combine:
[tex]\[
\frac{12}{20} + \left(-\frac{10}{20}\right) - \frac{15}{20}
\][/tex]
[tex]\[
= \frac{12 - 10 - 15}{20}
\][/tex]
[tex]\[
= \frac{12 - 10 - 15}{20}
\][/tex]
[tex]\[
= \frac{-13}{20}
\][/tex]
Therefore, the value of [tex]\(\frac{3}{5}+\left(-\frac{1}{2}\right)-\frac{2}{3} \div \frac{8}{9}\)[/tex] is
[tex]\[
\boxed{-\frac{13}{20}}
\][/tex]
