Alright, let's solve the expression step by step.
We need to solve the following expression:
[tex]\[
\left( \frac{2}{3} - \frac{1}{5} + \frac{1}{10} \right) \times 5 - \frac{3}{4} \times \frac{6}{5}
\][/tex]
### Step 1: Simplify the Expression Inside the Parentheses
First, we need to calculate [tex]\(\frac{2}{3} - \frac{1}{5} + \frac{1}{10}\)[/tex].
1. Calculate [tex]\(\frac{2}{3} - \frac{1}{5}\)[/tex]:
- The least common multiple of 3 and 5 is 15.
- Convert the fractions to have a common denominator:
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}
\][/tex]
- Subtract the fractions:
[tex]\[
\frac{10}{15} - \frac{3}{15} = \frac{7}{15}
\][/tex]
2. Add [tex]\(\frac{1}{10}\)[/tex] to the result:
- The least common multiple of 15 and 10 is 30.
- Convert the fractions to have a common denominator:
[tex]\[
\frac{7}{15} = \frac{7 \times 2}{15 \times 2} = \frac{14}{30}
\][/tex]
[tex]\[
\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}
\][/tex]
- Add the fractions:
[tex]\[
\frac{14}{30} + \frac{3}{30} = \frac{17}{30}
\][/tex]
So, the expression inside the parentheses simplifies to [tex]\(\frac{17}{30}\)[/tex].
### Step 2: Multiply by 5
Now, we multiply [tex]\(\frac{17}{30}\)[/tex] by 5:
[tex]\[
\frac{17}{30} \times 5 = \frac{17 \times 5}{30} = \frac{85}{30} = \frac{17}{6} \approx 2.833333333333333
\][/tex]
### Step 3: Calculate the Second Part of the Expression
We need to calculate [tex]\(\frac{3}{4} \times \frac{6}{5}\)[/tex]:
1. Multiply the fractions:
[tex]\[
\frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20} = \frac{9}{10} \approx 0.9
\][/tex]
### Step 4: Combine Both Parts
Finally, subtract the second part from the first part:
[tex]\[
\frac{17}{6} - \frac{9}{10}
\][/tex]
1. Equalize their denominators:
- The least common multiple of 6 and 10 is 30:
[tex]\[
\frac{17}{6} = \frac{17 \times 5}{6 \times 5} = \frac{85}{30}
\][/tex]
[tex]\[
\frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30}
\][/tex]
- Subtract the fractions:
[tex]\[
\frac{85}{30} - \frac{27}{30} = \frac{58}{30} = \frac{29}{15} \approx 1.933333333333333
\][/tex]
Thus, the final result of the expression is:
[tex]\[
\boxed{1.933333333333333}
\][/tex]
