Questions: 1/5(1+3/5-9/10)+(2+5(2/8-3)) 1/5(1+6/5)+(7(1/5)) 8/11+8/12=16/23 -2/5(15/20-8/10)-[9/2+5(2/8-2/4)] -2/5(7/10)-[14/7+(4)] - 9/15-+[18/11] -9/15-18/11=27/26

Transcript text: \[ \begin{array}{l} \frac{1}{5}\left(1+\frac{3}{5}-\frac{9}{10}\right)+\left[2+5\left(\frac{2}{8}-3\right)\right] \\ \frac{1}{5}\left(1+\frac{6}{5}\right)+\left[7\left(\frac{1}{5}\right)\right] \\ \frac{8}{11}+\frac{8}{12}=\frac{16}{23} \end{array} \] \[ \begin{array}{l} -\frac{2}{5}\left(\frac{15}{20}-\frac{8}{10}\right)-\left[\frac{9}{2}+5\left(\frac{2}{8}-\frac{2}{4}\right)\right] \\ -\frac{2}{5}\left(\frac{7}{10}\right)-\left[\frac{14}{7}+(4)\right] \\ - \frac{9}{15}-\left[+\frac{18}{11}\right] \\ -\frac{9}{15}-\frac{18}{11}=\frac{27}{26} \end{array} \]

Solution

Solution Steps

Step 1: Simplify the first expression

The first expression is \( \frac{1}{5} \left( \frac{3}{5} + \frac{9}{10} \right) + 2 + \left( \frac{2}{8} - 3 \right) \).

First, simplify inside the parentheses: \[ \frac{3}{5} + \frac{9}{10} \] Find a common denominator (10): \[ \frac{3}{5} = \frac{6}{10} \] \[ \frac{6}{10} + \frac{9}{10} = \frac{15}{10} = \frac{3}{2} \]

Next, simplify the second set of parentheses: \[ \frac{2}{8} - 3 = \frac{1}{4} - 3 = \frac{1}{4} - \frac{12}{4} = -\frac{11}{4} \]

Step 2: Multiply and add constants

Now, multiply and add the constants: \[ \frac{1}{5} \left( \frac{3}{2} \right) + 2 + \left( -\frac{11}{4} \right) \] \[ \frac{1}{5} \times \frac{3}{2} = \frac{3}{10} \]

Step 3: Combine all terms

Combine all the terms: \[ \frac{3}{10} + 2 - \frac{11}{4} \] Convert 2 to a fraction with a common denominator (20): \[ 2 = \frac{40}{20} \] Convert \(\frac{3}{10}\) and \(\frac{11}{4}\) to have a common denominator (20): \[ \frac{3}{10} = \frac{6}{20} \] \[ \frac{11}{4} = \frac{55}{20} \]

Combine all fractions: \[ \frac{6}{20} + \frac{40}{20} - \frac{55}{20} = \frac{6 + 40 - 55}{20} = \frac{-9}{20} \]