Answer :
Let's solve each of the given problems step by step:
### Problem 10:
[tex]\[ \left(\frac{1}{6} + \frac{2}{7} - \frac{1}{5}\right) - \left(\frac{1}{2} + \frac{1}{5}\right) \][/tex]
First, calculate the individual terms inside each parenthesis:
[tex]\[ \frac{1}{6} + \frac{2}{7} - \frac{1}{5} \][/tex]
[tex]\[ \frac{1}{2} + \frac{1}{5} \][/tex]
Now, perform the operations step by step and then combine results:
[tex]\[ \left(\frac{1}{6} + \frac{2}{7} - \frac{1}{5}\right) \approx 0.480952 - 0.2 = 0.280952 \][/tex]
[tex]\[ \frac{1}{2} + \frac{1}{5} \approx 0.5 + 0.2 = 0.7 \][/tex]
Subtract the results of these two calculations:
[tex]\[ 0.280952 - 0.7 \approx -0.447619 \][/tex]
So, the result for problem 10 is:
[tex]\[ -0.447619 \][/tex]
### Problem 11:
[tex]\[ \left(3 \frac{1}{5} + 2 \frac{1}{6} - \frac{2}{3}\right) - \left(1 \frac{2}{5} + \frac{3}{2}\right) \][/tex]
First, convert mixed numbers to improper fractions and calculate:
[tex]\[ 3 \frac{1}{5} = 3 + \frac{1}{5} \approx 3.2 \][/tex]
[tex]\[ 2 \frac{1}{6} = 2 + \frac{1}{6} \approx 2.167 \][/tex]
[tex]\[ 1 \frac{2}{5} = 1 + \frac{2}{5} \approx 1.4 \][/tex]
Now, finish the calculations inside the parenthesis:
[tex]\[ 3.2 + 2.167 - \frac{2}{3} \approx 3.2 + 2.167 - 0.667 \approx 4.7 \][/tex]
[tex]\[ 1.4 + \frac{3}{2} \approx 1.4 + 1.5 = 2.9 \][/tex]
Subtract the results:
[tex]\[ 4.7 - 2.9 \approx 1.8 \][/tex]
So, the result for problem 11 is:
[tex]\[ 1.8 \][/tex]
### Problem 12:
[tex]\[ \left(7 \frac{3}{5} + \frac{5}{7} - \frac{1}{5}\right) + \left(\frac{1}{25} - \frac{1}{3}\right) \][/tex]
Convert and solve each term individually:
[tex]\[ 7 \frac{3}{5} = 7 + \frac{3}{5} \approx 7.6 \][/tex]
[tex]\[ \frac{5}{7} \approx 0.714 \][/tex]
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
Calculate within the first parenthesis:
[tex]\[ 7.6 + 0.714 - 0.2 \approx 8.114 \][/tex]
Now, solve the second parenthesis:
[tex]\[ \frac{1}{25} - \frac{1}{3} \approx 0.04 - 0.333 \approx -0.293 \][/tex]
Add the results of these two parenthesis:
[tex]\[ 8.114 - 0.293 \approx 7.821 \][/tex]
So, the result for problem 12 is:
[tex]\[ 7.821 \][/tex]
### Problem 13:
[tex]\[ \left(\frac{2}{3} + \frac{1}{2} + \frac{1}{8}\right) - \left(\frac{1}{2} + \frac{2}{3} - \frac{1}{8}\right) - \left(\frac{1}{4} - \frac{1}{2}\right) \][/tex]
Calculate within each parenthesis individually:
[tex]\[ \frac{2}{3} + \frac{1}{2} + \frac{1}{8} \approx 0.667 + 0.5 + 0.125 \approx 1.292 \][/tex]
[tex]\[ \frac{1}{2} + \frac{2}{3} - \frac{1}{8} \approx 0.5 + 0.667 - 0.125 \approx 1.042 \][/tex]
[tex]\[ \frac{1}{4} - \frac{1}{2} \approx 0.25 - 0.5 \approx -0.25 \][/tex]
Perform the overall subtraction:
[tex]\[ \left(1.292 - 1.042\right) - (-0.25) \approx 0.25 + 0.25 \approx 0.5 \][/tex]
So, the result for problem 13 is:
[tex]\[ 0.5 \][/tex]
### Summary:
1. [tex]\(\left(\frac{1}{6}+\frac{2}{7}-\frac{1}{5}\right)-\left(\frac{1}{2}+\frac{1}{5}\right)\)[/tex] = [tex]\(-0.447619\)[/tex]
2. [tex]\(\left(3 \frac{1}{5}+2 \frac{1}{6}-\frac{2}{3}\right)-\left(1 \frac{2}{5}+\frac{3}{2}\right)\)[/tex] = [tex]\(1.8\)[/tex]
3. [tex]\(\left(7 \frac{3}{5}+\frac{5}{7}-\frac{1}{5}\right)+\left(\frac{1}{25}-\frac{1}{3}\right)\)[/tex] = [tex]\(7.821\)[/tex]
4. [tex]\(\left(\frac{2}{3}+\frac{1}{2}+\frac{1}{8}\right)-\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{8}\right)-\left(\frac{1}{4}-\frac{1}{2}\right)\)[/tex] = [tex]\(0.5\)[/tex]
### Problem 10:
[tex]\[ \left(\frac{1}{6} + \frac{2}{7} - \frac{1}{5}\right) - \left(\frac{1}{2} + \frac{1}{5}\right) \][/tex]
First, calculate the individual terms inside each parenthesis:
[tex]\[ \frac{1}{6} + \frac{2}{7} - \frac{1}{5} \][/tex]
[tex]\[ \frac{1}{2} + \frac{1}{5} \][/tex]
Now, perform the operations step by step and then combine results:
[tex]\[ \left(\frac{1}{6} + \frac{2}{7} - \frac{1}{5}\right) \approx 0.480952 - 0.2 = 0.280952 \][/tex]
[tex]\[ \frac{1}{2} + \frac{1}{5} \approx 0.5 + 0.2 = 0.7 \][/tex]
Subtract the results of these two calculations:
[tex]\[ 0.280952 - 0.7 \approx -0.447619 \][/tex]
So, the result for problem 10 is:
[tex]\[ -0.447619 \][/tex]
### Problem 11:
[tex]\[ \left(3 \frac{1}{5} + 2 \frac{1}{6} - \frac{2}{3}\right) - \left(1 \frac{2}{5} + \frac{3}{2}\right) \][/tex]
First, convert mixed numbers to improper fractions and calculate:
[tex]\[ 3 \frac{1}{5} = 3 + \frac{1}{5} \approx 3.2 \][/tex]
[tex]\[ 2 \frac{1}{6} = 2 + \frac{1}{6} \approx 2.167 \][/tex]
[tex]\[ 1 \frac{2}{5} = 1 + \frac{2}{5} \approx 1.4 \][/tex]
Now, finish the calculations inside the parenthesis:
[tex]\[ 3.2 + 2.167 - \frac{2}{3} \approx 3.2 + 2.167 - 0.667 \approx 4.7 \][/tex]
[tex]\[ 1.4 + \frac{3}{2} \approx 1.4 + 1.5 = 2.9 \][/tex]
Subtract the results:
[tex]\[ 4.7 - 2.9 \approx 1.8 \][/tex]
So, the result for problem 11 is:
[tex]\[ 1.8 \][/tex]
### Problem 12:
[tex]\[ \left(7 \frac{3}{5} + \frac{5}{7} - \frac{1}{5}\right) + \left(\frac{1}{25} - \frac{1}{3}\right) \][/tex]
Convert and solve each term individually:
[tex]\[ 7 \frac{3}{5} = 7 + \frac{3}{5} \approx 7.6 \][/tex]
[tex]\[ \frac{5}{7} \approx 0.714 \][/tex]
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
Calculate within the first parenthesis:
[tex]\[ 7.6 + 0.714 - 0.2 \approx 8.114 \][/tex]
Now, solve the second parenthesis:
[tex]\[ \frac{1}{25} - \frac{1}{3} \approx 0.04 - 0.333 \approx -0.293 \][/tex]
Add the results of these two parenthesis:
[tex]\[ 8.114 - 0.293 \approx 7.821 \][/tex]
So, the result for problem 12 is:
[tex]\[ 7.821 \][/tex]
### Problem 13:
[tex]\[ \left(\frac{2}{3} + \frac{1}{2} + \frac{1}{8}\right) - \left(\frac{1}{2} + \frac{2}{3} - \frac{1}{8}\right) - \left(\frac{1}{4} - \frac{1}{2}\right) \][/tex]
Calculate within each parenthesis individually:
[tex]\[ \frac{2}{3} + \frac{1}{2} + \frac{1}{8} \approx 0.667 + 0.5 + 0.125 \approx 1.292 \][/tex]
[tex]\[ \frac{1}{2} + \frac{2}{3} - \frac{1}{8} \approx 0.5 + 0.667 - 0.125 \approx 1.042 \][/tex]
[tex]\[ \frac{1}{4} - \frac{1}{2} \approx 0.25 - 0.5 \approx -0.25 \][/tex]
Perform the overall subtraction:
[tex]\[ \left(1.292 - 1.042\right) - (-0.25) \approx 0.25 + 0.25 \approx 0.5 \][/tex]
So, the result for problem 13 is:
[tex]\[ 0.5 \][/tex]
### Summary:
1. [tex]\(\left(\frac{1}{6}+\frac{2}{7}-\frac{1}{5}\right)-\left(\frac{1}{2}+\frac{1}{5}\right)\)[/tex] = [tex]\(-0.447619\)[/tex]
2. [tex]\(\left(3 \frac{1}{5}+2 \frac{1}{6}-\frac{2}{3}\right)-\left(1 \frac{2}{5}+\frac{3}{2}\right)\)[/tex] = [tex]\(1.8\)[/tex]
3. [tex]\(\left(7 \frac{3}{5}+\frac{5}{7}-\frac{1}{5}\right)+\left(\frac{1}{25}-\frac{1}{3}\right)\)[/tex] = [tex]\(7.821\)[/tex]
4. [tex]\(\left(\frac{2}{3}+\frac{1}{2}+\frac{1}{8}\right)-\left(\frac{1}{2}+\frac{2}{3}-\frac{1}{8}\right)-\left(\frac{1}{4}-\frac{1}{2}\right)\)[/tex] = [tex]\(0.5\)[/tex]
