Answer :
Alright, let's break down each part of the question and solve them step-by-step:
### 1. First Expression:
[tex]\[ \frac{5}{3} - \left(\frac{2}{5} \times \frac{7}{2}\right) - \frac{1}{3} \][/tex]
1.1 Calculate [tex]\(\frac{2}{5} \times \frac{7}{2}\)[/tex]:
[tex]\[ \frac{2}{5} \times \frac{7}{2} = \frac{2 \times 7}{5 \times 2} = \frac{14}{10} = \frac{7}{5} \][/tex]
1.2 Substitute back into the expression:
[tex]\[ \frac{5}{3} - \frac{7}{5} - \frac{1}{3} \][/tex]
1.3 Find a common denominator for the fractions, which is 15:
[tex]\[ \frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15} \][/tex]
[tex]\[ \frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15} \][/tex]
[tex]\[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \][/tex]
1.4 Substitute these fractions into the expression:
[tex]\[ \frac{25}{15} - \frac{21}{15} - \frac{5}{15} = \frac{25 - 21 - 5}{15} = \frac{-1}{15} \][/tex]
### Final answer for the first expression:
[tex]\[ -0.0667 \, (\text{approximately}) \][/tex]
### 2. Second Expression:
[tex]\[ \left(\frac{2}{3} \times 5 - \frac{3}{4}\right) \times \frac{7}{2} \][/tex]
2.1 Calculate [tex]\(\frac{2}{3} \times 5\)[/tex]:
[tex]\[ \frac{2}{3} \times 5 = \frac{2 \times 5}{3} = \frac{10}{3} \][/tex]
2.2 Substitute back into the expression:
[tex]\[ \left(\frac{10}{3} - \frac{3}{4}\right) \times \frac{7}{2} \][/tex]
2.3 Find a common denominator for the fractions, which is 12:
[tex]\[ \frac{10}{3} = \frac{10 \times 4}{3 \times 4} = \frac{40}{12} \][/tex]
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
2.4 Substitute these fractions into the expression:
[tex]\[ \left(\frac{40}{12} - \frac{9}{12}\right) = \frac{31}{12} \][/tex]
2.5 Multiply by [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[ \frac{31}{12} \times \frac{7}{2} = \frac{31 \times 7}{12 \times 2} = \frac{217}{24} = 9.0417 \, (\text{approximately}) \][/tex]
### Final answer for the second expression:
[tex]\[ 9.0417 \, (\text{approximately}) \][/tex]
### 3. Third Expression:
[tex]\[ \left[\left(\frac{-7}{3} \times \frac{4}{5}\right) - 2\right] \times \frac{5}{3} \][/tex]
3.1 Calculate [tex]\(\frac{-7}{3} \times \frac{4}{5}\)[/tex]:
[tex]\[ \frac{-7}{3} \times \frac{4}{5} = \frac{-7 \times 4}{3 \times 5} = \frac{-28}{15} \][/tex]
3.2 Substitute back into the expression:
[tex]\[ \left(\frac{-28}{15} - 2\right) \][/tex]
3.3 Convert the integer to a fraction with a common denominator:
[tex]\[ 2 = \frac{30}{15} \][/tex]
3.4 Substitute back into the expression:
[tex]\[ \left(\frac{-28}{15} - \frac{30}{15}\right) = \frac{-28 - 30}{15} = \frac{-58}{15} \][/tex]
3.5 Multiply by [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[ \frac{-58}{15} \times \frac{5}{3} = \frac{-58 \times 5}{15 \times 3} = \frac{-290}{45} = -6.4444 \, (\text{approximately}) \][/tex]
### Final answer for the third expression:
[tex]\[ -6.4444 \, (\text{approximately}) \][/tex]
### Summary of Results:
1. [tex]\(\frac{5}{3} - \left(\frac{2}{5} \times \frac{7}{2}\right) - \frac{1}{3} \approx -0.0667\)[/tex]
2. [tex]\(\left(\frac{2}{3} \times 5 - \frac{3}{4}\right) \times \frac{7}{2} \approx 9.0417\)[/tex]
3. [tex]\(\left[\left(\frac{-7}{3} \times \frac{4}{5}\right) - 2\right] \times \frac{5}{3} \approx -6.4444\)[/tex]
So the final results are:
[tex]\[ (-0.0667, 9.0417, -6.4444) \][/tex]
### 1. First Expression:
[tex]\[ \frac{5}{3} - \left(\frac{2}{5} \times \frac{7}{2}\right) - \frac{1}{3} \][/tex]
1.1 Calculate [tex]\(\frac{2}{5} \times \frac{7}{2}\)[/tex]:
[tex]\[ \frac{2}{5} \times \frac{7}{2} = \frac{2 \times 7}{5 \times 2} = \frac{14}{10} = \frac{7}{5} \][/tex]
1.2 Substitute back into the expression:
[tex]\[ \frac{5}{3} - \frac{7}{5} - \frac{1}{3} \][/tex]
1.3 Find a common denominator for the fractions, which is 15:
[tex]\[ \frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15} \][/tex]
[tex]\[ \frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15} \][/tex]
[tex]\[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \][/tex]
1.4 Substitute these fractions into the expression:
[tex]\[ \frac{25}{15} - \frac{21}{15} - \frac{5}{15} = \frac{25 - 21 - 5}{15} = \frac{-1}{15} \][/tex]
### Final answer for the first expression:
[tex]\[ -0.0667 \, (\text{approximately}) \][/tex]
### 2. Second Expression:
[tex]\[ \left(\frac{2}{3} \times 5 - \frac{3}{4}\right) \times \frac{7}{2} \][/tex]
2.1 Calculate [tex]\(\frac{2}{3} \times 5\)[/tex]:
[tex]\[ \frac{2}{3} \times 5 = \frac{2 \times 5}{3} = \frac{10}{3} \][/tex]
2.2 Substitute back into the expression:
[tex]\[ \left(\frac{10}{3} - \frac{3}{4}\right) \times \frac{7}{2} \][/tex]
2.3 Find a common denominator for the fractions, which is 12:
[tex]\[ \frac{10}{3} = \frac{10 \times 4}{3 \times 4} = \frac{40}{12} \][/tex]
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
2.4 Substitute these fractions into the expression:
[tex]\[ \left(\frac{40}{12} - \frac{9}{12}\right) = \frac{31}{12} \][/tex]
2.5 Multiply by [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[ \frac{31}{12} \times \frac{7}{2} = \frac{31 \times 7}{12 \times 2} = \frac{217}{24} = 9.0417 \, (\text{approximately}) \][/tex]
### Final answer for the second expression:
[tex]\[ 9.0417 \, (\text{approximately}) \][/tex]
### 3. Third Expression:
[tex]\[ \left[\left(\frac{-7}{3} \times \frac{4}{5}\right) - 2\right] \times \frac{5}{3} \][/tex]
3.1 Calculate [tex]\(\frac{-7}{3} \times \frac{4}{5}\)[/tex]:
[tex]\[ \frac{-7}{3} \times \frac{4}{5} = \frac{-7 \times 4}{3 \times 5} = \frac{-28}{15} \][/tex]
3.2 Substitute back into the expression:
[tex]\[ \left(\frac{-28}{15} - 2\right) \][/tex]
3.3 Convert the integer to a fraction with a common denominator:
[tex]\[ 2 = \frac{30}{15} \][/tex]
3.4 Substitute back into the expression:
[tex]\[ \left(\frac{-28}{15} - \frac{30}{15}\right) = \frac{-28 - 30}{15} = \frac{-58}{15} \][/tex]
3.5 Multiply by [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[ \frac{-58}{15} \times \frac{5}{3} = \frac{-58 \times 5}{15 \times 3} = \frac{-290}{45} = -6.4444 \, (\text{approximately}) \][/tex]
### Final answer for the third expression:
[tex]\[ -6.4444 \, (\text{approximately}) \][/tex]
### Summary of Results:
1. [tex]\(\frac{5}{3} - \left(\frac{2}{5} \times \frac{7}{2}\right) - \frac{1}{3} \approx -0.0667\)[/tex]
2. [tex]\(\left(\frac{2}{3} \times 5 - \frac{3}{4}\right) \times \frac{7}{2} \approx 9.0417\)[/tex]
3. [tex]\(\left[\left(\frac{-7}{3} \times \frac{4}{5}\right) - 2\right] \times \frac{5}{3} \approx -6.4444\)[/tex]
So the final results are:
[tex]\[ (-0.0667, 9.0417, -6.4444) \][/tex]
