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Sagot :
Certainly! Let's solve each part step-by-step.
### 1\) \(\left(\frac{-1}{3}\right)+\left(\frac{5}{6}-\frac{1}{4}\right)\)
First, calculate \(\frac{5}{6} - \frac{1}{4}\):
- Find a common denominator for \(\frac{5}{6}\) and \(\frac{1}{4}\). The common denominator is 12.
[tex]\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \][/tex]
[tex]\[ \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \][/tex]
Now, add \(\frac{-1}{3}\) and \(\frac{7}{12}\):
- Change \(\frac{-1}{3}\) to have a common denominator of 12.
[tex]\[ \frac{-1}{3} = \frac{-1 \times 4}{3 \times 4} = \frac{-4}{12} \][/tex]
[tex]\[ \frac{-4}{12} + \frac{7}{12} = \frac{3}{12} = \frac{1}{4} \][/tex]
So,
[tex]\[ \left(\frac{-1}{3}\right)+\left(\frac{5}{6}-\frac{1}{4}\right) = \frac{1}{4} \][/tex]
### 2\) \(\left(4 \frac{1}{9}-2 \frac{1}{3}\right)-\left(4 \frac{1}{3}-3 \frac{1}{9}\right)\)
Convert mixed fractions to improper fractions:
[tex]\[ 4 \frac{1}{9} = \frac{4 \times 9 + 1}{9} = \frac{37}{9} \][/tex]
[tex]\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} = \frac{21}{9} \][/tex]
[tex]\[ 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3} = \frac{39}{9} \][/tex]
[tex]\[ 3 \frac{1}{9} = \frac{3 \times 9 + 1}{9} = \frac{28}{9} \][/tex]
Now, perform the subtractions:
[tex]\[ \left(\frac{37}{9} - \frac{21}{9}\right) = \frac{16}{9} \][/tex]
[tex]\[ \left(\frac{39}{9} - \frac{28}{9}\right) = \frac{11}{9} \][/tex]
Subtract the two results:
[tex]\[ \frac{16}{9} - \frac{11}{9} = \frac{5}{9} \][/tex]
### 3\) \(\frac{1}{2} - \left[ \left( \left( \frac{1}{3} - \frac{7}{11} \right) \div \frac{2}{3} \right) + \frac{5}{11} \right]\)
First, calculate \(\frac{1}{3} - \frac{7}{11}\):
- Find a common denominator for \(\frac{1}{3}\) and \(\frac{7}{11}\). The common denominator is 33.
[tex]\[ \frac{1}{3} = \frac{11}{33} \][/tex]
[tex]\[ \frac{7}{11} = \frac{21}{33} \][/tex]
[tex]\[ \frac{11}{33} - \frac{21}{33} = \frac{-10}{33} \][/tex]
Now divide by \(\frac{2}{3}\):
[tex]\[ \frac{-10}{33} \div \frac{2}{3} = \frac{-10}{33} \times \frac{3}{2} = \frac{-30}{66} = \frac{-15}{33} = \frac{-5}{11} \][/tex]
Add \(\frac{-5}{11}\) and \(\frac{5}{11}\):
[tex]\[ \frac{-5}{11} + \frac{5}{11} = 0 \][/tex]
Finally, subtract from \(\frac{1}{2}\):
[tex]\[ \frac{1}{2} - 0 = \frac{1}{2} \][/tex]
### 4\) \(\left( \frac{1}{4} + \frac{1}{6} \right) \times \left( \frac{-1}{2} \right) \times \left( \frac{1}{6} - \frac{1}{4} \right)\)
First, calculate \(\frac{1}{4} + \frac{1}{6}\):
- Find a common denominator for \(\frac{1}{4}\) and \(\frac{1}{6}\). The common denominator is 12.
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{1}{6} = \frac{2}{12} \][/tex]
[tex]\[ \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \][/tex]
Next, calculate \(\frac{1}{6} - \frac{1}{4}\):
- Ensure a common denominator of 12.
[tex]\[ \frac{1}{6} = \frac{2}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{2}{12} - \frac{3}{12} = \frac{-1}{12} \][/tex]
Now calculate:
[tex]\[ \left( \frac{5}{12} \right) \times \left( \frac{-1}{2} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \left( \frac{5}{12} \times \frac{-1}{2} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \left( \frac{-5}{24} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \frac{5}{24} \times \frac{1}{12} = \frac{5}{288} \][/tex]
Simplify the fraction:
[tex]\[ \frac{5}{288} \][/tex]
### 5\) Evaluate the expression with \(x = -\frac{1}{2}\), \(y = -\frac{3}{4}\), \(z = \frac{2}{3}\):
[tex]\[ x + \left( y \times z - y - \frac{x}{z} \right) \][/tex]
First, calculate \(y \times z\):
[tex]\[ y \times z = (-\frac{3}{4}) \times \frac{2}{3} = -\frac{6}{12} = -\frac{1}{2} \][/tex]
Next, calculate \( \frac{x}{z} \):
[tex]\[ \frac{x}{z} = \frac{ -\frac{1}{2} }{ \frac{2}{3} } = -\frac{1}{2} \times \frac{3}{2} = -\frac{3}{4} \][/tex]
Now, substitute back:
[tex]\[ x + \left( -\frac{1}{2} - (-\frac{3}{4}) - \frac{-3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left( -\frac{1}{2} + \frac{3}{4} + \frac{3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left(-\frac{1}{2} + \frac{3}{4} + \frac{3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left(-\frac{1}{2} + \frac{6}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left( -\frac{1}{2} + \frac{3}{2} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \frac{1}{2} = 0 \][/tex]
After performing great caution:
[tex]\[ = \text{0.5 approximately, indicating work included fraction and robbed vigilant} \][/tex]
### 1\) \(\left(\frac{-1}{3}\right)+\left(\frac{5}{6}-\frac{1}{4}\right)\)
First, calculate \(\frac{5}{6} - \frac{1}{4}\):
- Find a common denominator for \(\frac{5}{6}\) and \(\frac{1}{4}\). The common denominator is 12.
[tex]\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \][/tex]
[tex]\[ \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \][/tex]
Now, add \(\frac{-1}{3}\) and \(\frac{7}{12}\):
- Change \(\frac{-1}{3}\) to have a common denominator of 12.
[tex]\[ \frac{-1}{3} = \frac{-1 \times 4}{3 \times 4} = \frac{-4}{12} \][/tex]
[tex]\[ \frac{-4}{12} + \frac{7}{12} = \frac{3}{12} = \frac{1}{4} \][/tex]
So,
[tex]\[ \left(\frac{-1}{3}\right)+\left(\frac{5}{6}-\frac{1}{4}\right) = \frac{1}{4} \][/tex]
### 2\) \(\left(4 \frac{1}{9}-2 \frac{1}{3}\right)-\left(4 \frac{1}{3}-3 \frac{1}{9}\right)\)
Convert mixed fractions to improper fractions:
[tex]\[ 4 \frac{1}{9} = \frac{4 \times 9 + 1}{9} = \frac{37}{9} \][/tex]
[tex]\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} = \frac{21}{9} \][/tex]
[tex]\[ 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3} = \frac{39}{9} \][/tex]
[tex]\[ 3 \frac{1}{9} = \frac{3 \times 9 + 1}{9} = \frac{28}{9} \][/tex]
Now, perform the subtractions:
[tex]\[ \left(\frac{37}{9} - \frac{21}{9}\right) = \frac{16}{9} \][/tex]
[tex]\[ \left(\frac{39}{9} - \frac{28}{9}\right) = \frac{11}{9} \][/tex]
Subtract the two results:
[tex]\[ \frac{16}{9} - \frac{11}{9} = \frac{5}{9} \][/tex]
### 3\) \(\frac{1}{2} - \left[ \left( \left( \frac{1}{3} - \frac{7}{11} \right) \div \frac{2}{3} \right) + \frac{5}{11} \right]\)
First, calculate \(\frac{1}{3} - \frac{7}{11}\):
- Find a common denominator for \(\frac{1}{3}\) and \(\frac{7}{11}\). The common denominator is 33.
[tex]\[ \frac{1}{3} = \frac{11}{33} \][/tex]
[tex]\[ \frac{7}{11} = \frac{21}{33} \][/tex]
[tex]\[ \frac{11}{33} - \frac{21}{33} = \frac{-10}{33} \][/tex]
Now divide by \(\frac{2}{3}\):
[tex]\[ \frac{-10}{33} \div \frac{2}{3} = \frac{-10}{33} \times \frac{3}{2} = \frac{-30}{66} = \frac{-15}{33} = \frac{-5}{11} \][/tex]
Add \(\frac{-5}{11}\) and \(\frac{5}{11}\):
[tex]\[ \frac{-5}{11} + \frac{5}{11} = 0 \][/tex]
Finally, subtract from \(\frac{1}{2}\):
[tex]\[ \frac{1}{2} - 0 = \frac{1}{2} \][/tex]
### 4\) \(\left( \frac{1}{4} + \frac{1}{6} \right) \times \left( \frac{-1}{2} \right) \times \left( \frac{1}{6} - \frac{1}{4} \right)\)
First, calculate \(\frac{1}{4} + \frac{1}{6}\):
- Find a common denominator for \(\frac{1}{4}\) and \(\frac{1}{6}\). The common denominator is 12.
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{1}{6} = \frac{2}{12} \][/tex]
[tex]\[ \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \][/tex]
Next, calculate \(\frac{1}{6} - \frac{1}{4}\):
- Ensure a common denominator of 12.
[tex]\[ \frac{1}{6} = \frac{2}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{2}{12} - \frac{3}{12} = \frac{-1}{12} \][/tex]
Now calculate:
[tex]\[ \left( \frac{5}{12} \right) \times \left( \frac{-1}{2} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \left( \frac{5}{12} \times \frac{-1}{2} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \left( \frac{-5}{24} \right) \times \left( \frac{-1}{12} \right) \][/tex]
[tex]\[ = \frac{5}{24} \times \frac{1}{12} = \frac{5}{288} \][/tex]
Simplify the fraction:
[tex]\[ \frac{5}{288} \][/tex]
### 5\) Evaluate the expression with \(x = -\frac{1}{2}\), \(y = -\frac{3}{4}\), \(z = \frac{2}{3}\):
[tex]\[ x + \left( y \times z - y - \frac{x}{z} \right) \][/tex]
First, calculate \(y \times z\):
[tex]\[ y \times z = (-\frac{3}{4}) \times \frac{2}{3} = -\frac{6}{12} = -\frac{1}{2} \][/tex]
Next, calculate \( \frac{x}{z} \):
[tex]\[ \frac{x}{z} = \frac{ -\frac{1}{2} }{ \frac{2}{3} } = -\frac{1}{2} \times \frac{3}{2} = -\frac{3}{4} \][/tex]
Now, substitute back:
[tex]\[ x + \left( -\frac{1}{2} - (-\frac{3}{4}) - \frac{-3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left( -\frac{1}{2} + \frac{3}{4} + \frac{3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left(-\frac{1}{2} + \frac{3}{4} + \frac{3}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left(-\frac{1}{2} + \frac{6}{4} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \left( -\frac{1}{2} + \frac{3}{2} \right) \][/tex]
[tex]\[ = -\frac{1}{2} + \frac{1}{2} = 0 \][/tex]
After performing great caution:
[tex]\[ = \text{0.5 approximately, indicating work included fraction and robbed vigilant} \][/tex]
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