Answer :
Sure, let's evaluate each of these expressions step-by-step.
### 1. [tex]\(\frac{4}{15} - \frac{4}{3}\)[/tex]
First, we need a common denominator. The least common multiple of 15 and 3 is 15.
[tex]\[ \frac{4}{3} = \frac{4 \cdot 5}{3 \cdot 5} = \frac{20}{15} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{4}{15} - \frac{20}{15} = \frac{4 - 20}{15} = \frac{-16}{15} \approx -1.0667 \][/tex]
### 2. [tex]\(-6 + \frac{1}{4}\)[/tex]
Convert [tex]\(-6\)[/tex] into a fraction with a denominator of 4:
[tex]\[ -6 = -6 \cdot \frac{4}{4} = \frac{-24}{4} \][/tex]
Now add the fractions:
[tex]\[ \frac{-24}{4} + \frac{1}{4} = \frac{-24 + 1}{4} = \frac{-23}{4} = -5.75 \][/tex]
### 3. [tex]\(\left(-\frac{1}{3}\right) + \left(-\frac{1}{7}\right)\)[/tex]
Find a common denominator. The least common multiple of 3 and 7 is 21.
[tex]\[ -\frac{1}{3} = -\frac{1 \cdot 7}{3 \cdot 7} = -\frac{7}{21} \][/tex]
[tex]\[ -\frac{1}{7} = -\frac{1 \cdot 3}{7 \cdot 3} = -\frac{3}{21} \][/tex]
Add the fractions:
[tex]\[ -\frac{7}{21} + -\frac{3}{21} = -\frac{7 + 3}{21} = -\frac{10}{21} \approx -0.4762 \][/tex]
### 4. [tex]\(-12 + \left(-\frac{1}{8}\right)\)[/tex]
Convert [tex]\(-12\)[/tex] into a fraction with a denominator of 8:
[tex]\[ -12 = -12 \cdot \frac{8}{8} = -\frac{96}{8} \][/tex]
Now add the fractions:
[tex]\[ -\frac{96}{8} + -\frac{1}{8} = -\frac{96 + 1}{8} = -\frac{97}{8} = -12.125 \][/tex]
### 5. [tex]\(\left(-\frac{5}{14}\right) + \left(-\frac{17}{18}\right) + \left(-\frac{23}{21}\right)\)[/tex]
We need a common denominator. The least common multiple of 14, 18, and 21 is 126.
[tex]\[ -\frac{5}{14} = -\frac{5 \cdot 9}{14 \cdot 9} = -\frac{45}{126} \][/tex]
[tex]\[ -\frac{17}{18} = -\frac{17 \cdot 7}{18 \cdot 7} = -\frac{119}{126} \][/tex]
[tex]\[ -\frac{23}{21} = -\frac{23 \cdot 6}{21 \cdot 6} = -\frac{138}{126} \][/tex]
Now add the fractions:
[tex]\[ -\frac{45}{126} + -\frac{119}{126} + -\frac{138}{126} = -\frac{45 + 119 + 138}{126} = -\frac{302}{126} \approx -2.3968 \][/tex]
### 6. [tex]\(\left(-\frac{11}{43}\right) - \left(-\frac{2}{3}\right) - \left(-\frac{29}{21}\right)\)[/tex]
Find a common denominator. The least common multiple of 43, 3, and 21 is 2709.
[tex]\[ -\frac{11}{43} = -\frac{11 \cdot 63}{43 \cdot 63} = -\frac{693}{2709} \][/tex]
[tex]\[ -\frac{2}{3} = -\frac{2 \cdot 903}{3 \cdot 903} = -\frac{1806}{2709} \][/tex]
[tex]\[ -\frac{29}{21} = -\frac{29 \cdot 129}{21 \cdot 129} = -\frac{3741}{2709} \][/tex]
Subtract the fractions:
[tex]\[ -\frac{693}{2709} - \left( - \frac{1806}{2709} \right) - \left( - \frac{3741}{2709} \right) \][/tex]
[tex]\[ 1.7918 \][/tex]
So, let's summarize the results:
1. [tex]\(\frac{4}{15} - \frac{4}{3} \approx -1.0667\)[/tex]
2. [tex]\(-6 + \frac{1}{4} = -5.75\)[/tex]
3. [tex]\(\left(-\frac{1}{3}\right) + \left(-\frac{1}{7}\right) \approx -0.4762\)[/tex]
4. [tex]\(-12 + \left(-\frac{1}{8}\right) = -12.125\)[/tex]
5. [tex]\(\left(-\frac{5}{14}\right) + \left(-\frac{17}{18}\right) + \left(-\frac{23}{21}\right) \approx -2.3968\)[/tex]
6. [tex]\(\left(-\frac{11}{43}\right) - \left(-\frac{2}{3}\right) - \left(-\frac{29}{21}\right) \approx 1.7918\)[/tex]
### 1. [tex]\(\frac{4}{15} - \frac{4}{3}\)[/tex]
First, we need a common denominator. The least common multiple of 15 and 3 is 15.
[tex]\[ \frac{4}{3} = \frac{4 \cdot 5}{3 \cdot 5} = \frac{20}{15} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{4}{15} - \frac{20}{15} = \frac{4 - 20}{15} = \frac{-16}{15} \approx -1.0667 \][/tex]
### 2. [tex]\(-6 + \frac{1}{4}\)[/tex]
Convert [tex]\(-6\)[/tex] into a fraction with a denominator of 4:
[tex]\[ -6 = -6 \cdot \frac{4}{4} = \frac{-24}{4} \][/tex]
Now add the fractions:
[tex]\[ \frac{-24}{4} + \frac{1}{4} = \frac{-24 + 1}{4} = \frac{-23}{4} = -5.75 \][/tex]
### 3. [tex]\(\left(-\frac{1}{3}\right) + \left(-\frac{1}{7}\right)\)[/tex]
Find a common denominator. The least common multiple of 3 and 7 is 21.
[tex]\[ -\frac{1}{3} = -\frac{1 \cdot 7}{3 \cdot 7} = -\frac{7}{21} \][/tex]
[tex]\[ -\frac{1}{7} = -\frac{1 \cdot 3}{7 \cdot 3} = -\frac{3}{21} \][/tex]
Add the fractions:
[tex]\[ -\frac{7}{21} + -\frac{3}{21} = -\frac{7 + 3}{21} = -\frac{10}{21} \approx -0.4762 \][/tex]
### 4. [tex]\(-12 + \left(-\frac{1}{8}\right)\)[/tex]
Convert [tex]\(-12\)[/tex] into a fraction with a denominator of 8:
[tex]\[ -12 = -12 \cdot \frac{8}{8} = -\frac{96}{8} \][/tex]
Now add the fractions:
[tex]\[ -\frac{96}{8} + -\frac{1}{8} = -\frac{96 + 1}{8} = -\frac{97}{8} = -12.125 \][/tex]
### 5. [tex]\(\left(-\frac{5}{14}\right) + \left(-\frac{17}{18}\right) + \left(-\frac{23}{21}\right)\)[/tex]
We need a common denominator. The least common multiple of 14, 18, and 21 is 126.
[tex]\[ -\frac{5}{14} = -\frac{5 \cdot 9}{14 \cdot 9} = -\frac{45}{126} \][/tex]
[tex]\[ -\frac{17}{18} = -\frac{17 \cdot 7}{18 \cdot 7} = -\frac{119}{126} \][/tex]
[tex]\[ -\frac{23}{21} = -\frac{23 \cdot 6}{21 \cdot 6} = -\frac{138}{126} \][/tex]
Now add the fractions:
[tex]\[ -\frac{45}{126} + -\frac{119}{126} + -\frac{138}{126} = -\frac{45 + 119 + 138}{126} = -\frac{302}{126} \approx -2.3968 \][/tex]
### 6. [tex]\(\left(-\frac{11}{43}\right) - \left(-\frac{2}{3}\right) - \left(-\frac{29}{21}\right)\)[/tex]
Find a common denominator. The least common multiple of 43, 3, and 21 is 2709.
[tex]\[ -\frac{11}{43} = -\frac{11 \cdot 63}{43 \cdot 63} = -\frac{693}{2709} \][/tex]
[tex]\[ -\frac{2}{3} = -\frac{2 \cdot 903}{3 \cdot 903} = -\frac{1806}{2709} \][/tex]
[tex]\[ -\frac{29}{21} = -\frac{29 \cdot 129}{21 \cdot 129} = -\frac{3741}{2709} \][/tex]
Subtract the fractions:
[tex]\[ -\frac{693}{2709} - \left( - \frac{1806}{2709} \right) - \left( - \frac{3741}{2709} \right) \][/tex]
[tex]\[ 1.7918 \][/tex]
So, let's summarize the results:
1. [tex]\(\frac{4}{15} - \frac{4}{3} \approx -1.0667\)[/tex]
2. [tex]\(-6 + \frac{1}{4} = -5.75\)[/tex]
3. [tex]\(\left(-\frac{1}{3}\right) + \left(-\frac{1}{7}\right) \approx -0.4762\)[/tex]
4. [tex]\(-12 + \left(-\frac{1}{8}\right) = -12.125\)[/tex]
5. [tex]\(\left(-\frac{5}{14}\right) + \left(-\frac{17}{18}\right) + \left(-\frac{23}{21}\right) \approx -2.3968\)[/tex]
6. [tex]\(\left(-\frac{11}{43}\right) - \left(-\frac{2}{3}\right) - \left(-\frac{29}{21}\right) \approx 1.7918\)[/tex]
