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Sagot :
Sure, let's go through each part step-by-step:
### Part (a)
[tex]\[ \frac{2}{5} + \frac{-8}{6} + \frac{4}{5} + \frac{5}{6} \][/tex]
First, let's arrange the fractions involving common denominators:
[tex]\[ \left(\frac{2}{5} + \frac{4}{5}\right) + \left(\frac{-8}{6} + \frac{5}{6}\right) \][/tex]
Calculate each grouped sum:
[tex]\[ \frac{2}{5} + \frac{4}{5} = \frac{2+4}{5} = \frac{6}{5} = 1.2 \][/tex]
[tex]\[ \frac{-8}{6} + \frac{5}{6} = \frac{-8+5}{6} = \frac{-3}{6} = -0.5 \][/tex]
Now, add these results together:
[tex]\[ 1.2 + (-0.5) = 0.7 \][/tex]
So, the sum for part (a) is:
[tex]\[ 0.7 \][/tex]
### Part (b)
[tex]\[ \frac{-11}{3} + \frac{-3}{4} + \frac{-11}{6} + \frac{3}{8} \][/tex]
Combine the negative fractions first:
[tex]\[ \left(\frac{-11}{3} + \frac{-11}{6}\right) + \left(\frac{-3}{4} + \frac{3}{8}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{-11}{3} + \frac{-11}{6} = \frac{-22}{6} + \frac{-11}{6} = \frac{-33}{6} = \frac{-11}{2} = -5.5 \][/tex]
[tex]\[ \frac{-3}{4} + \frac{3}{8} = \frac{-6}{8} + \frac{3}{8} = \frac{-6+3}{8} = \frac{-3}{8} = -0.375 \][/tex]
Now, add these results together:
[tex]\[ -5.5 + (-0.375) = -5.875 \][/tex]
So, the sum for part (b) is:
[tex]\[ -5.875 \][/tex]
### Part (e)
[tex]\[ \frac{13}{2} + \frac{-14}{3} + \frac{-21}{2} + \frac{7}{12} \][/tex]
Combine the fractions with common denominators:
[tex]\[ \left(\frac{13}{2} + \frac{-21}{2}\right) + \left(\frac{-14}{3} + \frac{7}{12}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{13}{2} + \frac{-21}{2} = \frac{13-21}{2} = \frac{-8}{2} = -4 \][/tex]
[tex]\[ \frac{-14}{3} + \frac{7}{12} = \frac{-56}{12} + \frac{7}{12} = \frac{-56+7}{12} = \frac{-49}{12} = -4.083333333333334 \][/tex]
Now, add these results together:
[tex]\[ -4 + (-4.083333333333334) = -8.083333333333334 \][/tex]
So, the sum for part (e) is:
[tex]\[ -8.083333333333334 \][/tex]
### Part (d)
[tex]\[ \frac{-8}{7} + \frac{-4}{9} + \frac{-11}{7} + \frac{5}{6} \][/tex]
Combine the fractions with common denominators:
[tex]\[ \left(\frac{-8}{7} + \frac{-11}{7}\right) + \left(\frac{-4}{9} + \frac{5}{6}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{-8}{7} + \frac{-11}{7} = \frac{-8-11}{7} = \frac{-19}{7} \approx -2.7142857142857144 \][/tex]
[tex]\[ \frac{-4}{9} + \frac{5}{6} = \frac{-24}{54} + \frac{45}{54} = \frac{21}{54} = \frac{7}{18} \approx 0.3888888888888889 \][/tex]
Now, add these results together:
[tex]\[ -2.7142857142857144 + 0.3888888888888889 = -2.325396825396825 \][/tex]
So, the sum for part (d) is:
[tex]\[ -2.325396825396825 \][/tex]
Therefore, the sums are:
(a) [tex]\( 0.7 \)[/tex]
(b) [tex]\( -5.875 \)[/tex]
(e) [tex]\( -8.083333333333334 \)[/tex]
(d) [tex]\( -2.325396825396825 \)[/tex]
### Part (a)
[tex]\[ \frac{2}{5} + \frac{-8}{6} + \frac{4}{5} + \frac{5}{6} \][/tex]
First, let's arrange the fractions involving common denominators:
[tex]\[ \left(\frac{2}{5} + \frac{4}{5}\right) + \left(\frac{-8}{6} + \frac{5}{6}\right) \][/tex]
Calculate each grouped sum:
[tex]\[ \frac{2}{5} + \frac{4}{5} = \frac{2+4}{5} = \frac{6}{5} = 1.2 \][/tex]
[tex]\[ \frac{-8}{6} + \frac{5}{6} = \frac{-8+5}{6} = \frac{-3}{6} = -0.5 \][/tex]
Now, add these results together:
[tex]\[ 1.2 + (-0.5) = 0.7 \][/tex]
So, the sum for part (a) is:
[tex]\[ 0.7 \][/tex]
### Part (b)
[tex]\[ \frac{-11}{3} + \frac{-3}{4} + \frac{-11}{6} + \frac{3}{8} \][/tex]
Combine the negative fractions first:
[tex]\[ \left(\frac{-11}{3} + \frac{-11}{6}\right) + \left(\frac{-3}{4} + \frac{3}{8}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{-11}{3} + \frac{-11}{6} = \frac{-22}{6} + \frac{-11}{6} = \frac{-33}{6} = \frac{-11}{2} = -5.5 \][/tex]
[tex]\[ \frac{-3}{4} + \frac{3}{8} = \frac{-6}{8} + \frac{3}{8} = \frac{-6+3}{8} = \frac{-3}{8} = -0.375 \][/tex]
Now, add these results together:
[tex]\[ -5.5 + (-0.375) = -5.875 \][/tex]
So, the sum for part (b) is:
[tex]\[ -5.875 \][/tex]
### Part (e)
[tex]\[ \frac{13}{2} + \frac{-14}{3} + \frac{-21}{2} + \frac{7}{12} \][/tex]
Combine the fractions with common denominators:
[tex]\[ \left(\frac{13}{2} + \frac{-21}{2}\right) + \left(\frac{-14}{3} + \frac{7}{12}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{13}{2} + \frac{-21}{2} = \frac{13-21}{2} = \frac{-8}{2} = -4 \][/tex]
[tex]\[ \frac{-14}{3} + \frac{7}{12} = \frac{-56}{12} + \frac{7}{12} = \frac{-56+7}{12} = \frac{-49}{12} = -4.083333333333334 \][/tex]
Now, add these results together:
[tex]\[ -4 + (-4.083333333333334) = -8.083333333333334 \][/tex]
So, the sum for part (e) is:
[tex]\[ -8.083333333333334 \][/tex]
### Part (d)
[tex]\[ \frac{-8}{7} + \frac{-4}{9} + \frac{-11}{7} + \frac{5}{6} \][/tex]
Combine the fractions with common denominators:
[tex]\[ \left(\frac{-8}{7} + \frac{-11}{7}\right) + \left(\frac{-4}{9} + \frac{5}{6}\right) \][/tex]
Calculate each part:
[tex]\[ \frac{-8}{7} + \frac{-11}{7} = \frac{-8-11}{7} = \frac{-19}{7} \approx -2.7142857142857144 \][/tex]
[tex]\[ \frac{-4}{9} + \frac{5}{6} = \frac{-24}{54} + \frac{45}{54} = \frac{21}{54} = \frac{7}{18} \approx 0.3888888888888889 \][/tex]
Now, add these results together:
[tex]\[ -2.7142857142857144 + 0.3888888888888889 = -2.325396825396825 \][/tex]
So, the sum for part (d) is:
[tex]\[ -2.325396825396825 \][/tex]
Therefore, the sums are:
(a) [tex]\( 0.7 \)[/tex]
(b) [tex]\( -5.875 \)[/tex]
(e) [tex]\( -8.083333333333334 \)[/tex]
(d) [tex]\( -2.325396825396825 \)[/tex]
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