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Simplify:
(i) $ \frac{8}{9}+\frac{-11}{5} $
(ii) $ 3+\frac{5}{-7} $
(iii) $ \frac{1}{-12} $ and $ \frac{2}{-15} $
(iv) $ \frac{-8}{19}+\frac{-4}{57} $
(v) $ \frac{7}{9}+\frac{3}{-4} $
(vi) $ \frac{5}{26}+\frac{11}{-39} $
(vii) $ \frac{-16}{9}+\frac{-5}{12} $
(viii) $ \frac{-13}{8}+\frac{5}{36} $
(ix) $ 0+\frac{-3}{5} $
(x) $ 1+\frac{-4}{5} $
(xi) $ \frac{-5}{16}+\frac{7}{24} $
To do:
We have to simplify the given expressions.
Solution:
(i) $\frac{8}{9}+\frac{-11}{5}=\frac{8\times5}{9\times5}+\frac{-11\times9}{5\times9}$ (LCM of 9 and 5 is 45)
$=\frac{40}{45}+\frac{-99}{45}$
$=\frac{40+(-99)}{45}$
$=\frac{-(99-40)}{45}$
$=\frac{-59}{45}$
Therefore, $\frac{8}{9}+\frac{-11}{5}=\frac{-59}{45}$.
(ii) $3+\frac{5}{-7}=\frac{3\times7}{1\times7}+\frac{-5\times1}{7\times1}$ (LCM of 7 and 1 is 7)
$=\frac{21}{7}+\frac{-5}{7}$
$=\frac{21+(-5)}{7}$
$=\frac{21-5}{7}$
$=\frac{16}{7}$
Therefore, $3+\frac{5}{-7}=\frac{16}{7}$.
(iii) $\frac{1}{-12} + \frac{2}{-15}=\frac{-1\times5}{12\times5}+\frac{-2\times4}{15\times4}$ (LCM of 12 and 15 is 60)
$=\frac{-5}{60}+\frac{-8}{60}$
$=\frac{-5+(-8)}{60}$
$=\frac{-(5+8)}{60}$
$=\frac{-13}{60}$
Therefore, $\frac{1}{-12}+\frac{2}{-15}=\frac{-13}{60}$.
(iv) $\frac{-8}{19} + \frac{-4}{57}=\frac{-8\times3}{19\times3}+\frac{-4\times1}{57\times1}$ (LCM of 19 and 57 is 57)
$=\frac{-24}{57}+\frac{-4}{57}$
$=\frac{-24+(-4)}{57}$
$=\frac{-(24+4)}{57}$
$=\frac{-28}{57}$
Therefore, $\frac{-8}{19}+\frac{-4}{57}=\frac{-28}{57}$.
(v) $\frac{7}{9} + \frac{3}{-4}=\frac{7\times4}{9\times4}+\frac{-3\times9}{4\times9}$ (LCM of 9 and 4 is 36)
$=\frac{28}{36}+\frac{-27}{36}$
$=\frac{28+(-27)}{36}$
$=\frac{28-27)}{36}$
$=\frac{1}{36}$
Therefore, $\frac{7}{9}+\frac{3}{-4}=\frac{1}{36}$.
(vi) $\frac{5}{26} + \frac{11}{-39}=\frac{5\times3}{26\times3}+\frac{-11\times2}{39\times2}$ (LCM of 26 and 39 is 78)
$=\frac{15}{78}+\frac{-22}{78}$
$=\frac{15+(-22)}{78}$
$=\frac{-(22-15)}{78}$
$=\frac{-7}{78}$
Therefore, $\frac{5}{26}+\frac{11}{-39}=\frac{-7}{78}$.
(vii) $\frac{-16}{9} + \frac{-5}{12}=\frac{-16\times4}{9\times4}+\frac{-5\times3}{12\times3}$ (LCM of 9 and 12 is 36)
$=\frac{-64}{36}+\frac{-15}{36}$
$=\frac{-64+(-15)}{36}$
$=\frac{-(64+15)}{36}$
$=\frac{-79}{36}$
Therefore, $\frac{-16}{9}+\frac{-5}{12}=\frac{-79}{36}$.
(viii) $\frac{-13}{8} + \frac{5}{36}=\frac{-13\times9}{8\times9}+\frac{5\times2}{36\times2}$ (LCM of 8 and 36 is 72)
$=\frac{-117}{72}+\frac{10}{72}$
$=\frac{-117+10}{72}$
$=\frac{-(117-10)}{72}$
$=\frac{-107}{72}$
Therefore, $\frac{-13}{8}+\frac{5}{36}=\frac{-107}{72}$.
(ix) $0+\frac{-3}{5}=\frac{0\times5}{1\times5}+\frac{-3\times1}{5\times1}$ (LCM of 1 and 5 is 5)
$=\frac{0}{5}+\frac{-3}{5}$
$=\frac{0+(-3)}{5}$
$=\frac{-(3-0)}{5}$
$=\frac{-3}{5}$
Therefore, $0+\frac{-3}{5}=\frac{-3}{5}$.
(x) $1+\frac{-4}{5}=\frac{1\times5}{1\times5}+\frac{-4\times1}{5\times1}$ (LCM of 1 and 5 is 5)
$=\frac{5}{5}+\frac{-4}{5}$
$=\frac{5+(-4)}{5}$
$=\frac{5-4}{5}$
$=\frac{1}{5}$
Therefore, $1+\frac{-4}{5}=\frac{1}{5}$.
(xi) $\frac{-5}{16}+\frac{7}{24}=\frac{-5\times3}{16\times3}+\frac{7\times2}{24\times2}$ (LCM of 16 and 24 is 48)
$=\frac{-15}{48}+\frac{14}{48}$
$=\frac{-15+14}{48}$
$=\frac{-(15-14)}{48}$
$=\frac{-1}{48}$
Therefore, $\frac{-5}{16}+\frac{7}{24}=\frac{-1}{48}$.
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