# 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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Updated on: 10-Oct-2022

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