Simplify:
Question:

Simplify:

(i) $\frac{8}{9}+\frac{-11}{6}$

(ii) $3+\frac{5}{-7}$

(iii) $\frac{1}{-12}+\frac{2}{-15}$

(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) $1+\frac{-4}{5}$

Solution:

(i) $\frac{8}{9}+\frac{-11}{6}$

L.C.M. of the denominators 9 and 6 is $18 .$

Now, we will express $\frac{8}{9}$ and $\frac{-11}{6}$ in the form in which they take the denominator 18 .

$\frac{8 \times 2}{9 \times 2}=\frac{16}{18}$

$\frac{-11 \times 3}{6 \times 3}=\frac{-33}{18}$

$\frac{8}{9}+\frac{-11}{6}=\frac{16}{18}+\frac{-33}{18}$

$=\frac{16+(-33)}{18}$

$=\frac{16-33}{18}$

$=\frac{-17}{18}$

(ii) $3+\frac{5}{-7}=\frac{3}{1}+\frac{-5}{7}$

L.C.M. of the denominators 1 and 7 is 7 .

Now, we will express $\frac{3}{1}$ in the form in which it takes the denominator 7 .

$\frac{3 \times 7}{1 \times 7}=\frac{21}{7}$

$\frac{3}{1}+\frac{-5}{7}=\frac{21}{7}+\frac{-5}{7}$

$=\frac{21+(-5)}{7}$

$=\frac{21-5}{7}$

$=\frac{16}{7}$

(iii) $\frac{1}{-12}+\frac{2}{-15}=\frac{-1}{12}+\frac{-2}{15}$

L.C.M. of the denominators 12 and 15 is $60 .$

Now, we will express $\frac{-1}{12}$ and $\frac{-2}{15}$ in the form in which they take the denominator 60 .

$\frac{-1 \times 5}{12 \times 5}=\frac{-5}{60}$

$\frac{-2 \times 4}{15 \times 4}=\frac{-8}{60}$

$\frac{-1}{12}+\frac{-2}{15}=\frac{-5}{60}+\frac{-8}{60}$

$=\frac{(-5)+(-8)}{60}$

$=\frac{-5-8}{60}$

$=\frac{-13}{60}$

(iv) $\frac{-8}{19}+\frac{-4}{57}$

L.C.M. of the denominators 19 and 57 is $57 .$

Now, we will express $\frac{-8}{19}$ in the form in which $i$ t takes the denominator 57 .

$\frac{-8 \times 3}{19 \times 3}=\frac{-24}{57}$

$\frac{-8}{19}+\frac{-4}{57}=\frac{-24}{57}+\frac{-4}{57}$

$=\frac{-24-4}{57}$

$=\frac{-28}{57}$

(v) $\frac{7}{9}+\frac{3}{-4}=\frac{7}{9}+\frac{-3}{4}$

L.C.M. of the denominators 9 and 4 is 36 .

Now, we will express $\frac{7}{9}$ and $\frac{-3}{4}$ in the form in which they take the denominator 36 .

$\frac{7 \times 4}{9 \times 4}=\frac{28}{36}$

$\frac{-3 \times 9}{4 \times 9}=\frac{-27}{36}$

So,

$\frac{7}{9}+\frac{-3}{4}=\frac{28}{36}+\frac{-27}{36}$

$=\frac{28+(-27)}{36}$

$=\frac{28-27}{36}$

$=\frac{1}{36}$

(vi), $\frac{5}{26}+\frac{11}{-39}=\frac{5}{26}+\frac{-11}{39}$

L.C.M. of the denominators 26 and 39 is 78 .

Now, we will express $\frac{5}{26}$ and $\frac{-11}{39}$ in the form in which they take the denominator 78 .

$\frac{5 \times 3}{26 \times 3}=\frac{15}{78}$

$\frac{-11 \times 2}{39 \times 2}=\frac{-22}{78}$

So,

$\frac{5}{26}+\frac{-11}{39}=\frac{15}{78}+\frac{-22}{78}$

$=\frac{15-22)}{78}$

$=\frac{-7}{78}$

(vii) $\frac{-16}{9}+\frac{-5}{12}$

L.C.M. of the denominators 9 and 12 is 36 .

Now, we will express $\frac{-16}{9}$ and $\frac{-5}{12}$ in the form in which they take the denominator 36 .

$\frac{-16 \times 4}{9 \times 4}=\frac{-64}{36}$

$\frac{-5 \times 3}{12 \times 3}=\frac{-15}{36}$

$\frac{-16}{9}+\frac{-5}{12}=\frac{-64}{36}+\frac{-15}{36}$

$=\frac{(-64)+(-15)}{36}$

$=\frac{-64-15}{36}$

$=\frac{-79}{36}$

(viii) $\frac{-13}{8}+\frac{5}{36}$

L.C.M. of the denominators 8 and 36 is 72 .

Now, we will express $\frac{-13}{8}$ and $\frac{5}{36}$ in the form in which they take the denominator 72 .

$\frac{-13 \times 9}{8 \times 9}=\frac{-117}{72}$

$\frac{5 \times 2}{36 \times 2}=\frac{10}{72}$

$\frac{-13}{8}+\frac{5}{36}=\frac{-117}{72}+\frac{10}{72}$

$=\frac{-117+10}{72}$

$=\frac{-107}{72}$

(ix) $0+\frac{-3}{5}$

Taking $L . C . M$. of the denominator:

$=\frac{0 \times 5-3}{5}$

$=\frac{-3}{5}$

(x) $1+\frac{-4}{5}=\frac{1}{1}+\frac{-4}{5}$

L.C.M. of the denominators 1 and 5 is 5 .

Now, we will express $\frac{1}{1}$ in the form in which $i t$ takes the denominator $5 .$

$\frac{1 \times 5}{1 \times 5}=\frac{5}{5}$

$\frac{1}{1}+\frac{-4}{5}=\frac{5}{5}+\frac{-4}{5}$

$=\frac{5-4}{5}$

$=\frac{1}{5}$

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