# Simplify each of the following and write as a rational number of the form:(i) $\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}$(ii) $\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9}$(iii) $\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8}$(iv) $\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15}$(v) $\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20}$(vi) $\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3$

To do:

We have to simplify and write the given expressions as  rational numbers.

Solution:

(i) $\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}$

LCM of $4, 6$ and $8=24$

$\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}=\frac{3(6)+5(4)+(-7)(3)}{24}$

$=\frac{18+20-21}{24}$

$=\frac{38-21}{24}$

$=\frac{17}{24}$

Therefore,

$\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}=\frac{17}{24}$.

(ii) $\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9}$

LCM of $3, 6$ and $9=18$

$\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9} = \frac{2(6)+(-5)(3)+(-7)(2)}{18}$

$=\frac{12-15-14}{18}$

$=\frac{12-29}{18}$

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

Therefore,

$\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9}=\frac{-17}{18}$.

(iii) $\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8}$

LCM of $2, 6$ and $8=24$

$\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8} = \frac{(-11)(12)+(7)(4)+(-5)(3)}{24}$

$= \frac{-132+28-15}{24}$

$=\frac{-147+28}{24}$

$=\frac{-119}{24}$

Therefore,

$\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8}=\frac{-119}{24}$.

(iv) $\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15}$

LCM of $5, 10$ and $15=30$

$\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15} = \frac{(-4)(6)+(-7)(3)+(-8)(2)}{30}$

$= \frac{-24-21-16}{30}$

$=\frac{-61}{30}$

Therefore,

$\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15}=\frac{-61}{30}$.

(v) $\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20}$

LCM of $10, 15$ and $20=60$

$\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20} = \frac{(-9)(6)+(22)(4)+(-13)(3)}{60}$

$= \frac{-54+88-39}{60}$

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

$=\frac{-1}{12}$

Therefore,

$\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20} =\frac{-1}{12}$.

(vi) $\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3$

LCM of $3, 2, 3$ and $1=6$

$\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3 = \frac{(5)(2)+(-3)(3)+(-7)(2)+3(6)}{6}$

$= \frac{10-9-14+18}{6}$

$=\frac{5}{6}$

Therefore,

$\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3 =\frac{5}{6}$.

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Updated on: 10-Oct-2022

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