Question

Add the following fractions $\dfrac{{ - 9}}{{10}},\dfrac{{22}}{{15}},\dfrac{{13}}{{ - 20}}$?

Answer 1

Hint – In this question 3 fractions have been given to us and we need to find the addition of the fractions, first of all take L.CM of the denominator part of all the fractions and then proceed by simplifying, this will get the answer.

Complete step by step answer:
Given fractions are
$\dfrac{{ - 9}}{{10}},\dfrac{{22}}{{15}},\dfrac{{13}}{{ - 20}}$
These fractions are also written as
$\dfrac{{ - 9}}{{10}},\dfrac{{22}}{{15}},\dfrac{{ - 13}}{{20}}$
So we have to add them
So first convert the denominator into the same denominator by taking the L.C.M of the denominator.
So the denominators are (10, 15 and 20)
So we have to take the L.C.M of these denominators.
So first factorize it we have
Factors of 10 are
$ \Rightarrow 10 = 2 \times 5$
Factors of 15 are
$ \Rightarrow 15 = 3 \times 5$
Factors of 20 are
$ \Rightarrow 20 = 2 \times 2 \times 5$
So the L.C.M of these numbers are,
$ \Rightarrow L.C.M = 2 \times 2 \times 3 \times 5 = 60$
So convert the denominators of all the fraction into 60 by appropriate multiplication
So first fraction is $\dfrac{{ - 9}}{{10}}$
So multiply and divide by 6 we have,
$ \Rightarrow \dfrac{{ - 9}}{{10}} = \dfrac{{ - 9 \times 6}}{{10 \times 6}} = \dfrac{{ - 54}}{{60}}$……………… (1)
So second fraction is $\dfrac{{22}}{{15}}$
So multiply and divide by 4 we have,
$ \Rightarrow \dfrac{{22}}{{15}} = \dfrac{{22 \times 4}}{{15 \times 4}} = \dfrac{{88}}{{60}}$……….. (2)
So third fraction is $\dfrac{{ - 13}}{{20}}$
So multiply and divide by 3 we have,
\[ \Rightarrow \dfrac{{ - 13}}{{20}} = \dfrac{{ - 13 \times 3}}{{20 \times 3}} = \dfrac{{ - 39}}{{60}}\]………………… (3)
Now add all three equations we have,
$ \Rightarrow \dfrac{{ - 54}}{{60}} + \dfrac{{88}}{{60}} + \dfrac{{ - 39}}{{60}}$
Now all the denominators value are same so we simplify add them we have,
$ \Rightarrow \dfrac{{ - 54}}{{60}} + \dfrac{{88}}{{60}} + \dfrac{{ - 39}}{{60}} = \dfrac{{ - 54 + 88 - 39}}{{60}} = \dfrac{{88 - 93}}{{60}} = \dfrac{{ - 5}}{{60}}$
Divide by 5 in numerator and denominator we have,
.$ \Rightarrow \dfrac{{ - 54}}{{60}} + \dfrac{{88}}{{60}} + \dfrac{{ - 39}}{{60}} = \dfrac{{ - 5}}{{60}} = \dfrac{{ - 1}}{{12}}$
So this is the required sum of the given fractions.

Note – Whenever we face such type of problems the key concept is simply to form a single fraction out of the given fractions given it is achieved by taking L.C.M of the denominator part for all the fractions, simple simplification will help you get on track to reach the answer.