Question

Solve: \[\dfrac{{( - 8)}}{{57}} + \dfrac{{( - 2)}}{{57}}\]

Answer 1

Hint: In the given question, we have to solve the fraction. We can solve it by first equalizing the denominator by finding out the LCM of the denominators and then rationalizing the equation. LCM of denominator can be found out using the formula \[\dfrac{{a \times b}}{{HCF(a,b)}}\] .

Complete step-by-step answer:
In order to solve fractions, we should use the following simple steps:
Make sure the bottom numbers (the denominators) are the same. If not, we will rationalize the denominators.
Add the top numbers (the numerators), put that answer over the denominator.
Simplify the fraction (if needed).
We can proceed to solve the give fraction as follows:
First, we will find the HCF of the denominator \[19\]and\[57\]i.e. \[HCF(19,57)\]:
Positive Factors of \[19\]will be \[19\]and \[1\] since it is a prime number.
Positive factors of \[57\]are: \[1,3,19\]and \[57\].
We can obtain the HCF by multiplying all the common factors of both the numbers.
\[19\]and \[1\] are common factors of both \[19\]and \[57\]. Thus-
\[HCF(19,57) = 1 \times 19 = 19\]
Now we can find out the LCM with the help of formula as follows:
\[LCM(a,b) = \dfrac{{a \times b}}{{HCF(a,b)}}\]
\[LCM(19,57) = \dfrac{{19 \times 57}}{{19}}\]
\[LCM(19,57) = 57\]
Now in order to solve the equation, we have to rationalize the denominators with the LCM and solve as follows:
 \[\dfrac{{( - 8)}}{{19}} + \dfrac{{( - 2)}}{{57}}\]
Multiplying numerator and denominator of \[\dfrac{{( - 8)}}{{19}}\]by \[3\]to rationalize, we get,
\[ = \dfrac{{( - 8) \times 3}}{{19 \times 3}} + \dfrac{{( - 2)}}{{57}}\]
\[ = \dfrac{{( - 24)}}{{57}} + \dfrac{{( - 2)}}{{57}}\]
Adding the denominators, we get,
\[ = \dfrac{{( - 24) + ( - 2)}}{{57}}\]
\[ = \dfrac{{( - 26)}}{{57}}\]
Thus, \[\dfrac{{( - 8)}}{{19}} + \dfrac{{( - 2)}}{{57}} = \dfrac{{( - 26)}}{{57}}\].

Note: LCM stands for Lowest Common Multiple and HCF stands for Highest Common Factor. Apart from the above given method to find LCM, we can find it directly by using multiplication method as follows:
We will write all the multiples till common multiple occurs-
\[19 \times 1 = 19\]
\[19 \times 2 = 38\]
\[19 \times 3 = 57\]
And \[57 \times 1 = 57\].
Since \[57\]occurs first and it is common for both the numbers, it will be the LCM.
This method is generally not preferred because we do not know how many multiples we have to write till we arrive at the solution.
We can alternatively solve the sum directly without LCM or HCF as follows by simplifying:
Cross multiplying both the denominators to equalize them-
\[ = \dfrac{{( - 8) \times 57}}{{19 \times 57}} + \dfrac{{( - 2) \times 19}}{{57 \times 19}}\]
\[ = \dfrac{{( - 456)}}{{1083}} + \dfrac{{( - 38)}}{{1083}}\]
\[ = \dfrac{{( - 456) + ( - 38)}}{{1083}}\]
\[ = \dfrac{{( - 494)}}{{1083}}\]
Dividing by common factor \[19\], we get,
\[ = \dfrac{{( - 26)}}{{57}}\].