Question

Solve the fraction $\dfrac{-8}{19}+\dfrac{(-2)}{57}$

Answer 1

Hint: In the question we are given an equation in which both terms are rational numbers. To understand this question properly we will learn the definition of rational numbers then we will learn the addition and subtraction rule of rational numbers which is the same as of integers then we will use these rules to get our required solution.

Complete step-by-step solution:
Integers:
Integers consist of all negative numbers, positive numbers, and zero.
Rational Numbers:
Rational numbers are the numbers which can be represented in the form $\dfrac{p}{q}$ where $p\And q$ are integers and $q\ne 0$ .
By definition we have understand that rational numbers consist of integers so for solving any equation in which rational numbers are present, we will use rules of integers
We will also use rules of fraction to solve an equation of rational numbers.
In integers if we want to add or subtract two numbers then there is a rule to perform these operations.
If we add both positive numbers then we will do basic addition.
Like if we want to add $8$ and $6$
Then we will simply do addition
$\begin{align}
  & 8+6 \\
 & \Rightarrow 14 \\
\end{align}$
$\therefore 14$ Is our required answer.
If we add two numbers and one number is positive and other number is negative then resultant operation will be of subtraction and sign of answer is of a number whose value is bigger irrespective of its sign
Example:
Solve $-12+15$
$\begin{align}
  &= -12+15 \\
 &= 3 \\
\end{align}$
$\therefore 3$ Is our required answer.
In this example one number is negative and other number is positive
So we have done subtraction and the sign of the answer is the same as the sign of a bigger value number irrespective of its sign as $15$ is greater than $12$ .
If both the numbers are negative numbers then the resultant operation will be of addition and sign will be negative
Example:
Add:$-12+(-8)$
$\begin{align}
  &= -12+(-8) \\
 &= -20 \\
\end{align}$
$\therefore -20$ Is our required answer.
Now let us proceed to our question
In the question we are given an equation and we have to find the value of it
Our given equation is
$\dfrac{-8}{19}+\dfrac{(-2)}{57}$
In the equation both terms are rational numbers so for solving these we will use rules of integers and fractions to solve this equation.
$\dfrac{-8}{19}+\dfrac{(-2)}{57}$
First we will try to make denominator of each term same, so for this we will take LCM of $19$ and $57$
LCM of $19$ and $57$ is $57$
As we can see, the denominator of second terms is $57$ so there is no need to change it. We only need to change first term
$\dfrac{-8}{19}$
As we know $19\times 3=57$
So, we will multiply both numerator and denominator by $3$
Then our first terms becomes
$\begin{align}
  &= \dfrac{-8\times 3}{19\times 3} \\
 &= \dfrac{-24}{57} \\
\end{align}$
So now our equation becomes
$\dfrac{(-24)}{57}+\dfrac{(-2)}{57}$
In simplified form we can write it as
$\begin{align}
  &= {(-24)}{57}+\dfrac{(-2)}{57} \\
 & = \dfrac{(-24)+(-2)}{57} \\
\end{align}$
Now we will solve numerator
$\begin{align}
  &= \dfrac{-24-2}{57} \\
 &= \dfrac{-26}{57} \\
\end{align}$
$\therefore \dfrac{-26}{57}$ Is our required answer.
Hence $\dfrac{-8}{19}+\dfrac{(-2)}{57}=\dfrac{-24}{57}$.

Note: In rational numbers there are two important terms which are additive identity and multiplicative identity. $0$ is called additive identity which means if we add or subtract any number from $0$ then we will get the number itself and $1$ is called multiplicative identity which means if we multiply or divide any number from $1$ then we will get the number itself.