Question

Add the following rational number: $-1$ and $\dfrac{3}{4}$.

Answer 1

Hint:
We can write -1 as a rational number. Then we can make the denominator of the two rational numbers by multiplying with the same number on both numerator and denominator. Then we can add the numerators as the denominators are the same. Then we can simplify it to get the required solution.

Complete step by step solution:
We need to add the rational numbers -1 and $\dfrac{3}{4}$
We know that a rational number is of the form $\dfrac{p}{q}$ , where p and q are integers.
So, we can write -1 as $ - 1 = \dfrac{{ - 1}}{1}$
We know that the value of the fraction remains the same when multiplied with the same number on the numerator and the denominator. So, we can multiply both the numerator and the denominator with 4.
$ \Rightarrow - 1 = \dfrac{{ - 1 \times 4}}{{1 \times 4}}$
Hence, we have
$ \Rightarrow - 1 = \dfrac{{ - 4}}{4}$ … (1)
Now, we can add the two rational numbers.
Let, $S = - 1 + \dfrac{3}{4}$
On substituting equation (1), we get the sum S as
$ \Rightarrow S = \dfrac{{ - 4}}{4} + \dfrac{3}{4}$
As the denominators are equal, we can add the numerators. So, the sum S will become
$ \Rightarrow S = \dfrac{{ - 4 + 3}}{4}$
On simplification, we get the sum S as
$ \Rightarrow S = \dfrac{{ - 1}}{4}$

Therefore, the required sum is $\dfrac{{ - 1}}{4}$

Note:
Alternate method to solve this problem is given by writing the sum and then by cross multiplying to make the denominator the same. Then we can add the numerator and simplify to get the required solution.
Let $S = - 1 + \dfrac{3}{4}$
We can write -1 as $ - 1 = \dfrac{{ - 1}}{1}$
$ \Rightarrow S = \dfrac{{ - 1}}{1} + \dfrac{3}{4}$
On taking the LCM by cross multiplying, we get
$ \Rightarrow S = \dfrac{{ - 4}}{4} + \dfrac{3}{4}$
As the denominators are equal, we can add the numerators. So, the sum S will become,
$ \Rightarrow S = \dfrac{{ - 4 + 3}}{4}$
On simplification, we get
$ \Rightarrow S = \dfrac{{ - 1}}{4}$
Therefore, the required sum is $\dfrac{{ - 1}}{4}$ .