Question

When a particular number is added to its own reciprocal, the resulting sum is $2.$ Find the number.
A. $ - 2$
B. $ - 1$
C. $ - \dfrac{1}{2}$
D. $1$
E. $2$

Answer 1

Hint: Here we need to proceed by letting the number to be a variable. Then we can try to make the equation from the statement that is given to us to solve that required variable.

Complete step-by-step answer:
Here we are given that when a number is added to its own reciprocal, the resulting sum is equal to $2$
So now we must know what the reciprocal of the number means. If the number is $x$ then we can say that when we divide $1$ by this number then it is turned into its reciprocal. So we can say that the reciprocal of this number which we have taken is $\dfrac{1}{x}$
So according to the given statement we need to find the number whose reciprocal’s sum to it is equal to $2$
So let the number be $x$
Reciprocal of this $ = \dfrac{1}{x}$
According to the statement we are given that sum of the number and the reciprocal is $2$
So now we can convert this statement into the equation form by writing that:
$ \Rightarrow$ $x + \dfrac{1}{x} = 2$
Now we need to solve this equation.
Taking the LCM which is $x,$ we get
$ \Rightarrow$ $\dfrac{{{x^2} + 1}}{x} = 2$
Simplifying this further we get:
$
 \Rightarrow {x^2} + 1 = 2x \\
 \Rightarrow {x^2} - 2x + 1 = 0 \\
 $
We also know that
$
 \Rightarrow {(x - 1)^2} = {x^2} - 2x + 1 \\
    \\
 $
So we can write the above equation after simplification that:
$
 \Rightarrow x - 1 = 0 \\
   \Rightarrow x = 1 \\
 $
Hence we get that the required number $x = 1$
So here we also needed to simplify the equation that is formed by converting the given statement.
So D is the correct option.

Note: Here the student can also apply the hit and trial methods by using the value of the required number from the options. Let us take the option A which is $ - 2$ then reciprocal will be $\dfrac{{ - 1}}{2}$
Now we can find the sum$ = - 2 - \dfrac{1}{2} = - \dfrac{5}{2} \ne 2$
Hence it is not the correct option. Similarly we can check for other options also to save time.