Question

What is the number if the sum of the number with its reciprocal is $\dfrac{{13}}{6}$?

Answer 1

Hint: Reciprocal of a number means, inverse of that number, i.e. $\dfrac{1}{{number}}$. Here take the number as ‘x’ and take the reciprocal which is $\dfrac{1}{x}$. Then do the sum of these two numbers. After doing the simple mathematical calculation, we will get a quadratic equation. THen we just need to solve the quadratic equation either by factorisation method or by using the formulae. We will get the required answer.

Complete step-by-step answer:
Let us consider the unknown number to be x.
Equation interpreted from the question:
$x + \dfrac{1}{x} = \dfrac{{13}}{6}$
To find: x
LHS: $x + \dfrac{1}{x}$
LHS has the addition sign, but to add the base is not the same. We need to make the base the same in order to move ahead with addition.
To make the base equal, we need to take LCM.
So, after taking the LCM, we get the LHS as,
$LHS:\dfrac{{{x^2} + 1}}{x}$
Therefore, the original equation becomes, $\dfrac{{{x^2} + 1}}{x} = \dfrac{{13}}{6}$
On cross multiplication, we get,
$6 \times ({x^2} + 1) = 13x$
On simplifying, we get,
$6{x^2} + 6 = 13x$
Taking all the terms on one side and equating them to 0, we get,
We get a binomial equation, i.e. $6{x^2} - 13x + 6 = 0$
$6{x^2} - 9x - 4x + 6 = 0$
$3x \times (2x - 3) - 2 \times (2x - 3) = 0 $
$(3x - 2) \times (2x - 3) = 0 $
Therefore,$(3x - 2) = 0 $
 and $(2x - 3) = 0 $
We get 2 equations after solving the binomial equation. Now we can solve for the value of x.
$(3x - 2) = 0\,and\,(2x - 3) = 0$
Therefore, we get $x = \dfrac{2}{3}\,and\,x = \dfrac{3}{2}$
But here x should have only one value. If you look closely at the values which we are getting for x, they are inverse or reciprocal of each other. So, you can choose any one of them equal to x, as
$\dfrac{2}{3} + \dfrac{3}{2} = \dfrac{3}{2} + \dfrac{2}{3}$.
Therefore, the number whose sum with its reciprocal is $\dfrac{{13}}{6}$ is $\dfrac{2}{3}$ or $\dfrac{3}{2}$.

Note: Read the question very carefully. After reading the question, which value is unknown and in what way we can find that value has to be clear. Interpreting the question into an equation has to be the important part of such sums. Once we form the equation, the rest is just solving. Most of the time binomial equations end up giving 2 values of x, we need to put the values of x in the original equation and check which one value to choose.