Question

The sum of a number and its reciprocal is ${\displaystyle \frac{10}{3}}$ . Find the number.

Answer 1

Hint:
Assume, the number is x and its reciprocal is${\displaystyle \frac{1}{x}}$. Then add them and equate their sum to${\displaystyle \frac{10}{3}}$.Now we will solve this obtained equation to form a quadratic equation. Once we get the quadratic equation, we will factorize it to get the value of x.

Complete step by step solution:
Given the sum of a number and its reciprocal is${\displaystyle \frac{10}{3}}$.
We have to find the number.
Let the number be x then its reciprocal will be${\displaystyle \frac{1}{x}}$.
Then according to question,
$\Rightarrow x+{\displaystyle \frac{1}{x}}={\displaystyle \frac{10}{3}}$
On taking LCM, we get-
$\Rightarrow {\displaystyle \frac{x^2+1}{x}}={\displaystyle \frac{10}{3}}$
On cross-multiplication, we get-
$\Rightarrow 3\left(x^2+1\right)=10x$
On simplifying, we get-
$\Rightarrow 3x^2+3=10x$
On rearranging, we get-
$\Rightarrow 3x^2-10x+3=0$ -- (i)
Now here the equation we obtained is in quadratic equation so, we can solve the quadratic equation to find the values of x.
On factorization we get-
$\Rightarrow 3x^2-9x-x+3=0$
On simplifying, we get-
$\Rightarrow 3x\left(x-3\right)-1\left(x-3\right)=0$
On further simplifying, we get-
$\Rightarrow \left(3x-1\right)\left(x-3\right)=0$
On equating either multiplication term to$0$, we gat-
$\Rightarrow x={\displaystyle \frac{1}{3}}\text{ or x = 3}$
If x=$3$ then its reciprocal is ${\displaystyle \frac{1}{3}}$
And if x=${\displaystyle \frac{1}{3}}$ then its reciprocal is $3$

Answer- The required number is $3$ and its reciprocal is ${\displaystyle \frac{1}{3}}$ or the required number is ${\displaystyle \frac{1}{3}}$and its reciprocal is $3$

Note:
Here we can also solve the quadratic equation using discriminant methods. If the equation in the standard form $a{x}^2+bx+c=0$ then we find the value of x using formula-
$\Rightarrow$ $x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
So on comparing the standard equation with eq. (i), we get-
$\Rightarrow$ a=$3$ , b=$-10$ and c=$3$
Then putting the values in the formula, we get-
$\Rightarrow x={\displaystyle \frac{-\left(-10\right)\pm \sqrt{{\left(-10\right)}^2-4\times 3\times 3}}{2\times 3}}$
On simplifying this we get-
$\Rightarrow x={\displaystyle \frac{10\pm \sqrt{100-36}}{6}}$
On further solving, we get-
$\Rightarrow x={\displaystyle \frac{10\pm \sqrt{64}}{6}}={\displaystyle \frac{10\pm 8}{6}}$
Now we can get two values of x by first taking the plus sign and solving for x. Then taking the negative sign and solving for x-
$\Rightarrow x={\displaystyle \frac{10+8}{6}}\text{ or }{\displaystyle \frac{10-8}{6}}$
On solving, we get-
$\Rightarrow x=3\text{ or }{\displaystyle \frac{1}{3}}$
Now we can find the reciprocal of the number.