Question

If the sum of the two numbers is \[9\] and the sum of their reciprocals is\[\dfrac{1}{2}\] . Find the numbers.

Answer 1

Hint: If \[x\] is a number then its reciprocal is $\dfrac{1}{x}$ .
Then frame the equations according to the given data.
If you get the equation of the general form $a{x^2} + bx + c = 0$ use the below formula to find x, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Stepwise Solution:
 Given: Sum of the two numbers is \[9\] .
The sum of their reciprocals is\[\dfrac{1}{2}\] .
Let the number be \[x\] and therefore its reciprocal becomes $\dfrac{1}{x}$ .
Let the other number be $y$ and therefore its reciprocal becomes $\dfrac{1}{y}$ .
According to the question,
The sum of the two numbers is \[9\]
$ \Rightarrow x + y = 9$
$ \Rightarrow y = 9 - x$ …………….. Equation 1
The sum of their reciprocals is\[\dfrac{1}{2}\]
$ \Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{2}$ …………… Equation 2
On substituting equation 1 in equation 2 we get,
$
  \dfrac{1}{x} + \dfrac{1}{{9 - x}} = \dfrac{1}{2} \\
   \Rightarrow \dfrac{{9 - x + x}}{{x\left( {9 - x} \right)}} = \dfrac{1}{2} \\
   \Rightarrow \dfrac{9}{{9x - {x^2}}} = \dfrac{1}{2} \\
   \Rightarrow 18 = 9x - {x^2} \\
   \Rightarrow {x^2} - 9x + 18 = 0 \\
$
Now the above equation is in the form of general quadratic equation $a{x^2} + bx + c = 0$ ,
solve for x using $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Comparing given equation with the general form of quadratic equation $a{x^2} + bx + c = 0$ we get,
From the above equation a is $1$ , b is $ - 9$ and c is $18$
\[
  x = \dfrac{{ - ( - 9) \pm \sqrt {{{( - 9)}^2} - 4 \times 1 \times 18} }}{{2 \times 1}} \\
   \Rightarrow x = \dfrac{{9 \pm \sqrt {81 - 72} }}{2} \\
   \Rightarrow x = \dfrac{{9 \pm \sqrt 9 }}{2} \\
   \Rightarrow x = \dfrac{{9 \pm 3}}{2} \\
\]
Therefore,
\[
  x = \dfrac{{9 + 3}}{2} \\
   \Rightarrow x = \dfrac{{12}}{2} \\
   \Rightarrow x = 6 \\
\]
0r
\[
  x = \dfrac{{9 - 3}}{2} \\
   \Rightarrow x = \dfrac{6}{2} \\
   \Rightarrow x = 3 \\
\]
If $x = 6$ then $y = 9 - 6 = 3$
If $x = 3$ then $y = 9 - 3 = 6$
Therefore the required numbers are $6,3$

Note: In such types of questions which involve wordy information they might ask you to frame equations and so we will need to have knowledge about solving equations. Most of the time solving equations leads to quadratic equation forms and so we need to have knowledge about the formula to find the value of x. As calculations play a critical role in these kinds of questions we will need to be vigilant about that while solving.