Question

The sum of two numbers is 8. Determine the numbers, if the sum of their reciprocals is $\dfrac{8}{{15}}$.

Answer 1

Hint: Let the two numbers be x and y and then use the information given in the question to form equations containing x and y. Then solve both equations to get the numbers.

Complete step-by-step answer:
Let us consider the two numbers are x and y.
Now, given in the question, the sum of the two numbers = 8 and the sum of their reciprocals = $\dfrac{8}{{15}}$.
Now, forming the equations using x and y and the information given in the question, we get-
$
  x + y = 8 - (1) \\
  \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{8}{{15}} - (2) \\
$
Using equation (1) we can write
$x = 8 - y$
substitute value of x in equation (2), we get-
$\dfrac{1}{{8 - y}} + \dfrac{1}{y} = \dfrac{8}{{15}}$
 solving this equation further,
$
  \dfrac{{y + 8 - y}}{{(8 - y)y}} = \dfrac{8}{{15}} \\
   \Rightarrow \dfrac{8}{{(8 - y)y}} = \dfrac{8}{{15}} \\
   \Rightarrow (8 - y)y = 15 \\
   \Rightarrow 8y - {y^2} = 15 \\
   \Rightarrow {y^2} - 8y + 15 = 0 \\
 $
Solving the quadratic equation, we get-
$
  {y^2} - 8y + 15 = 0 \\
  \Rightarrow {y^2} - 3y - 5y + 15 = 0 \\
 \Rightarrow y(y - 3) + 5(y - 3) = 0 \\
   \Rightarrow (y - 3)(y + 5) = 0 \\
$
This implies we have two values of y, $y = 3,y = - 5$.
For $y = 3,$we get $x = 8 - 3 = 5${since $x = 8 - y$}
For $y = - 5,$we get $x = 8 - ( - 5) = 13$
 $y = - 5,$ and $x = 13,$ doesn’t satisfy equation (2).
Hence, the two numbers are $x = 5$, $y = 3$.

Note: Whenever such a type of question, always for equations by using the details given in the question. Then, using the two equations formed, we can find the two unknown variables, as mentioned in the solution.