Question

The difference of two numbers is 5 and the difference of their reciprocals is \[\dfrac{1}{{10}}\] find the numbers.
A. 10,50
B. 10,5
C. 10,-5
D. 0,5

Answer 1

Hint: Assume two separate variables and then form two separate quotations using the properties mentioned in the question then solve those to get the values of the variables thus taken.

Complete step by step answer:
Let us assume that the 2 variables are x and y separately then we are given that the difference of two numbers is 5, which means that \[x - y = 5\]
And it is also given that the difference between the reciprocals of which is \[\dfrac{1}{{10}}\] , which means that \[\dfrac{1}{y} - \dfrac{1}{x} = \dfrac{1}{{10}}\]
Now if we look into the first equation we are getting
\[\begin{array}{l}
\therefore x - y = 5\\
 \Rightarrow - y = 5 - x\\
 \Rightarrow y = x - 5
\end{array}\]
Now putting this value in the second equation we will get
\[\begin{array}{l}
\therefore \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{{10}}\\
 \Rightarrow \dfrac{1}{{x - 5}} - \dfrac{1}{x} = \dfrac{1}{{10}}\\
 \Rightarrow \dfrac{{x - (x - 5)}}{{x(x - 5)}} = \dfrac{1}{{10}}\\
 \Rightarrow \dfrac{5}{{x(x - 5)}} = \dfrac{1}{{10}}\\
 \Rightarrow 50 = x(x - 5)\\
 \Rightarrow {x^2} - 5x - 50 = 0\\
 \Rightarrow {x^2} - 10x + 5x - 50 = 0\\
 \Rightarrow x(x - 10) + 5(x - 10) = 0\\
 \Rightarrow (x + 5)(x - 10) = 0\\
 \Rightarrow x = 10, - 5
\end{array}\]
So if i take the value of x as -5
Then the value of y will become \[y = x - 5 = - 5 - 5 = - 10\]
And if we take the value of x as 10
Then the value of y will be \[y = x - 5 = 10 - 5 = 5\]
In the options 10,5 is given which is the correct option here.

So, the correct answer is “Option B”.

Note: While creating the equations i have automatically imagined \[x > y\] which means \[\dfrac{1}{x} < \dfrac{1}{y}\] and that's why i took \[\dfrac{1}{y} - \dfrac{1}{x} = \dfrac{1}{{10}}\] because \[\dfrac{1}{x} - \dfrac{1}{y}\] will be a negative quantity and \[\dfrac{1}{{10}}\] is itself positive.