Question

The number 50 is divided into two parts such that the sum of their reciprocals is $\dfrac{1}{{12}}$ then these parts are
A.30 and 20
B.10 and 40
C.25 and 25
D.15 and 35

Answer 1

Hint: In this question first we will assume the two part of the number 50 to be a and b and then we will form two equations since their sum is 50 and the sum of their reciprocal is $\dfrac{1}{{12}}$ and then by solving these two equations we will find the unknown numbers.

Complete step-by-step answer:
Given the number is 50 which is divided into two parts
Now let us assume that the number 50 is divided into two parts \[a\] and \[b\] , hence we can write these numbers as
 \[a + b = 50 - - (i)\]
Now it is said that the reciprocal of these numbers \[a\] and \[b\] is equal to $\dfrac{1}{{12}}$, hence we can write this as
 \[\Rightarrow \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{{12}} - - (ii)\]
Now we further solve the equation (ii), we get
 \[\Rightarrow \dfrac{{a + b}}{{ab}} = \dfrac{1}{{12}}\]
Since we know \[a + b = 50\] from equation (i), hence we can further write equation (ii) as
 \[\dfrac{{50}}{{ab}} = \dfrac{1}{{12}}\]
By solving
 \[\Rightarrow ab = 12 \times 50 = 600\]
We can also write this as \[b = \dfrac{{600}}{a} - - (iii)\]
Now substitute b in equation (i), we get
 \[
\Rightarrow a + \dfrac{{600}}{a} = 50 \\
   \Rightarrow {a^2} - 50a + 600 = 0 \;
 \]
Hence by solving this quadratic equation by factorizing the middle terms we can write’
 \[
  {a^2} - 50a + 600 = 0 \\
   \Rightarrow {a^2} - 20a - 30a + 600 = 0 \\
   \Rightarrow a\left( {a - 20} \right) - 30\left( {a - 20} \right) = 0 \\
   \Rightarrow \left( {a - 20} \right)\left( {a - 30} \right) = 0 \;
 \]
So we get \[a = 20,30\]
Now if \[a = 20\] then \[b = \dfrac{{600}}{{20}} = 30\]
And if \[a = 30\] then \[b = \dfrac{{600}}{{30}} = 20\]
Hence we can say option (A) is correct since it matches with the second part of the question.
So, the correct answer is “Option A”.

Note: Another method to solve this question is to assume numbers to be x and 50-x and then finding their reciprocal whose sum is equal to $\dfrac{1}{{12}}$, so by solving the equation value of x can be fund which will be one subparts and other will be (50-x).