Question

Let s, m, and f denote the respective time taken by son, mother and father to complete the work. If$\dfrac{1}{s} + \dfrac{1}{m} = \dfrac{1}{{24}},\dfrac{1}{s} + \dfrac{1}{f} = \dfrac{1}{{18}}$and$\dfrac{1}{m} + \dfrac{1}{f} = \dfrac{1}{{36}}$ then find $\dfrac{1}{s} + \dfrac{1}{m} + \dfrac{1}{f}$and$\dfrac{1}{s},\dfrac{1}{m},\dfrac{1}{f}$respectively.
A. $\dfrac{1}{{16}},\dfrac{1}{{144}},\dfrac{5}{{144}},\dfrac{3}{{144}}$
B. $\dfrac{1}{{16}},\dfrac{5}{{144}},\dfrac{1}{{144}},\dfrac{3}{{144}}$
C. $\dfrac{1}{{16}},\dfrac{3}{{144}},\dfrac{1}{{144}},\dfrac{5}{{144}}$
D. None of the these

Answer 1

Hint: In this question, use the given information and name the given equations and use the addition operation to form a new equation and then use the subtraction operation for new equation and previous equation, use this information to approach the solution of the question.

Complete step by step answer:
As it is given so to solve this question, we first let
$\dfrac{1}{s} + \dfrac{1}{m} = \dfrac{1}{{24}}....eq\left( i \right)$
$\dfrac{1}{s} + \dfrac{1}{f} = \dfrac{1}{{18}}....eq\left( {ii} \right)$
$\dfrac{1}{m} + \dfrac{1}{f} = \dfrac{1}{{36}}....eq\left( {iii} \right)$
Now, we will have to add all the three equation $eq\left( i \right) + eq\left( {ii} \right) + eq\left( {iii} \right)$
$\dfrac{2}{s} + \dfrac{2}{m} + \dfrac{2}{f} = \dfrac{1}{{24}} + \dfrac{1}{{18}} + \dfrac{1}{{36}}$
$\dfrac{2}{s} + \dfrac{2}{m} + \dfrac{2}{f} = \dfrac{{3 + 4 + 2}}{{72}}$
$\dfrac{1}{s} + \dfrac{1}{m} + \dfrac{1}{f} = \dfrac{9}{{144}} = \dfrac{1}{{16}}....eq\left( {iv} \right)$
We have $eq\left( {iv} \right)$, so by subtracting $eq\left( i \right)$ from $eq\left( {iv} \right)$, we get
$\begin{gathered}
  \dfrac{1}{s} + \dfrac{1}{m} + \dfrac{1}{f} - \dfrac{1}{s} - \dfrac{1}{m} = \dfrac{1}{{16}} - \dfrac{1}{{24}} \\
  \dfrac{1}{f} = \dfrac{1}{{48}} = \dfrac{3}{{144}} \\
\end{gathered} $
For finding out the value of $\dfrac{1}{m}$, we need to subtract $eq\left( {ii} \right)$ from$eq\left( {iv} \right)$, we get
$\dfrac{1}{s} + \dfrac{1}{m} + \dfrac{1}{f} - \dfrac{1}{s} - \dfrac{1}{f} = \dfrac{1}{{16}} - \dfrac{1}{{18}}$
$\dfrac{1}{m} = \dfrac{1}{{144}}$
And last for the value of $\dfrac{1}{s}$, subtract$eq\left( {iii} \right)$from$eq\left( {iv} \right)$, we will get,
$\dfrac{1}{s} + \dfrac{1}{m} + \dfrac{1}{f} - \dfrac{1}{m} - \dfrac{1}{f} = \dfrac{1}{{16}} - \dfrac{1}{{36}}$
$\dfrac{1}{s} = \dfrac{5}{{144}}$

So, the correct answer is “Option B”.

Note: In the above question we had the three equations where the values where unknown and the trick behind these types of question is to use the operations like subtraction, addition between the given equations and due to which we able to form the new equation and then we used the substitution method to find the values of the unknown variables.