Question

A, B and C together can finish a piece of work in 12 days. A and C together work twice as much as B, A and B together work thrice as much as C. In what time (day) could each do it separately?
A.144/5, 36, 48
B.36, 48, 144/5
C.48, 144/5, 36
D.None of these

Answer 1

Hint: In this question, we need to determine the time taken (in days) by A, B and C to complete the work alone. For this, we will use the unitary method and the concept which states that the total work completed in a day is the summation of the work done by the individual workers in a day.

Complete step-by-step answer:
Let A, B and C complete the work while working alone be A, B and C respectively.
A, B and C complete a work in 12 days working together. So, the amount of work completed by A, B and C in a day is given as:
$\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{12}} - - - - (i)$
According to the question,
A and C together work twice as much as B alone. So, $\dfrac{1}{A} + \dfrac{1}{C} = 2\left( {\dfrac{1}{B}} \right) - - - - (ii)$
Also, A and B together work thrice as much as C. So, $\dfrac{1}{A} + \dfrac{1}{B} = 3\left( {\dfrac{1}{C}} \right) - - - - (iii)$
Substituting equation (ii) in the equation (i) to determine the time taken by B alone to complete the work.
$
  \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{12}} \\
  2\left( {\dfrac{1}{B}} \right) + \dfrac{1}{B} = \dfrac{1}{{12}} \\
  \dfrac{3}{B} = \dfrac{1}{{12}} \\
  B = 36{\text{ days}} \\
 $

Similarly, substituting equation (iii) in the equation (i) to determine the time taken by C alone to complete the work.
$
  \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{12}} \\
  3\left( {\dfrac{1}{C}} \right) + \dfrac{1}{C} = \dfrac{1}{{12}} \\
  \dfrac{4}{C} = \dfrac{1}{{12}} \\
  C = 48{\text{ days}} \\
 $

Now, substitute $B = 36{\text{ days}}$ and $C = 48{\text{ days}}$ in the equation $\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{12}}$ to determine the time taken by A alone to complete the work.
$
  \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{12}} \\
  \dfrac{1}{A} + \dfrac{1}{{36}} + \dfrac{1}{{48}} = \dfrac{1}{{12}} \\
  \dfrac{1}{A} = \dfrac{1}{{12}} - \dfrac{1}{{36}} - \dfrac{1}{{48}} \\
   = \dfrac{{12 - 4 - 3}}{{144}} \\
   = \dfrac{5}{{144}}{\text{ days}} \\
 $
Hence, the time taken by A alone to complete the work is $\dfrac{{144}}{5}{\text{ days}}$
So, the time taken (in days) by A, B and C to complete the work alone is $\dfrac{{144}}{5}{\text{, 36 and 48 days}}$ respectively.
Option A is correct.
So, the correct answer is “Option A”.

Note: Students must be careful while using the one-day method as all the calculations are done with the reciprocal terms. Also, the time taken by an alone person to complete the work is always more than the time taken by a group of persons.