Question

A and B can do a job in $12\,days$ and B and C can do it in $16\,days$. After A has been working for $5\,days$ and B for $7\,days$, C finishes rest of work in $13\,days$, in how many days can C do the above work alone .
$
  a)\,\,18\,days \\
  b)\,\,36\,days \\
  c)\,\,48\,days \\
  d)\,\,24\,days \\
 $

Answer 1

Hint: Assume A takes $a$ days, B takes $b$ days and C takes $c$ days to finish the work, then one day work for A, B and C is $\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}$ respectively.
Now you are given three conditions, solve them and get the answer

Complete step-by-step answer: Here according to question, A and B can do the job in $12\,days$ and the same job is done by B and C in $16\,days$.
So, let A alone do the work in $a\,days$
Let B alone do the work in $b\,days$
Let C alone do the work in $c\,days$
So we can say that one day work for A, B and C is $\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}$ respectively.
So first it is given that A and B can do the job in $12\,days$
$\therefore \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{{12}}$ $ \to \left( 1 \right)$
Also it is said that B and C can do it in $16\,days$
$\therefore \,\dfrac{1}{b} + \dfrac{1}{c} = \dfrac{1}{{16}}$ $ \to \left( 2 \right)$
And after that question says that if A works for $5\,days$ and B works for $7\,days$ but after A and B has completed the rest work has been completed by C then $13\,days$.
$\therefore \dfrac{5}{a} + \dfrac{7}{b} + \dfrac{{13}}{c} = 1$ $ \to \left( 3 \right)$
Mow we have three equations and three variables,
Now let’s add equation (1) and equation (2)
  $\therefore \dfrac{1}{a} + \dfrac{1}{b}$$ + \,\dfrac{1}{b} + \dfrac{1}{c}$$ = \dfrac{1}{{12}} + \dfrac{1}{{16}}$
$\dfrac{1}{a} + \dfrac{2}{b} + \dfrac{1}{c} = \dfrac{7}{{48}}$
Now if we multiply the first equation with $5$ and the second equation with $2$ and add them we will get
$
  5\left( {\dfrac{1}{a} + \dfrac{1}{b}} \right) + 2\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right) = \dfrac{5}{{12}} + \dfrac{2}{{16}} \\
  \dfrac{5}{a} + \dfrac{7}{b} + \dfrac{2}{c} = \dfrac{{26}}{{48}}\,\,\,\,\,\,\,\,\, \to \left( 4 \right) \\
 $
Now if we subtract equation (4) from equation (3)
$
  \left( {\dfrac{5}{a} + \dfrac{7}{b} + \dfrac{{13}}{c}} \right) - \left( {\dfrac{5}{a} + \dfrac{7}{b} + \dfrac{2}{c}} \right) = 1 - \dfrac{{26}}{{48}} \\
  \dfrac{{11}}{c} = \dfrac{{48 - 26}}{{48}} \\
  c = \dfrac{{11 \times 48}}{{22}} \\
  c = 24\,days \\
 $
Hence we can say that C alone will take $24\,days$ to complete the work.

Note: If this type of question is given you have to assume the total days to complete the work by A, B & C and then generate equations by using their one day work that is $\dfrac{1}{a},\,\dfrac{1}{b},\,\dfrac{1}{c}$ respectively.