Question

A, B and C can do a piece of work in 10, 15 and 20 days respectively. They began their work together but A left the work after 2 days and B left the work after 3 days. In how many days will C alone would finish the work?

Answer 1

$\textsf{Let the Work to be Completed by A , B , C be : W}$

$\textsf{Given : A can do the Given Work (W) in 10 days}$

$\implies \mathsf{The\;Amount\;of\;Work\;done\;by\;A\;in\;One\;Day = \dfrac{W}{10}}$

$\textsf{Given : B can do the Given Work (W) in 15 days}$

$\implies \mathsf{The\;Amount\;of\;Work\;done\;by\;B\;in\;One\;Day = \dfrac{W}{15}}$

$\textsf{Given : C can do the Given Work (W) in 20 days}$

$\implies \mathsf{The\;Amount\;of\;Work\;done\;by\;C\;in\;One\;Day = \dfrac{W}{20}}$

$\textsf{Given : A , B , C began their work together. But, A left the Work after 2 days}$

$\textsf{It means : The First Two days A , B , C worked together and completed a}\\\textsf{a certain amount of work.}$

$\textsf{The Work completed by A , B , C together at the end of Two days will be}\\\textsf{Work done by A , B , C together in One Day Multiplied by 2}$

$\underline{\textsf{Work completed by A , B , C together at the end of Two days :}}$

$\implies \mathsf{2 \times \bigg(\dfrac{W}{10} + \dfrac{W}{15} + \dfrac{W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{2W}{10} + \dfrac{2W}{15} + \dfrac{2W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{W}{5} + \dfrac{2W}{15} + \dfrac{W}{10}\bigg)}$

$\textsf{Given : B left the Work after 3 Days}$

$\textsf{It means : As A left the Work at end of 2nd Day and B left the Work at the}\\\textsf{end of 3rd Day, The Work done on the 3rd Day is the Work done only by B}\\\textsf{C together.}$

$\implies \mathsf{Work\;done\;by\;B\;and\;C\;together\;on\;the\;3rd\;Day = \bigg(\dfrac{W}{15} + \dfrac{W}{20}\bigg)}$

$\textsf{Total amount of Work completed by the end of 3rd Day will be : The Sum of} \\\textsf{Work done by A , B , C together in First Two days and Work done by B , C}\\ \textsf{together on the 3rd day}$

$\underline{\textsf{Total amount of Work completed by the end of 3rd Day :}}$

$\implies \mathsf{\bigg(\dfrac{W}{5} + \dfrac{2W}{15} + \dfrac{W}{10}\bigg) + \bigg(\dfrac{W}{15} + \dfrac{W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{W}{5} + \dfrac{2W}{15} + \dfrac{W}{15} + \dfrac{W}{10} + \dfrac{W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{W}{5} + \dfrac{3W}{15} + \dfrac{2W + W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{W}{5} + \dfrac{W}{5} + \dfrac{3W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{2W}{5} + \dfrac{3W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{4(2W)+ 3W}{20}\bigg)}$

$\implies \mathsf{\bigg(\dfrac{11W}{20}\bigg)}$

$\textsf{The Remaining Work which C should complete alone will be : The Work}\\\textsf{which is completed by the end of 3rd Day subtracted from the Total Work}$

$\implies \mathsf{Remaining\;Work\;which\;C\;should\;complete\;alone\;will\;be : \bigg(W - \dfrac{11W}{20}\bigg)}$

$\implies \mathsf{Remaining\;Work\;which\;C\;should\;complete\;alone = \bigg(\dfrac{20W -11W}{20}\bigg)}$

$\implies \mathsf{Remaining\;Work\;which\;C\;should\;complete\;alone = \bigg(\dfrac{9W}{20}\bigg)}$

$\mathsf{We\;know\;that : Amount\;of\;Work\;done\;by\;C\;in\;One\;Day = \dfrac{W}{20}}$

$\implies \mathsf{The\;Amount\;of\;Work\;done\;by\;C\;in\;9\;Days = \dfrac{9W}{20}}$

$\textsf{It means : C completes the remaining work alone in 9 days}$

${\textbf{Answer}} : \texttt{C alone would finish the work in \underline{\texttt{9 Days}}}$

Answer 2

Answer:

9 Days

Step-by-step explanation:

Let say work = W

A do in 10 days = W

A do in 1 day = W/10

B do in 15 days = W

B do in 1 day = W/15

C do in 20 days = W

C do in 1 day = W/20

A+B+C do in 1 Day = W/10  + W/15  + W/20

= (W/60) ( 6 + 4 + 3)

= 13W/60

Work done in 2 days by A , B & C = 2 × (13W/60)

= 13W/30

B & C 1 Day work = W/15 + W/20

=(W/60)(4+3)

= 7W/60

Total work done in 3 days ( 2 days ABC together & 1 day BC together)

13W/30  + 7W/60

= (W/60)(26 + 7)

= 33W/60

= 11W/20

Remaining Work to do

W - 11W/20

= (W/20)(20 -11)

= 9W/20

W/20 work done by C =  1 day

1 work done by C = 1/(W/20)

9W/20 work done by C  = {1/(W/20)} × (9W/20)

= 9 Days