Question

A can do a piece of work in 10 days and B in 15 days . How long will they take together to finish it ?
A. \[7{\text{ }}days\]
B. \[3{\text{ }}days\]
C. \[9{\text{ }}days\]
D. \[6{\text{ }}days\]

Answer 1

Hint:We have to find the number of days taken by both A and B to complete a piece of work together . We solve this question using the concept of solving linear equations . We will first find the amount of work in which both A and B alone can complete the piece of work in one day . Then to calculate the number of days in which both A and B can together complete the work we add the amount of work done by both A and B alone . Adding the reciprocals would give us the amount of work done by both in one day . Taking the reciprocal of the amount of work done by both gives us the number of days sin which both A and B together can complete the piece of work.

Complete step by step answer:
Given: A can do a piece of work in \[10{\text{ }}days\] and B can do the piece of work in \[15{\text{ }}days\]
So , the work done in one day by A is given as :
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A = \dfrac{1}{{10}}\]
Similarly , the work done in one day by B is given as :
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}B = \dfrac{1}{{15}}\]

Work done together by both A and B in one day is given as :
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B = Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A + Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}B\]
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B = \dfrac{1}{{10}}{\text{ + }}\dfrac{1}{{15}}\]
On , taking L.C.M. we get
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B = \dfrac{{3 + 2}}{{30}}\]
\[\Rightarrow Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B = \dfrac{5}{{30}}\]

Cancelling the terms , we get the value for Work done in one day by A and B as :
\[Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B = \dfrac{1}{6}\]
So , the number of days in which both A and B together complete the piece of work is given as :
\[Number{\text{ }}of{\text{ }}days{\text{ }}to{\text{ }}finish{\text{ }}work{\text{ }}together = \dfrac{1}{{Work{\text{ }}done{\text{ }}in{\text{ }}one{\text{ }}day{\text{ }}by{\text{ }}A{\text{ }}and{\text{ }}B}}\]
\[\Rightarrow Number{\text{ }}of{\text{ }}days{\text{ }}to{\text{ }}finish{\text{ }}work{\text{ }}together = \dfrac{1}{{\left( {\dfrac{1}{6}} \right)}}\]
On simplifying , we get the number of days in which both A and B together complete the piece of work is given as :
\[Number{\text{ }}of{\text{ }}days{\text{ }}to{\text{ }}finish{\text{ }}work{\text{ }}together = 6\]
Thus , the number of days taken by both A and B to finish the piece of work together is \[6{\text{ }}days\] .

Hence , the correct option is D.

Note:A mistake which we might to while solving such types of question is that we may directly add the number of days in which A and B can complete the piece to work separately for the number of days in which both can complete the work together , as \[number{\text{ }}of{\text{ }}days = 10 + 15\] . This is a possible point where it can be mistaken . Also , at the last we have to do the reciprocal of the sums of the reciprocals of the individuals otherwise again we would end up having wrong answers.