Time and work wages and chain rule is one of the most important and frequently tested topics in quantitative aptitude for competitive exams. It is asked in almost every major exam including CAT, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. A strong understanding of time and work wages and chain rule concept, formulas and tricks is essential for scoring well in these exams. In this post we cover three major subtopics. First is time and work which covers the basic definition of efficiency and work formula, three methods of solving problems — LCM method, percentage method and fraction method, ratio of efficiencies of two or more persons, and problems when persons leave work in between from beginning or from end. Second is wages which covers how total payment is divided among workers always in the ratio of work done by each person and not in ratio of time spent. Third is chain rule which covers problems involving multiple variables like number of workers, number of days, hours per day and efficiency all changing simultaneously using the formula N1 × D1 × H1 × E1 = N2 × D2 × H2 × E2 — all explained with multiple methods and solved examples.
Time & Work, Wages, Chain Rule
∴ efficiency = \(\frac{{work}}{{time}}\)
So we can say that work done in unit time by a person is called the efficiency of that person.
Hence, efficiency ∝ work
& efficiency ∝ \(\frac{1}{{time}}\)
So if A can complete a work in t₁ time
& B can completer a work in t₂ time
then \(\frac{{{A_{Eff.}}}}{{{B_{Eff.}}}} = \frac{{{t_2}}}{{{t_1}}}\)
● Problems can be solved by following three approaches as per given data value:
● LCM Method:
\({A_{eff}} = \frac{{Total\;Work}}{{time\;to\;complete\;total\;work\;by\;A}}\)
Similar for \({B_{eff}}\) & \({C_{eff}}\)
So we can say work done in unit time by A, B or C is called its efficiency.
because efficiency is inversely proportional to time taken to complete the work.
● Chain Rule:-
because the job of engine is to work coal.
● No. of Examiner₁ × days₁ × work rate₁ × efficiency₁ = No. of Examiner₂ × days₂ × work rate₂ × efficiency₂
(Q). A can complete \({\frac{2}{5}^{th}}\) of a work in 1 day. In how many days he can complete the same task.
Sol:
∴ \({A_{eff}} = \frac{2}{5}\)
time taken by A to complete the job = \(\frac{1}{{{A_{eff}}}} = \frac{5}{2}\) = 2.5 days Answer
(Q). A can complete a job in 10 days, B can do the same job in 20 days. In how many days working together they can complete the job.
Sol:
(Q). A can do a piece of job in 10 days. B can do it in 15 days & C can do it in 20 days. In how many days A, B and C can complete the whole work working togethers.
Sol:-
● If persons are leaving the work in between the completion of whole work:
⟹ whoever leaves the work from forward, he is marked minus and becomes absent to calculate complete time of completion of work
⟹ whoever leaves the work from behind, he is marked plus and becomes present to calculate complete time of completion of work.
(Q). A can do a work in 20 days and B can do it in 30 days. If A & B started working together but A left th job after 8 days. Find the time taken to complete the remaining job by B & overall time to complete the job.
Sol:
total work = 60
total work 8 days of (A + B) = 8 + (3 × 2)
= 40
Remaining work = 60 – 40 = 20
∴ time taken by B to do the Remaining job = \(\frac{{20}}{2}\)
= 10 days Answer
overall time to complete the work = 8 + 10 = 18 days Answer
Alternate:
∴ time taken to complete work = \(\frac{{60 – 8 \times 3}}{2}\) = 18 days Answer
∴ B worked for 18 – 8 = 10 days Answer
(Q). A can do a job in 20 days & B can do it in 30 days. If A & B started working together but B left th job 5 days before completion. Find the time taken to complete the job.
Sol:-
5 days work of A = 5 × 3 = 15
∴ remaining work which will be done by A & B together = 60 – 15 = 45
combined efficiency of A & B = 3 + 2 = 5
∴ time taken by A & B to do combined work = \(\frac{{45}}{9}\) = 9
∴ total time to complete the job = 9 + 5 = 14 days Answer
Alternate:
time to complete job = \(\frac{{60 + 5 \times 2}}{{3 + 2}}\) = 14 days Answer
(Q) Two workers A & B are engaged to do a piece of work. A working alone would take 12 hours more than they working together. If B worked alone, he would take \(1\frac{1}{3}\) hours more than when worked together. Find the time required to finish the work together ?
Sol:-
(Q). A and B are equally efficient, and each could individually complete a piece of work in 30 days. A & B started working together but A took ‘a’ days off after every four days of work while B took ‘a’ days off after every five days of work. If they started work on 01 august, 2022, on which date was the work completed ?
Sol:
If A & B would not have taken any leave then the work would have been completed in \(\frac{{30}}{2}\) = 15 days
But A & B are taking leaves after working few days so time taken will be more than 15 days.
Now in these 15 days A’s leaves = \(\left[ {\frac{{15}}{4}} \right]\) = 3
& in these 15 days B’s leave = \(\left[ {\frac{{15}}{5}} \right]\) = 3
B will take 3ʳᵈ leave after work completion so its 3ʳᵈ leave will not be counted
So B’s leaves will be counted 2
but since A & B are working together so work will be finished at the same time for A & B
So higher of A’s leaves & B’s leaves will be counted
Hence total time to complete the work as per described condition = 15 + 3 = 18 days Answer
∴ work will be completed in 18 august2022 Answer
Note: the person who is taking higher of the leaves ⟹ its leaves are added
(Q). A & B individually can do some work in certain number of days. A & B completed a work together. If A started the work 4 days later than the scheduled time & B started the work 13 days earlier than the scheduled time then the work would be completed 5 days earlier. What is the time taken by A to complete the work alone if B completes the work in 24 days.
Sol:
Concept:
When alligation method is applied on time we get ratio of efficiencies.
A & B started working together & work completes 5 days earlier than the scheduled time.
If we consider only A then work would have been completed 4 days late than the scheduled time but it completed 5 days earlier than scheduled time.
∴ A
time ⟹ 5 + 4
= 9
Similarly, If we consider only B then work would have been completed 13 days earlier than the scheduled time but it completed 5 days earlier than the scheduled time.
∴ B
time ⟹ 13 – 5
= 8
● wages will be divded into the ratio of their work done.