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.