- Efficiency * Total time taken = Total work
Assume the total work to be 1 unit.
If X can do the work in 'n' days, the fraction of work(efficiency) X does in a day is $$\frac{1}{n}$$ units/day. - If X can do the work in 'x' days, and Y can do the work in 'y' days, the number of days taken by both of them together to do the work is $$\frac{x*y}{x+y}$$
- If $$A_1$$ men can do $$B_1$$ work in $$C_1$$ days and $$A_2$$ men can do $$B_2$$ work in $$C_2$$ days, then $$\frac{A_1 C_1}{B_1}$$ =$$\frac{A_2 C_2}{B_2}$$
Note:
To simplify calculations, we try to get efficiencies as integers. We assume the total work to be some integral multiple of the total time taken.
Example: If X can do a work in 15 days and Y can do it in 12 days. Find the total days required to complete the work if X and Y both are working together.
Sol. We assume the work to be LCM(15,12)=60 units to get the efficiencies of both X and Y in integers.
Efficiency of X$$=\dfrac{60}{15}=4$$ units/day.
Efficiency of Y$$=\dfrac{60}{12}=5$$ units/day.
So, when they work together the efficiencies would add up$$=4+5=9$$units/day.
Let the time required to complete the work by X and Y together$$=a$$.
So, $$9a=60$$. $$a=\dfrac{60}{9}$$ days.
Work - Efficiency
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Question 1
Anu, Vinu and Manu can complete a work alone in 15 days, 12 days and 20 days, respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is
Solution
Let the total amount of work be 60 units.
Then Anu, Vinu, and Manu do 4, 5, and 3 units of work per day respectively.
On the 1st day, Anu and Vinu work. Work done on the 1st day = 9 units
On the 2nd day, Manu and Vinu work. Work done on the 2nd day = 8 units
This cycle goes on. And in 6 days, the work completed is 9+8+9+8+9+8 = 51 units.
On the 7th day, again Anu and Vinu work and complete the remaining 9 units of work. Thus, the number of days taken is 7 days.
Question 2
Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in 1 year, Akbar and Anthony can complete in 16 months, Anthony and Amar can complete in 2 years. If the person who is neither the fastest nor the slowest works alone, the time in months he will take to complete the project is
Solution
Correct Answer: 32
Let the total work be 48 units. Let Amar do 'm' work, Akbar do 'k' work, and Anthony do 'n' units of work in a month.
Amar and Akbar complete the project in 12 months. Hence, in a month they do $$\frac{48}{12}$$=4 units of work.
m+k = 4.
Similarly, k+n = 3, and m+n = 2.
Solving the three equations, we get $$m=\frac{3}{2},\ k=\frac{5}{2},\ n=\frac{1}{2}$$.
Here, Amar works neither the fastest not the slowest, and he does 1.5 units of work in a month. Hence, to complete the work, he would take $$\frac{48}{1.5}=32$$months.
Question 3
Anil can paint a house in 60 days while Bimal can paint it in 84 days. Anil starts painting and after 10 days, Bimal and Charu join him. Together, they complete the painting in 14 more days. If they are paid a total of ₹ 21000 for the job, then the share of Charu, in INR, proportionate to the work done by him, is
Solution
Let Entire work be W
Now Anil worked for 24 days
Bimal worked for 14 days and Charu worked for 14 days .
Now Anil Completes W in 60 days
so in 24 days he completed 0.4W
Bimal completes W in 84 Days
So in 14 Days Bimal completes = $$\frac{W}{6}$$
Therefore work done by charu = $$W-\frac{W}{6}-\frac{4W}{10}$$= $$\frac{26W}{10}$$=$$\frac{13W}{30}$$
Therefore proportion of Charu = $$\frac{13}{30}\times\ 21000$$=9100
Question 4
One day, Rahul started a work at 9 AM and Gautam joined him two hours later. They then worked together and completed the work at 5 PM the same day. If both had started at 9 AM and worked together, the work would have been completed 30 minutes earlier. Working alone, the time Rahul would have taken, in hours, to complete the work is
Solution
Let Rahul work at a units/hr and Gautam at b units/hour
Now as per the condition :
8a+6b =7.5a+7.5b
so we get 0.5a=1.5b
or a=3b
Therefore total work = 8a +6b = 8a +2a =10a
Now Rahul alone takes 10a/10 = 10 hours.
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