Fraction of work method:
If A can do a work in 'a' days, then work done by him in 1 day is $$\frac{1}{{\text{a}}}$$.
If B can do a work in 'b' days, then work done by him in 1 day is $$\frac{1}{{\text{b}}}$$.
Then, in one day, if A and B work together, then work done by them in 1 day,
$$\eqalign{
& = \frac{1}{{\text{a}}} + \frac{1}{{\text{b}}} \cr
& = \frac{{{\text{a}} + {\text{b}}}}{{{\text{ab}}}} \cr} $$
In this case, we take the total work to be done as 1. Hence, the work will be completed when 1 unit of work is completed.
Example:
If A can do a work in 10 days and B can do same work in 12 days, then the work will be completed in how many days.
Solution:
one day combined work of A and B,
=$$\frac{1}{{10}} + \frac{1}{{12}}$$
(A+B) = $$\frac{{12 + 10}}{{120}}$$ [LCM of 10 and 12 = 120]
= $$\frac{{22}}{{120}}$$
Then work will be completed in $$\frac{{120}}{{22}}$$ days.
This is reciprocal of one day's work done by A and B together.
Percentage of work done method:
Instead of taking the value of the total work as 1 unit of work, we can take total work as 100% work.
If A can do a work in a days, then work done by him in 1 day is $$\frac{{100}}{{\text{a}}}$$ % of work.
If B can do a work in b days, then work done by him in 1 day is $$\frac{{100}}{{\text{b}}}$$ % of work.
Then, in one day, if A and B work together, then work done by them in 1 day is $$\left( {\frac{{100}}{{\text{a}}} + \frac{{120}}{{\text{b}}}} \right)$$
This approach is very use full and time saving because thinking in terms of percentage gives a direct and clear picture of the actual quantum of work done.
Example:
If A can do a work in 10 days and B can do same work in 12 days, then the work will be completed in how many days.
Solution:
One day work of A = $$\frac{{100}}{{10}}$$ = 10%
One day work of B = $$\frac{{100}}{{12}}$$ = 8.33%
One day work of (A + B) = 10 + 8.33 = 18.33%
Hence, 100% work will require = $$\frac{{100}}{{18.33}}$$
At this point, we can guess that answer lies between 5-6 days. If we go through the options, then we can mark the closest answer.
They will complete 91.66 % in 5 days and rest 8.33% will be done in = $$\frac{{8.33}}{{18.33}} = \frac{5}{{11}}$$
That means 100% will be completed in $$5\frac{5}{{11}}$$ days.
Negative Work and Its Application:
Suppose, A and B are working to build a wall while C is working to break the wall. In such case, the wall is being built by while it is broken by C. here, we consider that A and B are doing positive work in building the wall while C is doing a negative work as he is breaking the wall.
Example:
A can build a wall in 10 days and B can build it in 5 days, while C can completely destroy the wall in 20 days. If they start working at the same time, in how many days will the work be completed?
Solution:
A's one day work = $$\frac{{100}}{{10}}$$ = 10%
B's one day work = $$\frac{{100}}{5}$$ = 20%
C's one day work = $$\frac{{100}}{{20}}$$ = 5%. [This work is taken as negative].
The net combined work per day= (10 + 20 - 5)% = 25%.
Hence work will completed in $$\frac{{100}}{{25}}$$ = 4 days.
Applications of negative work:
Pipe and Cisterns:-
The concept of negative work commonly appears as a problem based on pipes and cisterns, where there are inlet pipes and outlet pipes or leaks which working against each other.
If we consider the work to be filling a tank, the inlet pipe does positive work while the outlet pipe or leak does negative work.
Example:
Two pipes A and B require 12 min. and 15 min. respectively to fill a tank while; another pipe C requires 18 min. to empty it. In how many minute will the tank be filled if all the three pipes are opened together?
Solution:
In 1 minute, A fills the tank = $$\frac{{100}}{{12}}$$ = 8.33%
In 1 minute, B fills the tank = $$\frac{{100}}{{15}}$$ = 6.66%
In 1 minute, C empties the tank = $$\frac{{100}}{{18}}$$ = 5.55%
Net combined work of all pipe per minute = (8.33 + 6.66 - 5.55)% = 9.44%
Tank will be filled in = 10.5 minutes.
Efficiency and Its Application:
Work done per unit time is known as efficiency. It is also known as Work Rate.
Efficiency = $$\frac{{{\text{Work}}\,{\text{done}}}}{{{\text{Time}}}}$$ . . . . . (i)
Hence, Work done = efficiency × time.
From equation (i) means that if the work done is constant,
Then, We can say, →
Work rate is inversely proportional to time. That means, if A has taken more time, then its efficiency or work rate is less.
Work Equivalence:
The work equivalence is an application of the efficiency and formula,
Work rate × time = work done;
Example:
A contractor undertakes to build a wall in 50 days. He employs 50 people for the same. However, after 25 days he finds that the work is only 40% complete. How many more men need to be employed to:
a) Complete the work in time?
Solution:
The contractor has completed 40% work in 25 days.
If number of men working on the project remains constant, the rate of work also remains constant. Hence, to complete 100% work, he will have to complete the remaining 60% of the work.
He would require,
Work rate × time = work done
$$\eqalign{
& {\text{Time}} = \frac{{{\text{Work}}\,{\text{done}}}}{{{\text{Work}}\,{\text{rate}}}} \cr
& {\text{Time}} = \frac{{60 \times 25}}{{40}} \cr} $$
Time = 37.5 days.
That means he would require 37.5 more days to complete the work.
In order to complete work in time he has to complete the 60% work in 25 days. It is oblivious that he will have to increase the number of men working on the project. It can solve as
50 men working for 25 days → 50 × 25 = 1250 men-days.
1250 men-days resulted 40% work completion. Hence,
Work left = 60% = 1250 × 1.5 = 1875 men-days.
This has to complete in 25 days. Hence, number
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