Work to be done is usually considered as one
unit. It may be, constructing a wall or laying a road, filling up or emptying a
tank or cistern or eating certain amount of food. If there is more than one
person (or thing) carrying out the work, it is assumed that each person (or
thing) does the same amount of work each day and all the persons (or things) do
exactly the same amount of work. .
For example:
(i) If
a person completes the work in 4 days, he does 1/4th of the work on each day
and conversely, if a person can complete 1/4th of the work in one day, he can
complete the work in 4 days. .
(ii) If
a tap can fill a tank in 20 minutes, then in one minute, it can fill 1/20th
part of the tank. .
Wages earned by people doing a work together
are to be distributed in the ratio of the total work done by each of them. If a
group of people, working at different pace (i.e., the work done by each of them
per day is different), start and complete a work together, then the wages or
earnings have to be divided in the ratio of the work per day of each of them. .
Here, we follow what is known as “UNITARY
METHOD”, i.e., the time taken per “Unit Work” or number of persons required to
complete “Unit Work” or work completed by “Unit Person” in “Unit Time”, etc.,
is what is first calculated. .
For example, if 20 men take 30 days to
complete a work, then we have
1 man can complete (1/20)th work in 30 days. .
1 man can complete (1/600)th work in 1 day. .
20 men can complete (1/30)th work in 1 day. .
600 men can complete 1 work in 1 day. .
1 man can complete 1 work in 600 days. .
We should recollect the fundamentals on
variation (direct and inverse) here. .
- Work and
men are directly proportional to each other, i.e., if the work increases, the
number of men required to complete the work in the same number of days
increases, and vice versa. .
- Men and
days are inversely proportional, i.e., if the number of men increases, the
number of days required to complete the same work decreases and vice-versa. .
- Work and
days are directly proportional, i.e., if the work increases, the number of days
required to complete the work with the same number of men also increases and
vice versa. .
Worked out examples
example.1
A can do a work in 21 days and B in 28 days.
If they work together in how many days will they complete the work? .
Solution
In one day, A can do 
th of the work
and B can do 
th of the work.
In one day they together can complete 
of the total
work.
So they can complete the work in 12 days. .
example.2
A can do a work in 12 days, with the help of
B he can do the work in 10 days. In how many days can B alone do the work? .
Solution
In one day A and B together can do 
of the work.
A alone can do 
th of the work
So B can do 
of the work in
one day.
So B can do the work in 60 days.
Example.3
A and B can do a work in 12 days and 20 days
respectively. A started the work and left after 3 days. B immediately took over
and worked till the completion. In how many days was the total work completed? .
Solution
A can do of work in one
day. So he can do x 3 = 
th of the work
in 3 days. After 3 days 
of the work is
still left over.
B can do 
th of the work
in one day. So he can do 
of work in 
= 15 days
So total time taken is 15 + 3 = 18 days. .
example.4
A and B can complete a work in 6 days, B and
C in 8 days and A and C in 10 days. If all three work together, in how many
days will the work will be completed? .
Solution
One day
work of,
A and B = 
B and C = 
A and C = 
two days of work (A + B + C)
= 
one day of work (A + B + C) = 
So, they can complete the work in 
days i.e., 
days
Example.5
A can do a work in 6 days more than what B
takes to do the work. A worked for 7 days, then B takes the work and completed
it in 10 more days. In how many days each of them can complete the work
individually? .
Solution
Let B
take x days to do the work. .
A takes (x + 6) days to do the work. .
Work done by A in 7 days = 
Work done by B in 10 days = 
So 
= 1
= 1
17x + 60 = x2 + 6x
&⇒ x2 - 11x - 60 = 0
&⇒ x2 - 15x + 4x - 60 =
0
&⇒ x (x - 15) + 4 (x - 15) = 0
&⇒ x = 15
A can do the work in 21 days and B can do it
in 15 days.
