The document contains information about work formulas used to calculate the time taken to complete a task when multiple workers are involved. It provides formulas to calculate the time taken when: - Multiple groups work separately - Multiple workers work together - One worker assists others on alternate days It also includes example problems worked out using the formulas to find the time taken for various work scenarios.

1. C.S.VEERARAGAVAN
SIXPHRASE
sixphrase.veeraa1729@gmail.com

2.  MDW FORMULA
  If M1 persons can do W1 works in D1 days
and M2 persons can do W2 work in D2 days, then
we have M1D1W2 = M2D2W1
  If we include the working hours say T1 and
T2 for the two groups, then the relationship is
M1D1 T1 W2 = M2D2 T2W1

3.  If A can do a piece of work in x days and B
can do it in y days, then A and B working
together will do the same work in
𝑥𝑦
𝑥+𝑦
days.

4.  If A, B and C can do a work in x, y and z days
respectively, then all of them working
together can finish the work in
𝑥𝑦𝑧
𝑥𝑦+𝑦𝑧+𝑧𝑥
days

5.  If A and B together can do a piece of work
in x days and A alone can do it in y days,
then B alone can do the work in
𝑥𝑦
𝑦−𝑥
days.

6.  If A can do a piece of work in n days, then A’s
1 day’s work=
1
𝑛

7. A, B and C can do a piece of work in 11 days, 20 days and 55 days
respectively, working alone.
How soon can the work be done if A is assisted by B and C on alternate
days?
a) 7 days b) 8 days c) 9 days d) 10 days.
(A + B)’s 1 day’s work =
1 1 31
11 20 220
 
(A + C)’s 1 day’s work =
1 1 6
11 55 55
 
Work done in 2 days =
31 6 55 1
220 55 220 4
  
Therefore, Whole work will be done in 4 x 2 = 8 days.

8.  If A is thrice as good a workman as B, then
 Ratio of work done by A and B = 3:1 and
 Ratio of time taken by A and B to finish a
work = 1:3

9. 1. If A,B and C together can finish a piece of work
in 4 days; A alone can do it in 12 days and B in
18 days then C alone can do it in a)21days
b)16days c)14days d)9 days
(A + B + C)’s 1 day’s work =
(A + B)’s 1 day’s work = .
C’s 1 day’s work = .
C alone can finish the work in 9 days.
4
1
36
5
18
1
12
1

9
1
36
4
36
5
4
1


10. 4.A filling tap can fill a tank in 6 hours and an
emptying tap empty the tank in 12 hours. If both the
taps are opened simultaneously, in how many hours
will the tank be filled? (1)6 (2)12 (3)18 (4)24.
In 1 hour, the quantity of water filled by both the taps
working together is
1
6
−
1
12
=
1
12
th of the tank.
Therefore, it takes 12 hours to completely fill the tank.
Alternatively, the emptying tap takes twice the time as
the filling tap.
Thus when both the taps are open, the tank gets filled
at half the rate of the filling tap.
Hence it takes double the time as that of the filling
tap.
 6 hours x 2 = 12 hours.

11.  VI.Tap P can fill a tank in 3 hours and tap Q can
fill the tank in 4 hours. What fraction of the tank
can be filled by tap P alone in the time taken by
both working together to fill the tank?
1)
3
7
2)
2
7
3)
1
7
4)
4
7
 Ratio of efficiency of P and Q is
1
3
:
1
4
𝑖. 𝑒 4:3.
 In the time the tank is filled completely by both
taps working together tap P can fill
4
7
th of the
tank.
 Note: Here, the actual time taken to fill the tank
is not required.

12.  9.P and Q working together can complete a piece
of work in 24 days. The ratio of their efficiencies
is 2:3. In how many days can the faster person
complete, the work working alone? 1) 40 2) 50
3) 60 4) 90
 The ratio of efficiencies of the two persons is
2:3. so Q is faster. He completes
3
5
th of the work
by the time, the work completes.
 In 24 days, he completes
3
5
th of the work.
 Hence he completes total work in 24 x
5
3
=40
days.

13.  A can do a piece of work in 4 hours; B and C
together can do it in 3 hours, while A and C
together can do it in 2 hours. How long will B
alone take to do it?
 A. 8 hours B. 10 hours
 C. 12 hours D. 24 hours
A’s 1 hr work=
1
4
(B+C)’s 1hr work=
1
3
(A+C)’s 1hr work=
1
2
(A+B+C)’s 1 hr work =
1
4
+
1
3
=
7
12
So, B’s 1 hr work =
7
12
−
1
2
=
1
12
Hence B can finish the work in 12 hours.

14. A can do a certain work in the same time in
which B and C together can do it. If A and B
together could do it in 10 days and C alone in
50 days, then B alone could do it in:
 A. 15 days B. 20 days
 C. 25 days D. 30 days
 A’s 1 day work = (B+C)’s 1 day work  1
 (A+B)’s 1 day work =
1
10
& C’s 1 day work =
1
50
 (A+B+C)’s 1 day work =
1
10
+
1
50
=
6
50
 2
 By 1 & 2, 2 x A’s 1 day work =
6
50
 Hence A’s 1 day work =
3
50
 3
 B’s 1 day work =
1
10
−
3
50
=
2
50
=
1
25
 B alone can do it in 25 days.

15.  A does 80% of a work in 20 days. He then calls in B
and they together finish the remaining work in 3 days.
How long B alone would take to do the whole work?
 A. 23 days B. 37 days
 C. 37
1
2
days D. 40 days
 Whole work done by A in 20𝑋
100
80
= 25 days.
 1 −
80
100
i.e
20
100
=
1
5
th work completed by A+B in 3 days.
 Whole work done by A+B in 3x5=15 days.
 B’s 1 day work =
1
15
−
1
25
=
10
15𝑋25
=
2
75
 Hence B can finish the work in
75
2
days i.e.,37
1
2
days.

16.  A machine P can print one lakh books in 8 hours,
 machine Q can print the same number of books in 10 hours
 while machine R can print them in 12 hours. All the machines
are started at 9 A.M. while machine P is closed at 11 A.M.
 and the remaining two machines complete work.
 Approximately at what time will the work (to print one lakh
books) be finished ?
 A. 11:30 A.M. B. 12 noon
 C. 12:30 P.M. D. 1:00 P.M.
 (P+Q+R)’s 1 hour work =
1
8
+
1
10
+
1
12
=
37
120
 Work done in 2 hours by P,Q & R =
37
120
𝑋2 =
37
60
 Remaining work = 1–
37
60
=
23
60
.
 (Q+R)’s 1 hour work =
1
10
+
1
12
=
11
60
. In 2 hours =
22
60
≅
23
60
.

17. A is 30% more efficient than B. How much time will
they, working together, take to complete a job which
A alone could have done in 23 days?
 A. 11 days B. 13 days
 C. 20
3
17
days D. None of these
 Time taken by A & B = 100:130=10:13.
 Let job done by B alone be x.
 Then 10:13::23:x  x =
23𝑋13
10
=
299
10
 A’s 1 day work =
1
23
=
13
299
 B’s 1 day work =
10
299
 (A+B)’s 1 day work =
23
299
=
1
13
.
 Both could complete in 13 days.

18. Ravi and Kumar are working on an assignment. Ravi
takes 6 hours to type 32 pages on a computer, while
Kumar takes 5 hours to type 40 pages. How much time
will they take, working together on two different
computers to type an assignment of 110 pages?
 A. 7 hours 30 minutes B. 8 hours
 C. 8 hours 15 minutes D.8 hours 25 minutes
Number of pages typed by Ravi in 1 hour =
32
6
=
16
3
Number of pages typed by Kumar in 1 hour =
40
5
= 8.
Number of pages typed by both in 1hour=
16
3
+ 8 =
40
3
Therefore Time taken by both to type 110 pages
= 110𝑋
3
40
=
33
4
= 8
1
4
hours (or) 8 hours 15 minutes.

19.  If 6 men and 8 boys can do a piece of work in 10
days while 26 men and 48 boys can do the same
in 2 days, the time taken by 15 men and 20 boys
in doing the same type of work will be
 1man’s 1day work = x and 1 boy’s 1day work=y
 Then 6x + 8y =
1
10
and 26x + 48y =
1
2
 Solving we get x =
1
100
and y =
1
200
.
 15 men and 20 boys 1 days work =
15
100
+
20
200
=
1
4
.
 15 men and 20 boys can do the work in 4 days.

20. A can finish a work in 18 days and B can do the
same work in 15 days. B worked for 10 days and
left the job. In how many days, A alone can finish
the remaining work? A.5 B.5
1
2
C. 6 D. 8
B's 10 day's work =
1
15
𝑋10 =
2
3
Remaining work = 1 −
2
3
=
1
3
Now,
1
18
work is done by A in 1 day.
Therefore
1
3
work is done by A in
1
3
𝑋18 =6 days.

21. A can complete a piece of work working alone in 10
days. B can complete the work working alone in 15
days. If they work on alternate days, the work would be
completed in the
1)Least number of days if A starts the work.
2)Least number of days if B starts the work.
3) In the same time irrespective of who starts the work.
4) None of these.
If A starts the work,
Then 2 days work =
1
10
+
1
15
=
3+2
30
=
5
30
=
1
6
If B starts the work,
Then 2 days work =
1
10
+
1
15
=
3+2
30
=
5
30
=
1
6
In both cases work will be completed in 12 days. So
choice 3.

22. P can complete a piece of work, working alone
in 3 days. Q can complete the work working
alone in 5 days. If they work on alternative
days, the work would be completed in the
1)Least number of days if P starts the work.
2)Least number of days if Q starts the work.
3)In the same time irrespective of who starts
the work.
4) None of these.

23. P can complete a piece of work, working alone
in 3 days. Q can complete the work working
alone in 5 days. If they work on alternative
days, the work would be completed in the
If P starts the work,
2 days work =
1
3
+
1
5
=
8
15
. Remaining work =
7
15
3rd day P will work
1
3
=
5
15
.Remaining work =
2
15
4th day Q can complete the work.

24. Suppose two persons work on alternate days to
complete a work.
If the work done by both together in 1 day is
1
𝑛
th of the total work and n is an integer, then, the
work will be completed in the same time
irrespective of who starts the work.
However if n is not integer, then the exact time
to complete the work depend upon who starts
the work.