# Time and Work Tricks, Formulas | Top 15 Frequently Asked Problems

## Top 15 Time and Work Tricks with Concepts

### Introduction

Time and Work tricks are the most important and this topic is a priority-based chapter for any competitive exam like SSC, Bank PO, Clerks, CAT.

In this Time and work chapter, we will discuss how much time required to finish the given work by the number of people based on their number and capacity of work.

It means we need to find out that the given work whether finished or not within the stipulated time, if not how many workers we need to assign the work to complete given task within time. This is the main concept behind the Time and Work chapter.

### Time and Work Tricks – Keywords and their relation

The frequent keywords are Time and Work, Number of Persons, Capacity of person. The relation explained in the following way.
• The relation between Number of Person and Work Directly Proportional
(More persons can do more work and few men can do less work)
• The relation between Number of Person and Time –  Inversely Proportional
(More persons require less time and few men require more time)
• The relation between Work and Time – Directly Proportional
(More work require more time and less work  require less time)

### Time and Work Solved Examples

Concept 1: If a person can do a piece of work in “n” days (hours), then that
person’s 1 day’s (hour) work = 1/n

Example: Keerthi completes work in 35 days. What work will she can do in one day?
If a person can do piece of work in “n” days (hours), then that person’s 1 day’s (hour) work = 1/n.
Hence, n = 35 days,
Keerthi’s one day work = 1/n = 1/35.
Concept 2: If person’s 1 day’s (hour) work  1/n  then the person can complete work in “n” days (hours).

Example: Kavitha does 1/13 part of certain work in 1 day. In how many days, will she complete the whole work?
If person’s 1 day’s (hour) work  1/n  then person can complete work in “n” days (hours).
Hence, 1/n =  1/13
Kavitha can complete the work in 13 days.
Concept 3: If the person is n times efficient than the second person, then
work done by first person : second person = n : 1
and
time taken to complete a work by first person : second person = 1: n
Example : P can do a work 3 times faster than Q and therefore takes 40 days less than Q. FInd the time in which P and Q can complete the work individually?
If person is n times efficient than the second person, then time taken to complete a work by first person : second person = 1: n
Time taken to complete the work by P : Q = K : 3K
According to the questions, Time taken by Q – Time taken by P = 40
3K – K = 40
2K =40
K =20
Hence, the number of days required by P = 20 days and
the number of days required by Q = 60 days.

#### Time and Work Tricks

Concept 4: If ratio of numbers of men required to complete a work is m : n, then the ratio of time taken by them will be n : m
Example : If 12 men can finish a work in 20 days, then find the number of days required to complete the same work by 15 men?
If ratio of numbers of men required to complete a work is m : n, then the ratio of time taken by them will be n : m
Ratio of numbers of men m : n= 12:15 = 4:5
Ratio of time taken n:m = 5:4
Assume that 15 men can finish a work in “x” days. Then
20 : x = 5 : 4
x = 16 days.
Concept 5: If M1 persons can do W1 work in D1 days T1 hours in a day and M2 persons can do W2 work in D2 days working T2 hours in a day, then the relationship between them is
M1 D1 T1 W2 = M2 D2 T2 W1

Example : 10 men can make 20 toys in 12 days working 12 hr a day. Then, in how many days can 24 persons make 32 toys working 16 hr a day?
M1 = 10, M2=24, D1=12, D2=?
T1=12, T2= 16, W1= 20 and W2 =32
According to the formula of Time and Work
M1 D1 T1 W2 = M2 D2 T2 W1

D2 = M1 D1 T1 W2
M2 T2 W1

D2 = 10 x 12 x 12 x 32    = 12/2 = 6 days.
24 x 16 x 20

#### Time and Work Tricks

Concept 6: (I) If  A can do a piece of work in x days and B can do the same work in y days, then
(A+B) ‘s 1 day ‘s work  = 1/x + 1/y  = x + y/x y and
Time taken by (A+B) to complete the work  = x y/ x +y days

(II) If  A can do a piece of work in x days and B can do the same work in y days, C can do the same work in z days then for n persons, their
1 day ‘s work  = 1/X1 + 1/X2 +1/ X3 + ….+ 1/Xn
(X1, X2, X3, …., Xn  = indicates number of days taken by them to complete work)
and
Time taken by (A+B+C) to complete the work  = x y z / xy + yz + zx days.

Example 1 A can do a piece of work in 10 days and B can do the same work in 12 days. How long will they take to finish the work, if both work together?
Answer:  A’s 1 day work  = 1/10 and B ‘s 1 day work = 1/12

(A+B) ‘s 1 day ‘s work  = 1/x + 1/y  = x + y/x y

(A+B) ‘s 1 day ‘s work = 1/10 +1/12 = 6+5 / 60 = 11/60 day
(A+B) complete the whole work in 60/11 days.
(or)

Time taken by (A+B) to complete the work  = x y/ x +y days
= 12 x 10 /12 +10
=120/22
= 60/11 days
Example 2 A can do a piece of work in 4 days and B can do the same work in 8 days and C can do the same work in  12 days. How long will they take to finish the work, if both work together?
Answer:  A’s 1 day work  = 1/4 ;  B ‘s 1 day work = 1/8; C’s 1 day work = 1/12
According to formula
(A+B+C) ‘s 1 day ‘s work  = 1/X1 + 1/X2 +1/ X3

= 1/4 + 1/8 + 1/12
= 6 + 3 + 2 /24
= 11/24
Time is taken by (A+B+C) to complete the work = 24/11 days.

#### Time and Work Tricks

Concept 7: If  A and B can complete a work in x days and A alone can finish that work in y days, then Number of days required to complete the work by B = xy / y – x days.

Example: A and B together can do a piece of work in 12 days and A alone can do it in 18 days. In how many days can B alone can do it?
Answer:  (A+B ) ‘s 1 day work = 1/12 and A’s 1 day ‘s work = 1/18
B’s 1 day’s work  = (A+ B ) ‘s 1 day’s work – A’s 1 day’s work  = 1/12 -1/18
= 3-2 /36
= 1/36
=36 days
(or)
Number of days required to complete the work by B alone = xy / y – x days.
x = 12 and y =18
= 12 x18/ 18-12
= 36 days.
Concept 8: If A and B can do a piece of work in x days, B and C can do the same work in y days and A and C can do it in z days,
then working together A, B and C can finish the work in 2xyz / xy + yz + zx days.

Example  A and B  can do apiece of work in 3 days. B and c can do the same work in 9 days and A and C can do it in 12 days, then working together A, B and C can do that work in how many days?
Here  x = 3, y =9, z = 12
According to formula of Time and Work
Required time taken by A, B and C  =  2xyz / xy + yz + zx days.

= 2 x 3 x 9 x 12 / (3×9) + (9×12) + (3×12)
= 72/19 days.
Concept 9: If a1 men or b1 women can finish a work in D days, then
time taken by a2 men and b2 women to complete the work  = D (a1b1) / (a2b1 +a1b2) days.

Example: If  6 men or 8 women can reap a field in 86 days, how long will 14 men and 10 women take to reap it?
a1 = 6, b1 = 8, a2= 14, b2 = 10 and D = 86

Time taken to complete the work  = D (a1b1) / (a2b1 +a1b2) days.

=  86 x 6 x 8 /(14 x 8) + (10 x 6)
= 86 x 6 x 8/ 172
= 24 days.
Concept 10: If A can do work in 15 days and B  can do y% fast than A, then B will complete the work in 100x / 100 + y days

Example: Ravi can do a work in 15 days  and Veena is 50% more expert than Ravi  to complete the same work, then find total time taken to complete the work by Veena?
Answer:  Here , x = 15 days and y = 50%
Now, time taken by Veena = 100x / 100 + y
= 100 x 15/ 100 +50
= 1500/150
= 10 days.

#### Time and Work Tricks

Concept 11: If a men can do a piece of work in x days and b boys can do the same work in y days. then time taken to complete the same work by C men and d boys will be       1           days.
c/ax + d/by
Example  If 5 men can do a work in 2 days and 3 boys  can do the same work in 5 days, then find the time taken to complete the same work by 10 men and 3 boys?
Here a = 5, b =3, x =2, y =5, c=10 and d=3
According to formula of Time and Work
Time taken by 10 men and 3 boys  =       1           days.
c/ax + d/by
1                 days.
10/5 x 2 + 3/3 x 5
= 5/6 days.

#### Time and Work Tricks

Concept 12: If a1 men and b1 boys can complete a work in x days, while a2 men and b2 boys can finish the same work in y days, then
one day work of 1 man  = (yb2 – xb1)
one day work of 1 boy       (xa1 – ya2)

Example  If 12 men and 16 boys can finish a work in 5 days, while 13 men and 24 boys can finish the same work in 4 days. Compare the one day work of 1 man and 1 boy?
Answer: Here a1 = 12, b1 =16, x= 5, a2 =13, b2 =24 and y =4
According to formula of Time and Work
one day work of 1 man  = (yb2 – xb1)
one day work of 1 boy       (xa1 – ya2)
= 4 x 24 – 5 x 16
5 x12 – 4 x13
= 16/8
=2

Concept 13: If x takes a days more to complete a work than the time taken by (x +y) to do same work and y takes b days more than the time taken by (x+y) to do the same work, then (x+y) do the work in √ab days

Example: When A alone does a piece of work, he takes 16 days more than the time taken by (A+B) to complete the work, while B alone takes 9 days more than the time taken by (A+B) to finish the work. What time, A and B together will take to finish this work?
Here a = 16 and b =9
According to formula of Time and Work

Required time = √ab = √16 x 9
= 4 x 3 = 12 days.

Concept 14: A and B, each alone can do a piece of work in a and b days, respectively. Both begin together and if
(I) A leaves the work x days before its completion, then total time taken for completion of work will be given as
T = (a+x)b/ (a+b) days

(II) B leaves the work x days before its completion, then total time taken for completion of work will be given as
T = (b+x) a/(a+b) days

Example: A can do a piece of work in 10 days while B can do it in 15 days. They begin together but 5 days before the completion of the work, B leaves off. Find the total number of days for the work to be completed?
Answer: Here a = 10 days, b= 15 days, x = 5 and T =?
According to formula of Time and Work

Required time = (b+x) a/(a+b)
=( 15+5) 10/ 10+15
= 20 x 10/ 25
= 8 days.

Concept 15: A and B do a piece of work in a and b days, respectively. Both begin together but after some days, A leaves off and the remaining work is completed by B in x days. Then the time after which A left, is given by
T = (b- x)a/ a+b.

Example: A and B can do a piece of work in 40 days and 50 days, respectively. Both begin together but after a certain time, A leaves off. In this case, B finishes the remaining work in 20 days. After how many days did A leave?
Here a = 40 days, b = 50 days, x=20 and T=?
Required time =  (b- x)a/ a+b
= (50-20) 40/ 40+50
=30 x 40/9
= 40/3 days.

These are all concepts related to Time and Work tricks based questions frequently asked in exams.

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