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Time & Work Problems
Type I: Calculate Time to Complete Work by 2 or More People
In this type, you have to calculate time taken by 2 or more people to do a job. In the question, you will be given the time required by each member individually. You have to calculate the required time if they work together.
Below is an example question.
Example Question 1: Rahul takes 5 hours to do a job. Benny takes 8 hours to do the same job. How long should it take for Rahul and Benny, working together but independently, to do the same job?
Solution:
From the question, you can write down the below values
Part of work done by Rahul in 1 hour = 1/5 …value 1
Part of work done by Benny in 1 hour = 1/8 …value 2
Part of work done by Rahul and Benny together = value 1 + value 2
= 1/5 + 1/8
= 13/40
Now, you can calculate total hours required by both to complete the work using direct proportion table.
Hours Work
1 13/40
? 1 (1 represents full work)
Number of hours required by both to complete the work together = 1 x 1/ (13/40)
40/13 = 3 1/13 hours
Type II: Extension to Type 1 When Days and Hours are Given
In this type, you will see time as a measure of days and working hours per day. Below example will help you to understand better.
Example Question 2: Arjun can do a piece of work in 5 days of 8 hours each and Chinna can do it in 4 days of 6 hours each. How long will they take to do it working together 7 ½ hours a day?
Solution:
In question, you can see that Arjun can complete the work in 5 days working 8 hours per day.
Therefore, Arjun can complete the work in (5 days x 8 hours/day) = 40 hours
Similarly, Chinna can complete the work in (4 days x 6 hours/day) = 24 hours
Now, you have to proceed like type 1.
Part of work that Arjun can do in 1 hour = 1/40 … value 1
Part of work that Chinna can do in 1 hour = 1/24 … value 2
Part of work that Arjun and Chinna can do together in 1 hour = value 1 + value 2
= 1/40 + 1/24 = (3+5)/120 = 8/120
You can calculate total hours required by both to complete the work using direct proportion table
Hours Work
1 8/120
? 1 (1 represents full work)
Both will finish the work in 1 x 1 / (8/120) = 120/8 hours.
You have to calculate number of days required if they work 7 ½
or 15/2 hours each day. Now, you can calculate the number of days required using direct proportion table method.
Hours Days
15/2 1
120/8 ?
Therefore, if the friends work 15/2 hours each day, the total number of days to complete the work
= 120/8 x 2/15 = 2 days
= 120/8 x 2/15 = 2 days
Type III: Equations Based Time and Work Problems
In this type, you have to form equations based on question data. You have to then solve those equations to arrive at the solution. Now let us see an example.
Example Question 3: A and B can built a wall in 12 days, B and C can do it in 16 days and A and C can do it in 18 days. In how many days will A, B and C finish it separately?
Solution:
You have to assume the following.
Let A’s 1 day of work be X,
B’s 1 day of work be Y,
and C’s 1 day of work be Z.
Part 1: Form Equations Based on Question Data
From the question you know that A and B can build the wall in 12 days.
Part of work completed by A and B together in 1 day = 1/12
Or A’s 1 day of work + B’s 1 day of work = 1/12
Or X + Y = 1/12 … equation 1
You know that B and C can build the wall in 16 days
Part of work completed by B and C together in 1 day = 1/16
Or B’s 1 day of work + C’s 1 day of work = 1/16
Or Y + Z = 1/16 … equation 2
You also know that A and C can build the wall in 18 days
Part of work completed by A and C together in 1 day = 1/18
Or A’s 1 day of work + C’s 1 day of work = 1/18
Or X + Z = 1/18 … equation 3
Part 2: Let Us Solve The Equations
If you add equations 1,2 and 3, you will get the following.
2 (X + Y + Z) = 1/12 + 1/16 + 1/18
2 (X + Y + Z) = (12 + 9 + 8)/144 = 29/144
Or, X + Y + Z = 29/288 … equation 4
a) Subtract equation 1 from equation 4:
(X + Y + Z) – (X + Y) = 29/288 – 1/12
Or, Z = 29-24 / 288
Or, Z = 5/288
b) Subtract equation 2 from equation 4:
(X + Y + Z) – (Y + Z) = 29/288 – 1/16
Or, X = 29-18 / 288
Or, X = 11/288
c) Subtract equation 3 from equation 4:
(X + Y + Z) – (X + Z) = 29/288 – 1/18
Or, Y = 29-16 /288
Or, Y = 13/288
A’s 1 day of work = X = 11/288
Therefore, A can complete the work in 288/11 = 26 2/11 days
B’s 1 day of work = Y = 13/288
Therefore, B can complete the work in 288/13 = 22 2/13 days
C’s 1 day of work = Z = 5/288
Therefore, C can complete the work in 288/5 = 57 3/5 days
Type IV: Efficiency Based Time and Work Problems
In this type, efficiency of one worker compared to other worker/workers will be given. You have to use this efficiency data to solve this type. Now let us see an example.
Example Question 4: Sam is twice as good a workman as Raj and together they finish a painting work in 10 days. In how many days will Sam alone finish the work?
Solution:
Let us assume the following.
Sam’s 1 day work = X
and Raj’s 1 day work = Y.
From question you know that Sam is twice as good as workman as Raj. In other words, Sam is 2 times efficient than Raj. Therefore, you can form the below ration.
Sam’s 1 day work : Raj’s 1 day work = 2 : 1
Or X:Y = 2:1
Or, X = 2Y … equation 1
Sam and Raj can finish the work in 10 days.
Part of work completed by Sam and Raj in 1 day = 1/10
Or X + Y = 1/10 …equation 2
If you substitute equation 1 in equation 2, you will get.
2Y + Y = 1/10
3Y = 1/10
Y = 1/30 and
X = 2Y = 2/30 = 1/15
X = Sam’s 1 day work = 1/15
Therefore, Sam can complete the work in 15 days.
Type V: Calculate Time When Efficiency is Given in Percentage
This is a slight variation of type 4. In this type efficiency will be given in percentage. Below example will help you to understand this type better.
Example Question 5: Ram alone can fence the garden in 8 days. Bose is 50% more efficient than Ram. How many days does Bose alone take to fence the garden?
Solution:
Let Ram’s 1 day work be X
and let Bose’s 1 day work be Y
From the question, you know that Bose is 50% more efficient than Ram.
This means, if Ram’s does 1 unit of work in 1 day, Bose can do 150/100 x 1 = 1.5 units of work in 1 day
Or, X:Y = 1.5:1
OR X = 1.5Y …equation 1
In question, you can see that Ram can complete the work in 8 days
Or Ram’s 1 day work = X = 1/8
If you substitute X = 1/8 in equation 1, you will get,
Y = X/1.5 = 1/12
Therefore, Bose’s 1 day work = 1/12
Or, Bose can complete the work in 12 days.
Type VI: Calculate Time When Workers Leave in Between
In this type, you will find some workers leave in between and others will complete the work. Below example will help you to understand better.
Example Question 6: Saran can do a piece of work in 50 days. He works for 15 days and then Sanjay alone finishes the remaining work in 35 days. In how many days Sanjay alone can finish the work?
Solution:
Let Saran’s 1 day work be X
and Sanjay’s 1 day work be Y.
Saran can complete the work in 50 days.
Therefore, Saran’s 1 day work = X = 1/50.
Though Saran has the ability to complete the work in 50 days, he leaves in 15 days. From then onwards, Sanjay works to finish the remaining work.
Work done by Saran in 15 days = 15 x X = 15/50 = 3/10
You know that 1 represents full/complete work.
Therefore, remaining work = 1 – 3/10 = 7/10
Sanjay finishes this remaining 7/10 work in 35 days.
You have to calculate the time that Sanjay will take to finish the whole work. You can use direct proportion table method as shown below.
Days Work
35 7/10
? 1 (1 represents full work)
Therefore, Sanjay can complete the complete work in 1 x 35 / (7/10) = 10 x 35 / 7 = 50 days.
Type VII: Share of Salary Based on Work
In this type, you have to calculate the salary of each working member based on their amount of work. You will find the below example helpful.
Example Question 7: Sakshi and Saranya undertake a typist work for Rs.1000. Sakshi alone can complete it in 8 days while Saranya alone can complete it in 10 days. With the help of Ravi, they finish it in 4 days. Find the share of each.
Solution:
Let Sakshi’s 1 day work be X,
let Saranya’s 1 day work be Y
and Ravi’s 1 day work be Z.
Saskshi can complete the work in 8 days.
Therefore, Sakshi’s 1 day work = X = 1/8
Saranya can complete the work in 10 days.
Therefore, Saranya’s 1 day work = Y = 1/10
When all 3 work together, they complete the work in 4 days.
Therefore, Part of work done by all 3 together = ¼
Or, X + Y + Z = ¼
Z = ¼ – (X + Y) = ¼ – (1/8 + 1/10) = 1/40
To find salary share of each working member, remember the following rule.
Ratio of the salaries between members = Ratio of 1 day (or 1 hour) work of the members.
Based on the above rule,
Salary of Sakshi : Saranya : Ravi = X:Y:Z = 1/8 : 1/10 : 1/40
= 4 : 5 : 20
Sakshi’s share = 1000 x 4/(4+5+20) = 1000 x 4/29 = 137.93
Saranya’s share = 1000 x 5/(4+5+20) = 1000 x 5/29 = 172.41
Ravi’s share = 1000 x 20/(4+5+20) = 1000 x 20/29 = 689.66
Note: If you have doubt on above calculation, you can refer to type2.
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