Today I'm going to discuss a very important topic i.e. Time and Work of quantitative aptitude. In almost every exam at least 23 question are asked every time. In this chapter, I will tell you about a definite relationship between Time and work and easy method to solve the problems.
Work done(W) = Number of days(Time taken)(T or D) `times` Number of men(M)
`W = D times M`
More number of men take less time to complete certain job i.e. both are inversely proportional
By summarizing, we get
`(W_1)/(W_2) = ((M_1)/(M_2)) times ((D_1)/(D_2))`
Lets start solving some examples:
Example1: `10` men can cut `8`trees in `16` days. In how many days can `6`men cut `10`trees?
Solution: This is a very simple example. You are given:
`W_1` = `8`
`W_2` = `10`
`M_1` = `10`
`M_2` = `6`
`D_1` = `16`
`D_2` = `?`
Using formula,
`(W_1)/(W_2) = ((M_1)/(M_2)) times ((D_1)/(D_2))`
`8/(10) = (10)/6 times (16)/(D_2)`
⇒`D_2` = `33.3`
Example2: If a person can complete his work in `5` days. What will be his efficiency?
Solution: Number of days a person take to complete his work = `5`
⇒ He is doing 1/5 th work per day (converted into fraction)
Convert it into percentage:
`(100)/5` = `20`%
Therefore, his efficiency is 20%.
Summarizing, If a person can do his job in `n` days, efficiency will be
Efficiency = `(100)/n` %
Example3: A can do a job in `10` days. B can do a job in `5` days. In how many days they can complete the job if they work together?
Solution: Consider the above table
A's efficiency = `10`% `((100)/(10))`
B's efficiency = `20`%
A+ B efficiency = `10` + `20` = `30`%
This means, In one day A and B together can do `30`% of work.
Therefore,Number of days A and B together take to do 100% of work = `(100)/(30)`
⇒`3.33` days
Example4: A and B together can do a job in `4` days. If A can do job in `12` days if he works alone, then how many days B alone take to complete the job?
Solution: A+B take = `4` days
⇒ A+B's efficiency = `25`% i.e. they together do `25`% of work in one day
A takes = `12` days
⇒ A's efficiency = `8.33`%
B's efficiency = (A+B)  (A)
⇒ `25`%  `8.33`% = `16.66`%
This means, B can do `16.66`% work in one day
Therefore, to complete the job he will take = `(100)/(16.66)` days
⇒ `6`days
Example5: A and B can do job in `8` days. B and C can do same job in `12` days. A, B and C together can do same job in `6` days. In how many days A and C together can complete the job?
Solution: You are given that:
A+B's efficiency = `12.5`%
B+C's efficiency = `8.33`%
A+B+C's efficiency = `16.66`%
we need to find A+C
Consider, `2(A+B+C) = (A+B) + (B+C) + (C+A)`
⇒`2(16.66) = 12.5+ 8.33 + (C+A)`
⇒ C+A = `12.49` = `12.5`%
Therefore, A and C takes= `(100)/(12.5)` = `8` days
Hope you all understand this topic. I will soon update questions for your practice.
Three main factors of Time and Work
There is a definite relationship between Time and Work. In this concept, there are only three factors:
There is a fundamental relationship between these three, discussed as follows:Work done(W) = Number of days(Time taken)(T or D) `times` Number of men(M)
`W = D times M`
Some basic points
More number of men can do more work i.e. both are directly proportionalMore number of men take less time to complete certain job i.e. both are inversely proportional
By summarizing, we get
`(W_1)/(W_2) = ((M_1)/(M_2)) times ((D_1)/(D_2))`
Lets start solving some examples:
Example1: `10` men can cut `8`trees in `16` days. In how many days can `6`men cut `10`trees?
Solution: This is a very simple example. You are given:
`W_1` = `8`
`W_2` = `10`
`M_1` = `10`
`M_2` = `6`
`D_1` = `16`
`D_2` = `?`
Using formula,
`(W_1)/(W_2) = ((M_1)/(M_2)) times ((D_1)/(D_2))`
`8/(10) = (10)/6 times (16)/(D_2)`
⇒`D_2` = `33.3`
Concept of efficiency
This means, "How much work one person can do in one day (expressed in percentage)"
For example: A person can do a job in `2`days
⇒ He can do `50`% work in one day
Therefore, his efficiency will be `50`%
Just a 2step concept
This concept involves two steps to calculate efficiency:
 Convert into fraction i.e. per day work
 Multiply with 100 i.e. convert into percentage
Example2: If a person can complete his work in `5` days. What will be his efficiency?
Solution: Number of days a person take to complete his work = `5`
⇒ He is doing 1/5 th work per day (converted into fraction)
Convert it into percentage:
`(100)/5` = `20`%
Therefore, his efficiency is 20%.
Summarizing, If a person can do his job in `n` days, efficiency will be
Efficiency = `(100)/n` %
Note: Negative efficiency cancels the positive efficiency
For Example: Positive efficiency = `5`%
Negative efficiency = `1.5`%
Net efficiency = `5 1.5` = `3.5`%
As we all know, in competitive exams time management is very important. I suggest you to learn the fractions till `15`.
Number of days required to complete work

Work/Day

Efficiency (%)

N

1/n

100/n

1

1

100

2

1/2

50

3

1/3

33.33

4

1/4

25

5

1/5

20

6

1/6

16.66

7

1/7

14.28

8

1/8

12.5

9

1/9

11.11

10

1/10

10

11

1/11

9.09

12

1/12

8.33

13

1/13

7.69

14

1/14

7.14

15

1/15

6.66

Example3: A can do a job in `10` days. B can do a job in `5` days. In how many days they can complete the job if they work together?
Solution: Consider the above table
A's efficiency = `10`% `((100)/(10))`
B's efficiency = `20`%
A+ B efficiency = `10` + `20` = `30`%
This means, In one day A and B together can do `30`% of work.
Therefore,Number of days A and B together take to do 100% of work = `(100)/(30)`
⇒`3.33` days
Example4: A and B together can do a job in `4` days. If A can do job in `12` days if he works alone, then how many days B alone take to complete the job?
Solution: A+B take = `4` days
⇒ A+B's efficiency = `25`% i.e. they together do `25`% of work in one day
A takes = `12` days
⇒ A's efficiency = `8.33`%
B's efficiency = (A+B)  (A)
⇒ `25`%  `8.33`% = `16.66`%
This means, B can do `16.66`% work in one day
Therefore, to complete the job he will take = `(100)/(16.66)` days
⇒ `6`days
Example5: A and B can do job in `8` days. B and C can do same job in `12` days. A, B and C together can do same job in `6` days. In how many days A and C together can complete the job?
Solution: You are given that:
A+B's efficiency = `12.5`%
B+C's efficiency = `8.33`%
A+B+C's efficiency = `16.66`%
we need to find A+C
Consider, `2(A+B+C) = (A+B) + (B+C) + (C+A)`
⇒`2(16.66) = 12.5+ 8.33 + (C+A)`
⇒ C+A = `12.49` = `12.5`%
Therefore, A and C takes= `(100)/(12.5)` = `8` days
Hope you all understand this topic. I will soon update questions for your practice.