20
th
of the work. Since B does the work in 15 days, everyday he does 1
15
th
of the work. If both work together, everyday they complete 1 + 1 = 7
20 15 60
th
of the work. So number of day required to finish the work = 60/7 = approximately 8.5 days
Approach 2: LCM Approach.
In this approach we assume the work as a numeric quantity, usually the LCM of the days taken individually (this will avoid working with any fractions). After assuming the work, we find the individual rate of working.
Assuming work as LCM of 15 and 20 i.e. 60 units …
Aatish does 60 units in 20 day i.e. his rate of working is 3 units per day. Azad does 60 units in 15 day i.e. his rate of working is 4 units per day. When both work simultaneously, the effective rate is 3 + 4 = 7 units per day. Since total work is 60 units, day, taken will be 60/7 i.e. approximately 8.5 days.
E.g. 1: Parul is twice as efficient as Parag. If both working together complete a work in 10 days,
how many days would each have taken individually? Approach 1: Adding Per day’s Work:
Since Parul is twice as efficient as Parag, if Parag takes 2x days to complete a work, Parul will take x days to complete the same work.
Thus, work done by both of them in 1 day will be 1 +1 i.e. 3
2 2
th th th
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Since both work together they finish a work in 10 days, work done in 1 day will be 1
10
th
of the work.
Equating the two, 3 = 1 ⇒ 2x 10 x = 15
So Parag would have taken 30 days to complete the work independently and Parul would have taken 15 days.
Approach 2: LCM Approach.
Assuming work as 10 units and since it is done in 10 days, the rate of doing work is 1 unit per day.
Now if Parag does x unit’s of work in 1 day, Parul being twice as efficient will do 2x units of work in 1 day.
Thus, x + 2x = 1 i.e. = 1 3
x
Parag does 1/3 units per day and to do 10 units will require 101
3
= 30 days. Parul does 2/3 units per day and to do 10 units will require 102
3
= 15 days.
E.g. 2: Ishan can do a certain job in 18 days. He works at it for 10 days and then Bharat alone
completes it in 4 days. In how many days will Ishan and Bharat working together, finish the work?
Approach 1: Adding Per day’s Work:
Since Ishan does the work in 18 days, it means he does 1
18
th
of the work everyday. So in 10 days, he does 10
18
th
of the work. Remaining work is 8
18
th
Since Bharat completes the rest of the work in 4 days, everyday he does 818=1 4 9
th
of the work. So if they both work together, everyday they do 1 +1 1=
18 9 6
th
of the work. Hence working together, they will finish the work in 6 days.
Approach 2: LCM Approach.
Assume the work as 18 units. Since Ishan does the work in 18 days, his rate of working is 1 unit per day. When he works for 10 days, he will complete 10 units of work.
The remaining 8 units of work is done by Bharat in 4 days i.e. his rate of working is 2 units per day.
Together their rate is 1 + 2 = 3 units per day and to complete 18 units of work, they will require 18/3 = 6 days.
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E.g. 3: A man and two boys can do a piece of work in 10 days. Two men and a boy can do the
same work in 8 days. In how many days can a man and a boy together complete the work? Approach 1: Adding Per day’s Work:
A man and two boys complete 1
10
th
of the work everyday. Two men and a boy complete 1
8
th
of the work everyday.
Then, three men and three boys together will complete 1+ 1 = 9 8 10 40
th
of the work everyday. This means that one boy and one man will together do 9 ×1 = 3
40 3 40
th
of the work everyday. So they need 40/3 = 13.33 days to finish the work.
Approach 2: LCM Approach.
Let the work be LCM of 10 and 8 i.e. 40 units.
Further let each man do m units per day and each boy do b units per day.
Since a man and two boys can do 40 units in 10 days i.e. 4 units per day, hence m + 2b = 4 Since two men and a boy can do 40 units in 8 days i.e. 5 units per day, hence 2m + b = 5 Adding the two 3(m + b) = 9 i.e. m + b = 3
Thus, 1 man and 1 boy can do 3 units per day and to do 40 units, they would take 40/3 days.
E.g. 4: One man can build a bridge in 100 days. One woman can build the same bridge in 120
days. One child can destroy the bridge completely in 200 days. If two men, three women and two children are working simultaneously on the bridge, in how many days will it be complete? Approach 1: Adding Per day’s Work:
One man does 1
100
th
of the work everyday; One woman does 1
120
th
of the work everyday; One child destroys 1
200
th
of the work everyday. Two men do 1
50
th
of the work everyday; three women do 1
40
th
of the work everyday and two children destroy 1
100
th
of the work everyday.
Hence in all, the work done per day = 1 + 1 − 1 = 4 5 2+ − = 7 50 40 100 200 200
th
of the work. Hence no of days needed to finish the work = 200/7 = 28.56 days
Approach 2: LCM Approach.
Assume the work to be done as LCM of 100, 120 and 200 i.e. 600 units.
Since 1 man does 600 units in 100 days, one man’s rate of working is 6 unit’s per day. Since 1 woman does 600 units in 120 days, one woman’s rate of working is 5 unit’s per day. Since 1 child destroys 600 units in 200 days, one child’s rate of working is 2 unit’s per day.
112 | Time & Work
When 2 men, 3 women and 2 children work together, the net rate of working will be 2 × 6 + 3 × 5 – 2 × 3 = 12 + 15 – 6 = 21 units per day.
Thus, days taken to do 600 units will be 600/21 i.e. 200/7 days.
E.g. 5: Pipe A running alone can fill a cistern in 10 hours. Pipe B running alone can fill a cistern
in 12 hrs. In how many hours will both running together fill the cistern? Every hour Pipe A fills 1
10
th
of the cistern. Every hour Pipe B fills 1
12
th
of the cistern.
This means that every hour, when both are running together they will fill 1 + 1 =11 10 12 60
th
of the tank.
Hence the two pipes running together need 60/11 = 5.45 hours
Exercise
1. Six men can do a piece of work in 15 days. How many men are needed to complete the work in 12 days if they are half as efficient as the six men?
a. 8 b. 12 c. 15 d. 10
2. Two men and four women do a job in 3 days and three men and one women do the same job in 4 days. How long will one man and two women take to finish the same work?
a. 5 b. 6 c. 8 d. 10
3. A and B can do a work in 8 days; B and C can do it in 12 days; A, B and C together can finish it in 6 days. In how many days will A and C together do it?
a. 7 b. 8 c. 6 d. 5
4. A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together they can finish the work in 4 days. In how many days can B alone do the work?
a. 12 b. 10 c. 11 d. 14
5. A can do a piece of work in 14 days while B can do it in 21 days. They begin together but 3 days before the completion of the work, A leaves off. What is the total number of days required to complete the work?
a. 10 b. 10.2 c. 10.5 d. 11
6. A can do a work in 3 days while B can do the same work in 2 days. Both of them finish the work together and get Rs. 200. What is the share of A?
a. Rs. 120 b. Rs. 100 c. Rs. 80 d. Rs. 60
7. Twenty women can do a work in 16 days. Sixteen men can complete the same work in fifteen days. What is the ratio between the capacity of a man and a woman?
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www.takshzila.com 8. 12 men complete a work in 9 days. After they have worked for 6 days, 6 more men join them. How
many days will they take to complete the remaining work?
a. 3 b. 2 c. 1 d. 1.5
9. A man, a woman and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist 1 man and 1 woman to complete the job in 1
4
th
of a day?
a. 5 b. 41 c. 24 d. 36
10. Four men and six women can complete a work in 8 days, while three men and seven women can complete it in 10 days. In how many days will 10 women complete it?
a. 40 b. 20 c. 10 d. 30
11. A cistern normally fills up in 10 hours. However it takes 12 hours when there is a leak in its bottom. If the cistern is full, in what time shall the leak empty it?
a. 20 hrs b. 60 hrs c. 30 hrs d. 15 hrs
12. Two taps running simultaneously fill a tank. The first tap could have filled it in 8 hrs by itself, the second tap could have filled it in 24 hrs by itself. But due to a leak in its bottom, there was a delay in filling it up by 1 hr. Find the time in which the leak will empty a full tank.
114 | Assignment: Time Speed Distance & Work