*Wednesday, May 15th, 2019*

Work and men are directly proportional to each other, i.e. if the work increases the number of men required increases to complete the work in the same number of days and vice versa.

Men and days are inversely proportional, i.e., if the number of men increases, the number of days required to complete the same work decreases and vice versa.

Work and days are directly proportional i.e., if the work increases, then number of days required also increases if the work is to be completed with the same number of men and vice versa.

The concept of **MANDAYS** is very important and useful here. The number of men multiplied by the number of days that they take to complete the work will give the number of **MANDAYS** required to do the work. The total number of **MANDAYS** required to complete a specific task is always constant. So if we change one of the variables: men or days â€“ the other will change accordingly so that their product will remain constant. The two variable men are days are inversely proportional to each other.

Also: If M_{1 }men can do W_{1 }work in D_{1 }Â days working H_{1 }hours per day and M_{2 }men can do W_{2 }Â work in D_{2 }days working H_{2 }hours per day, then

(M1 D1 H1 )/W1 = (M2 D2 H2)/W2

**Example 1.** 4 men or 5 women can construct a wall in 82 days. How long will it take 5 men and 2 women to do the same?

**Sol:** Given, 4m=5w where m is work done by one man in one day and w is work done by one woman in one day;

Therefore 1m=5w/4

Now, 5m+4w= 5(5w/4) +4w = 41w/4

If 5w can do the work in 82 days, then 41w/4 will do in 5w x 82 x 4/41w = 40 days

**ExampleÂ 2.** If 9 men and 12 boys can do a piece of work in 4 days and 4 men and 16 boys can do the same piece of work in 6 days, how long will 6 men and 24 boys take to complete the same work?

**Sol:** Given 9m + 12b can do the work in 4 days and 4m + 16b can do the same work in 6 days therefore, 4(9m + 12b) = 6(4m + 16b)

â‡’ m=4b

Now, 9m + 12b= 9(4b) +12b = 48b i.e. 48b are taking 4 days to complete this job.

We have to find for: 6m + 24b i.e. 6(4b) + 24b = 48b.

Hence, 6 men and 24 boys will take 4 days to complete this job.

**Example 3.** X is 3 times as fast as & and is able to complete the work in 40 days less than y. Find the time in which they can complete the work together.

**Sol:** If Y does the work in 3 days, X does it in 1 day i.e. the difference is 2 days. But the actual difference is 40 days.

If difference is 2 days, X takes 1 day and Y takes 3 days, if difference is 40 days (i.e. 20 times), X will take 20 days and Y will take 40 days.

Therefore, Total time together = (20 x 60)/(20+60) = 15 days

**Example 4.Â **A and B separately can do a piece of work in 20 and 24 days. They work on alternate day starting with B on the first day. In how many days will the work be completed?

**Sol:** Work done by A and B in the period of 2 days =

1/20 + 1/24 =Â 11/120

If we consider â€˜10â€™such time periods ( we are considering 10 periods because in the fraction 11/120, the numerator 11 goes 10 times in the denominator 120)

Work done= 10 x11/120 = 11/12

Remaining work = 1- 11/12 =Â 1/12

Now it is Bâ€™s turn and he can complete 1/24^{th} of the work in 1 day. So, he can complete the balance 1/12^{th} of work. So after 1 dayâ€™s work by B, balance work = 1/12 -1/24 = 1/24

Now, it is Aâ€™s turn and he can complete 1/20^{th} of the work in one day.

So to complete 1/24^{th} of work, he will take 20/24= 5/6^{th} of a day.

Total time taken = 10×2+1+ 5/6

=131/6 days

When the people doing some work earn some money together for doing the work then this money has to be shared by the people doing the work.

In general, money earn should be shared by the people doing the work together in the ratio of the TOTAL WORK done by each of them.

For e.g. If A does 2/5^{th} of the work, then he should get 2/5^{th} of the total earnings for the work. If the remaining 3/5^{th} of the work is done by B and C in the ratio of 1:2, then the remaining 3/5^{th} of the earnings (after paying A) should be shared by B and C in the ratio of 1:2. Suppose Rs.500 is paid to A, B and C together for doing the work, then A will get Rs.200 (which is 2/5 of Rs.500), B will get Rs.100 and C Rs.200 (because the remaining Rs.300 after paying A is to be divided in the ratio 1:2 between B and C).

When people work for the same number of the days each then the ratio of total work done will be the same as the work done by each of them PER DAY. Hence, if all the people involved work for the same number of days then the earnings can directly be divided in the ratio of work done per day by each of them.

For e.g.

A, B and C can do a piece of work in 4, 5, and 7 days respectively. They got Rs.415 for the job. What is Aâ€™s share?

Sol: Since they work for the same number of days, the ratio in which they share the money in the ratio work done per day i.e. 1/4 : 1/5: 1/7 = 35: 28: 20

Hence, Aâ€™s share is (35/83) x 415 = Rs.175

Now letâ€™s raise the level of questions:

**Question 1.** P can complete a certain work in 18 days, Q and R can complete the same work in 9 days and 6 days respectively. P starts the work and after working for 3 days, he is joined by Q and they work together for another 3 days. Then they are joined by R and together they complete the work. What percentage of work is done by P?

**Sol:** The amount of work done by P, Q and R is: 1/18, 1/9, 1/6 respectively.

For the first 3 days, P alone will do the work i.e. work done = 3 x 1/18 = 1/6

Then the next days, P and Q both will work i.e. work done = (1/18+ 1/9)3 = 1/2

Therefore, remaining work = 1- (1/6+ 1/2) = 1/3

To do 1/3 of work, no of days = (1/3)/((1/18+1/9+1/6) ) = 1

i.e. P will do 1 work for 1 more days, hence total days the P will work = 3+3+1 = 7

Therefore the percentage of work done by P : 7/18 x 100 = 38.88

**Question 2.** A can do a piece of work in 20 days working 7 hours a day. The work is started by A and on the second day one man whose capacity to do the work is that of A, joined. On the third day another man whose capacity is thrice that of A joined. This manner of working is continued till the work gets completed. In how many days will the work be completed, if everyone works for 4 hours a day?

**Sol:**Â Â A can complete the work in 20 days working 7 hours a day i.e. in 20 x7 = 140 hours.

Hence, the number of days in which A will complete the work working 4 hours a day = 140/4 = 35 days.

On 1^{st} day: Amount of work done = 1/35

On 2^{nd} day: Other man will join whose capacity is twice of A, therefore amount of work done =Â Â 1/35 +2/35 = 3/35

On the 3^{rd} day: Another man will join whose capacity is thrice of A, therefore amount of workdone = 1/35 + 2/35 +3/35 = 6/35

This process will continueâ€¦

This is forming a series.

1/35, 3/35, 6/35â€¦ whose common difference is also changing.

We have to find the number of terms here which would be equal to 35 (total units of work done).

On solving we will get, n =5.

**Question 3.** The ratio of the efficiencies of P, Q, and R is 2:3:5. The total wages of P, Q, and R working for 14, 24, and 20 days respectively are Rs. 6000. Find the total wages of the three, when P works for 9 days, Q for 14 days and R for 8 days.

**Sol:** Let P can do 2 units of work, Q can 3 and R can do 5 units of work per day.

Hence the number of units of P did: 14 x 2= 28 units

Q did: 24 x 3= 72 units

R did: 20 x 5 = 100 units

Total units = 200

Price of each unit = 6000/200 = Rs 30

Now P will work for 9 days i.e. 18 units;

Q will work for 14 days i.e. 42 units;

R will work for 8 days i.e. 40 units;

Hence, total number of units = 100 units;

And total price = Rs.3000

In CAT exam, the questions can be make more complicated by many ways such as alternate working days, efficiency ratio, and many other combinations. Make sure you practice them hard and gain a good command over these questions.

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