# Time & Work

• Concepts:

If A can do a piece of work in ‘a’ number of days, then in one day (1/a)th of the work is done. Similarly if an inlet pipe fills a cistern in ‘a’ hours, then (1/a)th part is filled in one hour. If an outlet pipe empties a cistern in ‘a’ hours, then (1/a)th part is emptied in 1 hour.

If A is ‘x ‘ times as good a workman as B , then he will take (1/x)th of the time taken by B to do the same work. If Pipe A is ‘x’ times bigger than pipe B, then pipe A will take (1/x)th of the time taken by pipe B to fill the cistern.

If A and B can do a piece of work in ‘x’ and ‘y’ days respectively, then working together they will take (xy/x+y) days to finish the work and one day they will finish (x+y/xy)th part of the work. If A and B fill a cistern in ‘m’ and ‘n’ hours respectively then together they will take (mn/m+n) hours to fill the cistern and in one hour (m+n/mn)th part of the cistern will be filled. Similarly A and B empty a cistern in ‘m’ and ‘n’ hours, respectively then together they will take (m+n/mn) hours to empty the cistern.

If an outlet pipe empties the cistern in ‘n’ hours and an inlet pipe fills a cistern in ‘m’ hours then the net part filled in 1 hour when both the pipes are opened is (1/m – 1/n) and the cistern will get filled in (mn/n-m) hours. For the Cistern to get filled its necessary that mn the cistern will never get filled. In general Net part filled of a cistern = (Sum of work done by inlets) – (Sum of work done by outlets)

If two pipes A and B can fill a cistern in ‘x’ minutes and if A alone can fill it in ‘a’ minutes more than ‘x’ minutes and B alone can fill it in ‘b’ minutes more than x minutes then x= root(ab). If two men A and B together can finish a job in x days and if A working alone takes ‘a’ days more than A and B working together and B working alone takes ‘b’ days more than A and B working together then x = root(ab).

Questions

If 30 men working 7 hours per day can do a work in 18 days. In how many days will 21 men working 8 hours a day do the same work?

Total work = 30 * 7 * 18
Total work is equal, hence 30 * 7 * 18 = 21 * 8 * x
Where x is the number of days required for 21 men to finish the work.
X = 30 * 7 * 18 / 21 * 8 = 22.5 days

Two men and 7 boys can do a piece of work in 14 days. 3 men and 8 boys can do it in 11 days. In how many days can 8 men and 6 boys do a work 3 times as big as the first?

2/M+7/B = 1/14 (Work done by 2 men and 7 boys In one day) …… (1)
3/M+8/B = 1/11 ………… (2)
(1) * 3  6/M + 21/B = 3/14 …….. (3)
(2) * 2  6/M+ 16/B = 2/11 ………. (4)
(3)-(4)  5/B = 5/154
1/B = 1/154  One boy will take 154 days to complete the work.
2/M = 1/14 – 1/22 = 11-7/154 = 4/154
1/M = 2/154 = 1/77  One man will take 77 days to complete the work.
In case of 8 men and 6 boys  8/77 + 6/154 = 16+6/154 = 22/154
Number of days required for 8 men and 6 boys to complete the same work = 154/22 = 7 day
Since work is increased 3 times, required answer is 3*7 = 21 days.

To do a piece of work B takes 3 times as long as A and C together and C twice as long as A and B together. If the three together can complete the work In 10 days, how long would each take by himself?

3 times B’s daily work = (A+C)’s daily work
4 times B’s daily work = (A+B+C)’s daily work = 1/10
B’s daily work = 1/40
Hence B takes 40 days.
2 times C’s daily work = (A+B)’s daily work
3 times C’s daily work = (A+B+C)’s daily work = 1/10
C’s daily work = 1/30
Hence C takes 30 days to complete the work.
A’s daily work = 1/10 – (1/30+1/40) = 1/10 – 7/120 = 5/120 = 1/24
A takes 24 days to complete the work.

P can do a piece of work in 10 days, which Q can finish in 15 days. If they work at it on alternate days with P beginning In how many days the work will be finished.

The work done by P and Q in two days = 1/10+1/15 = 5/30 = 1/6
The work done in 12 days = 6/6 = 1
Hence work will be finished in 12 days.

Pipes A and B can fill a cistern in 20 and 30 minutes and C can empty it in 15 minutes. If the three are opened and closed one after the other successively for 1 min each in that order, how soon will the cistern be filled?

Part filled in 3 minutes = 1/20+1/30 – 1/15 = 3+2-4/60 = 1/60
Part filled in 165 minutes = 55/60
Part filled in 166 minutes = 55/60 + 1/20 = 58/60
Part filled in 167 minutes = 58/60+1/30 = 60/60 = 1
Hence it will be filled in 167 minutes.

A water tank is three fifth full. Pipe A can fill a tank in 8 minutes and pipe B can empty it in 5 minutes. If both the pipes are open how long will it take to empty/fill the tank completely?

Tank’s 3/5 th part is filled .
In one minute 1/5 – 1/8 = 3/40 part will be removed
So number of minutes required to empty the tank = 3/5 *40/3 = 8 minutes.

Two pipes A and B can separately fill a cistern in 15 and 20 minutes respectively and waste pipe C can carry off 10 litres per minute. If all the pipes are opened when the cistern is full, it is emptied in 2 hours. How many litres does the cistern hold?

When cistern is full, all pipes are opened and its emptied In 2 hours.
Let total volume is X.
In one minute pipe A will fill X/15 th of the Cistern.
In one minute pipe B will fill X/20 th of the cistern.
In one minute pipe C will empty 10 Litres.
Hence X+120(X/15+X/20 -10) = 0
X + 8X + 6X - 1200 = 0
15X = 1200
X = 80 litres.

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