StreamsIn water,the direction along the stream is called DownStream. And, the direction against the stream is called UpStream. (Speed of water is sometimes mentioned as speed of current (or) Rate of Current)If the speed of a boat or man in still water is u or Ssw km/hr, and the speed or Rate of the stream is v or Rs or Ss km/hr, then : (u>v)

DownStream Speed(a,Dss) = (u+v)=(Ssw+Rs) km/hrUpStream Speed (b,Uss) = (u-v)=(Ssw-Rs)km/hrIf the DownStream Speed is a or Dss km/hr, and the UpStream Speed is b or Uss km/hr, then

Speed in Still Water(u,Ssw)Rate of Stream (v, Rs)

A man can row u km/hr in still water. If in a stream which is flowing at v km/hr, it takes him t hrs to row to a place and back, the distance between the two places is :

A man rows a certain distance DownStream in t1 hrs and returns (UpStream) the same distance in t2 hrs. If the stream flows at the rate of v km/hr then :Speed of the man in still water is =

If in the above case speed of man in still water is u km/hr then :Speed of the Stream is

Ex: A man can row 30 km upstream and 44 km downstream in 10 hrs. Also, he can row 40 km upstream and 55 km downstream in 13 hrs. Find the rate of the current and the speed of the man in still water.Sol: (By use of multiple cross multiplication)Arrange the given figures in the following form:Upstream - DownStream - Time30 441040 5513Upstream Speed of manDownStream Speed of man

Note: How do the denominators of above two formulae differ? For upstream speed we use the figures of downstream speed and time and For downstream speed we use the figures of upstream speed and time. Numerator remain the same in both Formulae.Ex:If a man's rate with current is 12 km/hr and rate of current is 1.5 km/hr, then :=12 2*1.5 = 9 km/hrEx:A boat travels upstream from B to A and downstream from A to B in 3 hours. If the speed of boat in still water is 9 km/hr and the speed of the current is 3 km/hr, the distance between A and B is: (Consider T = Total Time)Distance=Ex:The speed of a boat in still water is 6 km/hr and the speed speed of stream is 1.5 km/hr. A man rows to a place at a distance of 22.5 km and comes back to the starting point. Find the total time taken by him.

Ex:A can row a certain distance down a stream in 6 hrs and return the same distance in 9 hrs. If the stream flows at the rate of km/hr, find how far he can row in an hour in still water.

Ex:The current of the stream runs at the rate of 4 km/hr. A boat goes 6 km and back to the starting point in 2 hrs. The speed of the boat in still water is ____km/hr.Let Speed of boat in still water = x km/hrSpeed of current = 4 km/hr ; Speed of Upstream=(x-4)km/hr ; Speed of DownStream=(x+4) km/hr,Now We reject negative Value, So Speed of boat in still water=8km/hrDirect Formula =>

Time and Distance

If the speed of a body is changed in the ratio a:b, then the ratio of time taken changes in the ratio b:a.

If a certain distance is covered at x km/hr and the same distance is covered at y km/hr then the average speed during the whole journey is

If two trains or two bodies start at the same time from points A and B towards (Opposite Directions) each other and after crossing they take a and b sec in reaching B and A respectively, then

If the Ratio of speeds of A and B is a:b, then the ratio of the times taken by them to cover the same distance is

Ex:A man covers a certain distance between his house and office on scooter. Having an average speed of 30 km/hr, he is late by10 min. However, with a speed of 40 km/hr, he reaches his office 5 min earlier. Find the distance between his house and office.

Note:10 min late and 5 min earlier making difference of (10+5)=15 min. As the other units are in km/hr, the difference in time should also be changed into hours.If both timings are late:10 min late and 5 min late, then difference between difference in arrival timings=10-5=5 min

If one time is late and another is on time:15 min late and on time, i.e. 0 min, then difference in arrival timings =10-0=10 min.

Ex:A boy goes to school at a speed of 3 km/hr and returns to the village at a speed of 2 km/hr. If he take 5 hrs in all, what is the distance between the village and the school?

Ex:A motor car does a journey in 10 hrs, the the first half at 21 km/hr and the second half at 24 km/hr. Find the distance.

Note: Half of the journey means Half of the Distance not the TimeEx:A man travels 600 km by train at 80 km/hr, 800 km by ship at 40 km/hr, 500 km by aeroplane at 400 km/hr and 100 km by car at 50 km/hr. What is the average speed for entire distance?

Ex:Two guns were fired from the same place at an interval of 12 min but a person in the train approaching the place hears the second shot 10 min after the first. The speed of the train, if speed of sound is 330 m/s, is Assume that the person were static:In that case, he would hear the sound in 12 mins, but he heard the sound in 10 minutes, and that's because during these 10 minutes the person was moving towards the sound. Thus these 10 minutes of person moving "saved" the sound its 2 minutes of moving.Let the distance between the shooting point and person when the second shot was made was d. If the person were static, this distance would be covered in 12 mins.So, d =12 * s (s speed of sound)= (12 * 60) * 330 [Converting 12 mins into seconds] ----- (i)Since the person moves towards the sound, he heard the sound 10 minutes, which means that the distance d was covered in 10 minutes at their combined speed (i.e. the sound also traveled and the person also)Their combined speed was s + p (p is the speed of person or train)So, d = 10 * (s + p) = (10 * 60) * (330 + p) [Converting 10 mins into seconds] ----- (ii)Comparing (i) and (ii) Hence p = 66 m/s.Ex: A carriage driving in a fog passed a man who was walking at the rate of 3 km/hr in the same direction. He could see the carriage for 4 minutes and it was visible to him upto a distance of 100 meters. What was the speed of the carriage?

Ex:The Distance between two stations, Delhi and Amritsar, is 450 km. A train starts at 4 PM from Delhi and moves towards Amritsar at an average speed of 60 km/hr. Another train starts from Amritsar at 3:20 PM and moves towards Delhi at at an average speed of 80 km/hr. How far from Delhi will the two trains meet and at what time?Note:(1) In this Problem we have to find the two trains meeting point distance is from Delhi from which train 'A' moves towards Amritsar. So, consider 'x' as the distance from Delhi to trains meeting point (the distance covered by train 'A' from Delhi to meeting point). (2) If they asked the meeting point distance from Amritsar from which train 'B' moves towards Delhi, then consider 'x' as distance from Amritsar to trains meeting point (the distance covered by train 'B' from Amritsar to meeting point). In this we consider Note(1) Meeting Point

Train ATrain BDelhi Amritsarfor Note(1) x450-x

for Note(2) 450-x xThe difference between the two trains journey time is 40 min (i.e. ). Second train starts moving after 40 min of first train. Here the distance is same for both trains but the journey time depends on speed of train, so we get equation as(we consider Note(1))For Note(2) we can just subtract 170 km from total distance= 450-170 = 280 km.To find at what Time we need to calculate time taken by train 'A' to cover 170 km => Train 'A' Starts at 4:00 PM= 4+2 hrs 50 min=06:50 PM.For Note(2) To find at what Time we need to calculate time taken by train 'B' to cover 280 kmTrain B starts at 3:20 PM, so meeting time is equal to 3hr 20 min + 3 hr 30 min = 06:50 PM.Formula:

Ex: Walking of his usual speed, a person 10 min late to his office. Find his Usual time to cover the distance.Ex: Running of his usual speed, a person improves his timing by 10 minutes. Find his usual time to cover the distance.Ex:A train travelling 25 km/hr leaves Delhi at 9 AM and another train travelling 35 km/hr starts at 2 PM in the same direction. How many km from Delhi will they be together?\Ex: Two men A and B walk from P to Q, a distance of 21 km, at 3 km/hr and 4 km/hr respectively. B reaches Q, returns immediately and meets A at R. Find the distance from P to R.In this problem A travels PR, B travels PQ+QR, i.e. both A and B together Completed 2PQ distance, R is any point between P and Q. Now the rates of A and B are 3:4 and they have walked 42 km.

PRQNote:When the ratio of speeds of A and B is a:b, then in this case:

Ex:Two cars A and B started moving from the same point at the same time towards opposite direction(A towards North and B is towards South). If the speed of car A is 34.5 km/h and that of car B is 41.5 km/h, after how much time will they be 684 km apart?(in hrs)

Their relative Speed = 34.5+41.5=76 ; Distance between them is 684 km , ; BAIf they are moving in same direction and distance=686 km ; Relative speed =41.5-34.5=7 km/h, Ex:Two runners cover the same distance at the rate of 15 km/hr and 16 km/hr respectively. Find the distance travelled when one takes 16 min longer than the other.Formula:

Ex: A man rode out a certain distance by train at the rate of 25 km/hr and walked back at the rate of 4 km/hr. The whole journey took 5 hours 48 minutes. What distance did he ride?

Formula: Note: Here Total Time during both types of journey is given whereas in the previous example the Diffrence in Time between both types of journey were given.Ex: A man takes 8 hrs to walk to certain place and ride back. However, he could have gained 2 hrs, if he had covered bothways by riding. How long would he have taken to walk both ways?Walking Time+Riding Time=8 hrs (1); 2 Riding Time=8-2=6 hrs (2) ; From (1) & (2) => 2 Walking Time=10 hrs

Ex: A man takes 12 hrs to walk to certain place and ride back. However, if he walks both ways he needs 3 hrs more. How long would he have taken to ride both ways? Both Way Riding Time = 12-3 = 9 hrs.

Ex: A man leaves a point P at 6 AM and reaches the point Q at 10 AM. Another man leaves the point Q at 8 AM, and reaches point P at 12 noon. At what time do they meet?Let the Distance PQ=A km ; And they meet x hrs after the first man starts.

They meet x hrs after the first man starts. The second man, as he starts 2 hrs late, meets after (x-2) hrs from his start. Therefore, the distance travelled by the second man

Formula: Since both the persons take equal time of 4 hrs to cover the distance, their meeting time will be exactly in the middle of 6 AM and 12 noon, i.e. at 9AM. But what happens when they take different times? In that case the following formula is useful

Ex: A person covers a distance in 40 minutes if he run at a speed of 45 km/h on an average. Find the speed at which he must run to reduce the time of journey to 30 minutes. => 45*40= S2 *30 =>S2=60k km/hr

Ex:Without any stoppage a person travels a certain distance at an average speed of 80 km/h, and with stoppages he covers the same distance at an average speed of 60 km/hr. How many minutes per hour does he stop.

Ex: One aeroplane started 30 minutes later than the scheduled time from a place 1500 km away from its destination. To reach the destination at the scheduled time the pilot had to increase the speed by 250 km/hr. What was the speed of the aeroplane per hour during the journey?Let it takes x hrs in second case.

Neglect negative Values.Therefore, the plane takes hrs in second case,i.e.in normal case.Thus, normal speed = Quicker MethodWe arrange the given information in Two columns as below.

The Ratio is continued until we get the two ratios such that their cross-products give the distance between the points.Thus, we find our answer as:With the speed of 1000 km/hr the plane takes hrs and with the speed of 750 km/hr the plane takes 2 hrs.Therefore, normal speed is 750 km/hr

Ex:An aeroplane scheduled on hour later than its scheduled time from a place 1200 km away from its destination. To reach the destination at the scheduled time the pilot had to increase the speed by 200 km/hr. What was the speed of the aeroplane per hour in normal case?

Thus, the normal time is 3 hrs and normal speed is 400 km/hrEx:A train was late by 6 minutes. The driver increased its speed by 4 km/hr. At the next station, 36 km away, the train reached on time. Find the original speed of train.

Thus the Normal speed is 36 km/hr.

Ex:When a man travels equal distance at speeds V1 and V2 km/hr, his average speed is 4 km/hr. But when he travels at these speeds for equal times his average speed is 4.5 km/hr. Find the difference of the two speeds.Suppose the equal distance=D km. Then time taken with V1 and V2 speeds are respectively.

In second case, average speed= =4.5 kn/hrThat is; V1+V2=9 and V1*V2==18

Direct Formula:

Ex: A person travels for 3 hrs at the speed of 40 km/hr and for 4.5 hrs at the speed of 60 km/hr. At the end of it, he finds that he has covered of total distance. At what average speed should he travel to cover the remaining distance in 4 hrs?

Ex: A person travelled 120 km by steamer, 450 km by train and 60 km by horse. It took 13 hrs 30 mins. If the rate of the train is 3 times that of the horse and 1.5 times that of the steamer, find the rate of the train per hour.Suppose the speed of horse= x km/hr. Then speed of the train=3x km/hr, and speed of steamer=2x km/hr

Quicker method: Arrange the information like: TrainSteamerHorseDistance 450 km120 km60 kmSpeed 3 2 1Total Time=13.5 hrs

Ex: A man covers a certain distance on scooter. Had he moved 3 km/h faster, he would have taken 40 minutes less. If he had moved 2 km/h slower, he would have taken 40 minutes more. Find the distance (in km) and original speed.Suppose the Distance is D km and the initial speed is x km/hr.

Quicker method: In the above question when time reduced in arrival (40 minutes) is equal to the time increased in arrival (40 minutes) then

Ex: A, B and C can walk at the rates of 3,4 and 5 km/hr respectively. They start from poona at 1, 2, 3 o'clock respectively. When B catches A, B sends him back with a message to C. When will C get the message?

A takes lead of 3 km from B.Relative speed of A and B=4-3=1 km/hr.B catches A after = 3 hr, i.e., at 2+3=5 o'clock.A returns at 5 o'clock and from a distance of 3*4=12 km from poona.In the mean time C covers a distance of 5*2=10 km from poona.Thus, A and C are 12-10=2 km apart at 5 o'clock.Relative speed of A and C=3+5=8 km/hr.Thus they meet after = hr=15 min.Thus, C will get the message at 5:15 o'clockEx:A thief is spotted by a policeman from a distance of 200 meters. When the policeman starts the chase, the thief also starts running. Assuming the speed of the thief 10 km/hr, and that of the policeman 12 km/hr, how far will have the thief run before he is overtaken?Relative Speed=12-10=2 km/hrThe thief will caught after== hrdistance covered by the thief before he gets caught=10*=1 km.Quicker Method:

TrainsRelative Speed:When two trains or two bodies are moving in opposite directions with u m/s and v m/s, then their relative speed is sum of their speeds i.e. (u+v) m/s.

When two trains or two bodies are moving in same direction with u m/s and v m/s, where u>v, then their relative speed is difference of their speeds i.e.(u-v) m/s.

If two trains or two bodies start at the same time from points A and B towards (Opposite Directions) each other and after crossing they take a and b sec in reaching B and A respectively, then

Distance:When a train of length l meters passes a platform of length b meters, it should travel the length equal to sum of the lengths of Train and Platform, i.e.(l+b) meters.

When a train of length l meters passes a pole or man it should travel the length equal to the length of the Train i.e. l meters.

Time:Time taken by a train of length l meters to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover l meters.

Time taken by a train of length l meters to pass a stationary object of length b meters is equal to the time taken by the train to cover (l+b) meters.

If two trains of length a meters and b meters are moving in opposite directions at u m/s and v m/s, then time taken by the trains to cross each other is

If two trains of length a meters and b meters are moving in same directions at u m/s and v m/s, then time taken by the faster train to cross slower train is (Such that u>v)

Ex:Two trains 121 meters and 99 meters in length respectively are running in opposite directions, one at the rate of 40 km/hr and the other at the rate of 32 km/hr. In what time will they be completely clear of each other from the moment they meet?Realative Speed (Opposite Direction)=40+32=72 km/hr=20 m/sec., Total Distance=121+99=220 meters.Realative Speed (Opposite Direction)=40-32=8 km/hr=; Total Distance=121+99=220 meters.

Ex: A train 110 meters in length travels at 60 km/hr. In what time will it pass a man who is walking at 6 km an hour (i)against it ; (ii)in the same direction?(We have to consider length of the man or pole is zero)(i)Relative speed=60+6=66 km/hr=Required Time==6 sec(ii)Relative speed=60-6=54 km/hr=15 m/sec.Required time==Ex: Two trains are moving in the same direction at 50 km/hr and 30 km/hr. The faster train crosses a man in the slower train in 18 seconds. Find the length of the faster train.Relative speed=(50-30) km/hrDistance covered in 18 sec at this speed==100 metersLength of faster train= 100 metersEx: Two trains start at the same time from Hyderabad and Delhi and proceed towards each other at the rate of 80 km/hr and 95 km/hr respectively. When they meet, it is found that one train has travelled 180 km more than the other. Find the distance between Delhi and Hyderabad.

Ex:Two trains for Delhi leave Jaipur at 08:30 a.m. and 09:00 a.m. And travel at 60 and 75 km/hr respectively. How many km from Jaipur will the two trains meet?

Ex: Without stoppage a train travels at an average speed of 75 km/hr and with stoppages it covers the same distance at an average speed of 60 km/hr. How many minutes per hour does the train stop?

Ex: A train passes by a stationary man standing on the platform in 7 seconds and passes by the platform completely in 28 seconds. If the length of th platform is 330 meters, what is the length of the train?(In this Problem man = Stationary Man)

Ex: Two stations A and B are 110 km apart on a straightline. One train starts from A at 8 a.m. And travels towards B at 40 km/hr. Another train starts from B at 10 a.m. And travels towards A at 50 km/hr. At what time will they meet?formulaformulaformulaEx: A train leaves Delhi for Amritsar at 2:45 pm and goes at the rate of 50 km/hr. Another train leaves Amritsar for Delhi at 1:35 pa and goes at the rate of 60 km/hr. If the distance between Delhi and Amritsar is 510 km, at what distance from Delhi will the two trains meet?

Direct Formula:

Ex: Two trains of length 100 m and 80 m respectively run on parallel lines of rails. When running in the same direction the faster train passes the slower train in 18 sec, but when they are running in opposite directions with the same speed as earlier, they pass each other in 9 seconds. Find the speed of each train.Let the speeds of the trains be x m/s and y m/s. When they are moving in the same direction, the relative speed=(x-y) m/s Solving we get, x=15 m/s and y=5 m/s.Formula:

Ex: A train overtakes two persons who are walking in the same direction as the train is moving, at the rate of 2 km/hr and 4 km/hr and passes them completely in 9 and 10 seconds respectively. Find the speed and length of train.

Ex: A train passes a pole in 15 seconds and passes a platform 100m long in 25 seconds. Find its length.

Ex:Two trains are running in opposite direction with speeds of 62 km/hr and 40 km/hr respectively. If the length of one train is 250 meters and they cross each other in 18 seconds, the length of the other train in meters.

Ex:A goods train and a passenger train are running on parallel tracks in the same direction. The driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 60 seconds. Whereas a passenger on the passenger train marks that he crosses the goods train in 40 seconds. If the speeds of the trains be in the ratio of 1:2, find the ratio of their lengths.Let speeds of two trains are x m/s and 2x m/s respectively. And Let lengths of two trains are A m and B m respectively.

Quicker Approach: The man in the passenger train crosses the goods train in 40 seconds. This implies that the man in the goods train can observe that the passenger train passes his in 60-40=20 seconds. (This is only because relative velocity for both the persons are the same.)Therefore, we may conclude that a person takes double the time to cross the goods train than to cross the passenger train. Thus the ration of their lengths=4020=2:1.Ex: A train after travelling 50 km meets with an accident and then proceeds at of its former speed and arrives at its destination 35 minutes late. Had the accident occurred 24 km further, it would have reached the destination only 25 minutes late. The speed of the train is___.Quicker Approach:We may conclude that the speeds of the train upto 50 km are the same in both the cases. And also, the speeds after (50+24=)74 km are the same in both the cases. Thus the difference in time (35 min-25 min=10 min) is only due to the difference in speeds for the 24 km journey.Now, if the speeds of the train is x km/hr then Direct Formula:

Ex: A train covers a distance between stations A and B in 45 minutes. If the speed is reduced by 5 km/hr, it will cover the same distance in 48 minutes. What is the distance between the two stations A and B (in km)? Also, find the speed of the train.

Direct Formula:

Ex: Two places P and Q are 162 km apart. A train leaves P for Q and at the same time another train leaves Q for P. Both the trains meet 6 hrs after they start moving. If the train travelling from P to Q travels 8 km/hr faster than other train, find the speed of the two trains.

Formula:Ex: Two trains A and B start from Delhi and Patna towards Patna and Delhi respectively. After Passing each other they take 4 hours 48 minutes and 3 hours 20 minutes to reach Patna and Delhi respectively. If the train fro Delhi is moving at 45 km/hr then find the speed of the other train.Formula:formulaformulaEx: The speed of two trains are in the ratio x : y. They are moving in the opposite directions on parallel tracks. The first train crosses a telegraph pole in 'a' seconds where as the second train crosses a telegraph pole in 'b' seconds. Find the time taken by the trains to cross each other completely.

Ex: The speed of two trains are in the ratio 7 : 9. They are moving in the opposite directions on parallel tracks. The first train crosses a telegraph pole in 4 seconds where as the second train crosses a telegraph pole in 6 seconds. Find the time taken by the trains to cross each other completely.

Ex: The speeds of the two trains are in the ratio 3:4. They are going in opposite directions along parallel tracks. If each takes 3 seconds to cross a telegraph post, find the time taken by th trains to cross each other completely?

Ex: A train 75 meters long overtook a person who was walking at the rate of 6 km/hr, and passed him in formula Subsequently it overtook a second person, and passed him in formula At what rate was the second person travelling?Formula:formula

Ex: Two Trains running at the rates of 45 and 36 km/hr respectively, on parallel rails in opposite directions, are observed to pass each other in 8 seconds, and when they are running in the same direction at the same rate as before, a person sitting in the faster train observes that he passes the other train in 30 seconds. Find the lengths of the trains.Relative Speed of two trains=45+36=81 km/hr(Two trains are moving Opposite Directions)

Length of both the TrainsNow, when two trains are moving in the same direction, the relative speedThe man sitting in the faster train passes the length of the slower train in 30 seconds.Length of the Slower Train= Length of the Faster Train= 180-75=105 m.Formula: Ex:Two trains measuring 100 and 80 meters respectively, run on parallel lines of rails. When travelling in opposite directions they are observed to pass each other in 9 seconds, but when they are running in the same direction at the same rates as before, the faster train passes the other in 18 seconds. Find the speed of the two trains in km/hr.

Ex: A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 km/h and 4 km/hr respectively and passes them completely in 9 and 10 seconds respectively. The length of the train is in meters.

Pipes and CisternsInlet: A pipe connected with a tank or reservoir or cistern, that fills it.

Outlet:A pipe connected with a tank or reservoir or cistern, emptying it.

If a pipe can fill a tank in 'x' hours, then the part filled in 1 hour =

If a pipe can empty a tank in 'y' hours, then the part emptied in 1 hour =

If a pipe can fill a tank in 'x' hours and another pipe can empty the full tank in 'y' hours, then net part filled in 1 hour, when both the pipes are opened (y>x) =

Time Taken to fill the tank, when both pipes are opened =

If a pipe can fill a tank in 'x' hours and another pipe can empty the full tank in 'y' hours, then net part emptied in 1 hour, when both the pipes are opened (x>y) =

Time Taken to empty the tank, when both pipes are opened =

If a pipe can fill the tank in 'x' hours and another can fill the same tank in 'y' hours, then net part filled in 1hour, when both pipes are opened =

Time Taken to fill the tank =

If a pipe fills a tank in 'x' hours and another fills the same tank in 'y' hours, but a third one empties the full tank in 'z' hours, and all of them are opened together, the net part filled in 1 hour =

Time Taken to fill the tank =

A pipe can fill a tank in 'x' hours. Due to a leak in the bottom it is filled in 'y' hours. If the tank is full, the time taken by the leak to empty the tank =

Ex: In what time would a cistern be filled by three pipes whose diameters are 1 cm, cm and 2 cm running together, when the largest alone will fill it in 61 minutes, the amount of water flowing in by each pipe being proportional to the square of its diameter?In 1 minute the pipe of 2 cm diameter fills of the cistern.In 1 minute the pipe of 1 cm diameter fills of the cistern.In 1 minute the pipe of diameter fills of the cistern.

In 1 minute =of the cistern filled.Therefore, The whole is filled in 36 minutesNote: We are given that amount of water flowing is proportional to square of the diameter of the pipe. Since 2 cm diameter fills of the cistern, => 1 cm Diameter fills of the cistern=> Diameter fills of the cistern.

Ex: Two pipes A and B can fill a tank in 24 minutes and 32 minutes respectively. If both the pipes are opened simultaneously, after how much time should B be closed so that the tank is full in 18 minutes?Let B be closed after 'x' minutes. Then, part filled by (A+B) in x min.+Part filled by A in (18-x) min. = 1.

Direct Formula: Pipe B should be closed after Note: If they asked for pipe A (should be closed after how much TIme), then Direct formula becomes

Ex: If two pipes function simultaneously, the reservoir is filled in 12 hrs. One pipe fills the reservoir 10 hrs faster than the other. How many hours does the faster pipe take to fill the reservoir?Let the first pipe fills the tank in 'x' hrs. Then the slower pipe fills the tank in (x+10) hrs.When both of them are opened, the reservoir will be filled in But x can't be hence the faster pipe will fill the reservoir in 20 hrs.Ex: Three pipes A,B and C can fill a cistern in 6 hrs. After working together for 2 hours, C is closed and A and B fill the cistern in 8 hrs. Then find the time in which the cistern can be filled by pipe C.A+B+C can fill in 1 hr = of cistern.A+B+C can fill in 2 hrs =of cistern.Unfilled part is filled by A+B in 8 hrs(A+B) can fill the cistern in = 12 hrs.And we have (A+B+C) can fill the cistern in 6 hrs.C=(A+B+C)-(A+B) can fill the cistern in

Ex: A tank has a leak which would empty it in 8 hrs. A tap is turned on which admits 6 liters a minute into the tank, and it is now emptied in 12 hrs. How many liters does the tank hold?The filler tap can fill the tank in Capacity of tank=24*60*6=8640 liters.Ex: A tank is normally filled in 8 hours but takes 2 hours longer to fill because of a leak in its bottom. If the cistern is full, in hoe many hrs will the leak empty it?Suppose the leak can empty the tank in x hrs.Then part of cistern filled in 1 hrCistern will be filled inNow, It is clear from the question that the filler pipe fill the tank in 8 hrs and if both the filler and the leak work together, the tank is filled in 10 hrs. Therefore, the leak will empty the tank in Ex: A pipe can fill a tank in 12 minutes and another pipe in 15 minutes, but a third pipe can empty it in 6 minutes. The first two pipes are kept open for 5 minutes in the beginning and then the third pipe is also opened. In what time is the cistern emptied?Cistern filled in 5 minutes =Net work done by 3 pipes in 1 minute part is emptied in Ex: If three taps are opened together, a tank is filled in 12 hrs. One of the taps can fill it in 10 hrs and another in 15 hrs. How does the third tap work?We have to find the nature of the third tap whether it is a filler or a waste pipe.Let it be a filler pipe which fills in x hrs.

Ex: A, B and C are three pipes connected to a tank. A and B together fill the tank in 6 hrs. B and C together fill the tank in 10 hrs. A and C together fill the tank in hrs. In how much time will A, B and C fill the tank separately?A+B fill in 6 hrs. ; B+C fill in 10 hrs. ;

2(A+B+C) fill in A+B+C fill the tank in 5 hrsNow, A[=(A+B+C)-(B+C)] fills in Similarly, B fills in and C fills in Ex: Two pipes can separately fill a tank in 20 hrs and 30 hrs respectively. Both the pipes are opened to fill the tank but when the tank is full a leak develops in the tank through whichof the water supplied by both the pipes leak out. What is the total time taken to fill the tank?Time taken by the two pipes to fill the tank

of tank is filled inNow, of the supplied water leaks out=> The filler pipes are only as efficient as earlier.=> the work of (12-4=) 8 hrs will be completed now in => Total Time=4+12=16 hrs.Quicker Method: Since of supplied water leaks out, the leakage empties the tank in 12*3=36 hrs. Now, time taken to fill the tank by the two pipes and the leakageTime taken by the two pipes and the leakage to fill of the tankTotal Time=4 hrs+12 hrs=16 hrs.

Ex: Two Pipes A and B can fill a cistern in 60 minutes and 75 minutes respectively. There is also an outlet C. If all the three pipes are opened together, the tank is full in 50 minutes. How much time will be taken by C to empty the full tank?Work done by C in 1 min.

Ex: A tap can fill a tank in 6 hours. After half the tank is filled , three more similar taps are opened. What is the total time taken to fill the tank completely?Time taken by one tap to fill half the tank= 3hrs.Part filled by the four taps in 1 hourRemaining part=>Total Time=3 hrs 45 min.Ex: A water tank is two-fifth full. Pipe A can fill a tank in 10 minutes and pipe B can empty it in 6 minutes. If both pipes are open, how long will it take to empty or fill the tank completely?Clearly, Pipe B is faster than pipe A and so, the tank will be emptied. Part to be emptiedPart emptied by (A+B) in 1 minute

So, the tank will be emptied in 6 min.Ex: Three pipes A,B and C can fill a tank from empty to full in 30 minutes, 20 minutes and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P,Q and R respectively. What is the proportion of solution R in the tank after 3 minutes?Part filled by (A+B+C) in 3 minutesPart filled by C in 3 minutesRequired RatioEx: Two taps A and B can fill a tank in 5 hours and 20 hours respectively. If both taps are open then due to leakage, it took 30 minutes more to fill the tank. If the tank is full, how long will it take for the leakage alone to empty the tank?Part filled by (A+B) in 1 hour=So, A and B together can fill the tank in 4 hours.Work done by the leak in 1 hourleak will empty the tank in 36 hours.By direct formula:Ex: One pipe can fill a tank three times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the slower pipe alone will be able to fill the tank in:Let the slower pipe alone fill the tank in x minutes.Then, faster pipe will fill it in minutes .

Ex: A tank is filled in 5 hours by three pipes A,B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?Suppose pipe A alone takes x hours to fill the tank.Then, pipes B and C will take hours respectively to fill the tank.

So pipe A in 35 hrs, Pipe B in 17.5 hrs and Pipe C in 8.75 hrs alone can fill the empty tank.Ex: A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is:Suppose, first pipe alone takes x hours to fill the tank. Then, second and third pipes will take (x-5) and (x-9) hours respectively to fill the tank.

Ex: Two pipes A and B can fill a tank in 12 minutes and 15 minutes respectively. If both the taps are opened simultaneously, and the tap A is closed after 3 minutes, then how much more time will it take to fill the tank by tap B?Part fille in 3 min. Remaining partPart filled by B in 1 min

Ex: Two pipes A and B can fill a tank in 15 hours and 20 hours respectively while a third pipe C can empty the full tank in 25 hours. All the three pipes are opened in the beginning. After 10 hours, C is closed. In how much time, will the tank be full?Part filled in 10 hoursRemaining part(A+B)'s 1 hour work

Ex: Two pipes A and B can fill a cistern in 12 minutes and 15 minutes respectively. While a third pipe C can empty the full tank in 6 minutes. A and B kept open for 5 minutes in the beginning and then C is also opened. In what time is the cistern emptied?Part filled in 5 min. Part emptied in 1 min. when all the pipes are openedNow, part is emptied in 1 min. part will be emptied in Ex: Three taps A, B and C can fill a tank in 12, 15 and 20 hours respectively. If A is open all the time and B and C are open for one hour each alternately, the tank will be full in:(A+B)'s 1 hour's work(A+C)'s 1 hour's workPart filled in 2 hrsPart filled in 6 hrs.Remaining PartNow, it is the turn of A and B andpart is filled by A and B in 1 hour.Total time taken to fill the tank=(6+1) hrs=7 hrs.

Ex: A booster pump can be used for filling as well as for emptying a tank. The capacity of the tank isThe emptying capacty of the pump ishigher than its filling capacity and the pump needs 8 minutes lesser to empty the tank than it needs to fill it. What is the filling capacity of the pump?Let the filling capacity of the pump beThen, emptying capacity of the pumpSo,Ex: A leak in the bottom of a tank can empty the full tank in 8 hours. An inlet pipe fills water at the rate of 6 liters a minute. When the tank is full, the inlet is opened and due to the leak, the tank is empty in 12 hours. How many liters does the cistern hold?Work done by the inlet in 1 hourWork done by the inlet in 1 min.Volume of Volume of whole = (1440*6) liters = 8640 liters.Ex: Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallon per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:Work done by the waste pipe in 1 minute Volume of Volume of whole =(3*40) gallons=120 gallons.

Work and WagesNote: Wages are distributed in Proportion to the work done and in indirect (or inverse) proportion to the time taken by the individual.

Ex: A can do a work in 6 days and B can do the same work in 5 days. The contract for the work is Rs 220. How much shall B get if both of them work together?Method I:

Method II:As wages are distributed in inverse proportion of number of days, their share should be in the ratio 5:6.B's Share

Ex: A man can do a work in 10 days. With the help of a boy he can do the same work in 6 days. If they get Rs 50 for that work, what is the share of that boy?The boy can do the work in Man's Share : Boy's Share=15:10=3:2Man's Share Ex: A, B and C can do a work in 6, 8 and 12 days respectively. Doing that work together they get an amount of Ra 1350. What is the share of B in that amount?Direct Method: A's Share : B's Share : C's Share=B's time * C's time : A's time * C's time : A's time * B's time =96:72:48=4:3:2Ex: A,B and C contract a work for Rs 550. Together, A and B are supposed to doof the work. How much does C get?

C's ShareEx: Two men undertake to do a piece of work for Rs 200. One alone could do it in 6 days, the other in 8 days. With the asssistance of a boy they finish it in 3 days. How should the money be divided?

The boy's 3 days workThe share will be in the ratio1 st man's Share2 nd man's ShareThe boy's ShareEx: Wages for 45 women amount to Rs 15525 in 48 days. How many men must work 16 days to receive Rs 5750, the daily wages of a man being double those of a woman?Wage of a women for a dayThus, wage of a man for a day

Ex: A and B undertake to do a work for Rs 56. A can do it alone in 7 days and B in 8 days. If with the assistance of a boy they finish the work in 3 days then the boy gets Rs___.A's 3 days work+B's 3 days work+Boy's 3 days work=1

Ex: A sum of money is sufficient to pay A's wages for 21 days or B's wages for 28 days. The money is sufficient to pay the wages of both for___days.sum of money is sufficient to pay A wages for 21 days money sufficient to pay A for 1 day =sum of money is sufficient to pay B wages for 28 days money sufficient to pay B for 1 day =money sufficient to pay A & B for 1 day =thus, The same money is sufficient to pay the wages of both for 12 daysFormula:

Ex: 3 men and 4 boys can earn Rs 756 in 7 days. 11 men and 13 boys can earn Rs 3008 in 8 days. In what time will 7 men with 9 boys earn Rs 2480?(3m+4b) in 1 day earn(11m+13b) in 1 day earnFrom (1), we see that to earn Re 1 in 1 day there should bepersons. Similarly, from (2), to earn Re 1 in 1 day there should bepersons.And also;Since both the LHS and the RHS denote the same newline quantity: Number of persons earning Re 1 in 1 day.Now, from (1)(3m+4b) in 1 day earn Rs 108or, in 1 day earn Rs 108or,in 1 day earn Rs 1081m in 1 day earnsThus, we get that a man earns Rs 20 daily and a boy earns(7m+9b) earn Rs (7*20+9*12)=Rs 248 in 1 day.(7m+9b) earn Rs 2480 in 10 days.Quicker method:Using Cross-Multiplication-Division Rule.

Now,

Ex: A,B and C together earn Rs 1350 in 9 days. A and C together earn Rs 470 in 5 days. B and C together earn Rs 760 in 10 days.Find the daily earning of C.

Time and WorkNote: If persons can doworks indays andpersons can doworks indays then we have

More men less days and conversely more days less men.

More men More work and conversely more work more men.

More days more work and conversely more work more days.

If we include the working hours (sayand) for two groups then the relation ship is

Again, if the efficiency (sayand) of the persons in two groups is different then the relationship is

If A can do a piece of work in x days, and B can do it in y days then A and B working together will do the same work in

If A,B and C can do a work in x,y and z days respectively then all of them working together can finish the work in

If and B together can do a piece of work in x days and A alone can do it in y days, then B alone can do the work in

Ex: A can do a piece of work in 5 days. How many days will he take to complete 3 works of same type?

Ex: 16 men can do a piece of work in 10 days. How many men are needed to complete the work in 40 days?

Ex: 40 men can cut 60 trees in 8 hrs. If 8 men leave the job how many trees will be cut in 12 hours?

Ex: 5 men can prepare 10 toys in 6 days working 6 hrs a day. Then in how many days can 12 men prepare 16 toys working 8 hrs a day?

By Rule of Fraction:We have to find number of days, so write given number of days first.

Number of men increases => Work will be done in less days => multiplying factor should be less than 1, which is

Number of toys increases => it will take more days => multiplying factor should be more than 1, which is

Number of working hours increases => it will take less days => multiplying factor should be less than 1, which is

Note: In rule of fraction the multiplying factor is decided such that how the known variable changes the value of unknown variable.(Increase or Decrease). Ex: more men less days, here men known, days unknown variable, number of men inversely proportional to number of days, so, the fraction is always less than 1.

Ex: A can do a piece of work in 5 days, B can do it in 6 days and C can do it in 12 days. (i) How long it will take if A and B work together? (ii) How long it will take if A, B and C work together?

Ex: A and B together can do a piece of work in 6 days and A alone can do it in 9 days. In how many days can B alone do it?

Ex: A and B can do a piece of work in 12 days, B and C in 15 days, C and A in 20 days. How long would each take separately to do the same work?

Ex: Two women, Ganga and Saraswati, working separetely can mow a field in 8 and 12 hrs respectively. If they work in stretches of one hour alternately, Ganga beginning at 9a.m., when will the mowing be finished?In the first hour Ganga mowsformulaof the field. In the second hour Saraswati mowsformulaof the field.In the first 2 hoursformulaof the field mown.In 8 hrsformulaof the field is mown.(we calculated for 4 pairs of hours only because if we calculate for 5 pairs of hours, the work done is more than 1.)Remaining Field to be mownformulaof the field.In the 9th hour Ganga mowsformulaof the field.Saraswati will finish the mowing offormulaof the field informulaThe total time required is formulaThus the work will be finished at formula