Selecting tasks for collaboration and productivity.

Tasks for working together and productivity

Tasks of this type usually contain information about the performance by several subjects (workers, mechanisms, pumps, etc.) of some work, the volume of which is not indicated and is not sought (for example, reprinting a manuscript, manufacturing parts, digging trenches, filling a reservoir through pipes and etc.). It is assumed that the work being performed is carried out evenly, i.e. with constant productivity for each subject. Since we are not interested in the amount of work performed (or the volume of a swimming pool being filled, for example), the volume of all work is. or basin is taken as a unit. Timet, required to complete all the work, and P is the manufacturerlabor intensity, that is, the amount of work done per unit of time, are related

ratioP= 1/t .It is useful to know the standard scheme for solving typical problems.

Let one worker do some work in x hours, and another worker in y hours. Then in one hour they will complete 1/xand 1/ypart of the work. Together in one hour they will complete 1/x +1/ ypart of the work. Therefore, if they work together, then all the work will be done in 1/ (1/x+ 1/ y)

Solving collaboration problems is challenging for students, so when preparing for the exam, you can start by solving the most simple tasks. Let's consider the type of problems for which it is enough to enter only one variable.

Task 1. One plasterer can complete a task 5 hours faster than another. Both of them together will complete this task in 6 hours. How many hours will it take each of them to complete the task?

Solution. Let the first plasterer complete the task inxhours, then the second plasterer will complete this task inx+5 hours. In 1 hour of joint work they will complete 1/x + 1/( x+5) tasks. Let's make an equation

6×(1/x+ 1/( x+5))= 1 orx² - 7 x-30 = 0. Solving given equation,we getx= 10 andx= -3. According to the conditions of the problemx– the value is positive. Therefore, the first plasterer can complete the job in 10 hours, and the second in 15 hours.

Problem 2 . Two workers completed the job in 12 days. How many days can each worker complete the job if one of them took 10 days more than the other to complete the entire job?

Solution . Let the first worker spend on all the workxdays, then the second- (x-10) days. In 1 day of working together they complete 1/x+ 1/( x-10) tasks. Let's make an equation

12×(1/x+ 1/( x-10)= 1 orx²- 34x+120=0. Solving this equation, we getx=30 andx= 4. The conditions of the problem are satisfied only byx=30. Therefore, the first worker can complete the job in 30 days, and the second worker in 20 days.

Task 3. In 4 days of joint work, 2/3 of the field was plowed with two tractors. In how many days could it take to plow the entire field with each tractor, if the first one can plow it 5 days faster than the second one?

Solution. Let the first tractor spendto complete the task x days, then the second - x + 5 days. During 4 days of joint work, both tractors plowed 4×(1/ x + 1/( x +5)) tasks, that is, 2/3 of the field. Let's create the equation 4×(1/ x + 1/ ( x +5)) = 2/3 orx² -7x-30 = 0. . Solving this equation, we getx= 10 andx= -3. According to the conditions of the problemx– the value is positive. Therefore, the first tractor can plow a field in 10 hours, and the second in 15 hours.

Problem 4 . Masha can print 10 pages in 1 hour. Tanya can print 4 pages in 0.5, and Olya can print 3 pages in 20 minutes. How can the girls distribute 54 pages of text among themselves so that each one works for the same amount of time?

Solution . According to the condition, Tanya prints 4 pages in 0.5 hours, i.e. 8 pages in 1 hour, and Olya – 9 pages in 1 hour. Designated by X hours - time, during which the girls worked, we get the equation

10X + 8X + 9X = 54, from which X = 2.

This means that Tanya must print 20 pages, Tanya must print 16 pages, and Olya must print 18 pages.

Task 5. Using two duplicating machines operating simultaneously, you can make a copy of a manuscript in 20 minutes. In what time can this work be completed on each machine separately, if it is known that when working on the first one it will take 30 minutes less than when working on the second?

Solution. Let X min be the time required to complete the copy on the first machine, then X+30 min-time work on the second device. Then 1/X copies are performed by the first machine in 1 minute, and 1/(X+30) copies - second machine.

Let's make the equation: 20× (1/X + 1/(X+30)) = 1, we getX²-10X-600= 0. From where X = 30 and X = - 20. The conditions of the problem are satisfied by X = 30. We got: 30 minutes - the time for the first device to make a copy, 60 minutes for the second.

Task 6. Firm A can fulfill some order for the production of toys 4 days faster than firm B. How long can each firm complete this order if it is known that when working together, they complete an order 5 times larger in 24 days?

Solution. Designated by X days - time, required by company A to complete the order, then X + 4 days is the time for company B. When drawing up the equation, it is necessary to take into account that in 24 days of joint work not 1 order, but 5 orders will be completed. We get 24× (1/X + 1/( X+4)) = 5. Whence it follows 5 X²- 28X-96 = 0. Solving the quadratic equation we get X = 8 and X = - 12/5. The first company can complete the order in 8 days, company B in 12 days.

When solving the following problems, you need to enter more than one variableand solve systems of equations.

Problem 7 . Two workers are doing some work. After 45 minutes of joint work, the first worker was transferred to another job, and the second worker completed the rest of the work in 2 hours 15 minutes. In what time could each worker individually complete all the work, if it is known that the second one will need 1 hour more to do this than the first?

Solution. Let the first worker complete all the work in x hours, and the second worker in y hours. From the conditions of the problem we have x = y -1. 1 hour first

the worker will perform 1/xpart of the work, and the second - 1/ypart of the work.T.To. they worked together for ¾ hours, then during this time they completed ¾ (1/x + 1/ y)

part of the work. For2 and 1/4hour of work the second completed 9/4× (1/y) part of the work.T.To. all the work is done, then we compose the equation ¾ (1/x+1/ y)+9/4×1/y=1 or

¾ ×1/x+ 3 ×1/y =1

Substituting the valuexinto this equation, we get ¾× 1/ (y-1)+ 3×1/y= 1. We reduce this equation to quadratic 4y2 -19у + 12 =0, which has

solutions from 1 = h andat 2 = 4 hours. The first solution is not suitable (both slavesOwho only worked together for ¾ hours!). Then y = 4 and x =3.

Answer. 3 hours, 4 hours.

Task 8. The pool can be filled with water from two taps. If the first tap is opened for 10 minutes and the second for 20 minutes, the pool will be filled.

If the first tap is opened for 5 minutes, and the second for 15 minutes, then 3/5 will be filled swimming pool

How long does it take to fill the entire pool from each tap separately?

Solution. Let it be possible to fill the pool from the first tap in x minutes, and from the second tap in y 1 minute. The first tap fills part of the pool, and the second . In 10 minutes from the first tap it will be filled part of the pool, and in 20 minutes from the second tap - . T.To. the pool will be filled, we get the first equation: . We compose the second equation in the same way (fills the entire pool, but only its volume). To simplify the solution of the problem, we introduce new variables: Then we have linear system equations:

10u + 20v =1,

the solution of which will be u = v = . From here we get the answer: x = min, y = 50 min.

Task 9 . Two people do the work. First one worked time during which the second one does all the work. Then the second one worked time in which the first one would have completed the remaining work. Both of them only completed all the work. How much time does it take for each person to complete this work if it is known that if they work together they will do it in3 h36 min?

Solution. Let us denote by x hours and y hours the time it takes the first and second to complete all the work, respectively. Then And

Those parts of the work that they perform for1 hourWorking (by condition) time, the first one will complete part of the work. Will remain unfulfilled part of the work that the first would have spent on hours. According to the second condition, 1 works/3 this time. Then he will do part of the work. Together they only completed all the work. Therefore, we get the equation . Working together for1 they'll both do an hour + part of the work. Since, according to the conditions of the problem, they will do this work in3 h36 min (that is, sa 3 hours), then for1 they'll do it in an hour all the work. Hence 1/x + 1/ y = 5/18. Denoting in the first equation , we get a quadratic equation

6 t 2 - 13 t + 6 = 0 , whose roots are equalt 1 =2/3 , t 2 =3/2. Since it is unknown who works faster, we consider both cases.

A)t = => y = X. Substitute y into the second equation: Obviously this is not a solution

tasks, since together they do the work in more than 3 hours.

b) t=3/2 => y=3/2 x. From the second equation we have 1/x+2/3× 1/x=5/18.From herex=6,y =9.

Task 10. Water enters the reservoir from two pipes of different diameters. On the first day, both pipes, working simultaneously, supplied 14m 3 water. On the second day it was included only small pipe. She served 14 m 3 water, working 5 hours longer than on the first day. On the third day, work continued for the same amount of time as on the second, but both pipes worked first, delivering 21 m 3 water. And then only a large pipe worked, supplying another 20 m 3 water. Find the productivity of each pipe.

Solution. In this problem there is no abstract concept“volume of the reservoir”, and specific volumes of water that flow through the pipes are indicated. However, the method for solving the problem actually remains the same.

Let the smaller and larger pipes pump x and y m in 1 hour3 water. Working together, both pipes supply x + y m3 water.

Consequently, on the first day the pipes worked 14/(x+ y) hours. On the second day, the small pipe worked 5 hours more, i.e. 5+14/(x+ y) . For this

time she served 14 m 3 water. From here we get the first equation 14 or 5+14/(x+ y)=14/ x. On the third day both pipes worked together21/(x+ y) hours, and then the large pipe worked for 20/xhours. The total time of the pipes coincides with the operating time of the first pipe on the second day, i.e.

5+14/( x+ y) =21/( x+ y)+ 20/ x. Since the left sides of the equation are equal, we have . Freed from the denominators, we get homogeneous equation 20 x 2 +27 xy-14 y 2 =0. Dividing the equation byy 2 and designatingx/ y= t, we have 20t 2 +27 t-14=0. From the two roots of this quadratic equation (t 1 = , t 2 = ) according to the meaning of the problem is suitable onlyt= . Hence,x= y. Substitutingxinto the first equation, we findy=5. Thenx=2.

Task 11. Two teams, working together, dug the trench in two days. After that, they began to dig a trench of the same depth and width, but 5 times longer than the first one. At first, only the first team worked, and then only the second team, completing one and a half times less work than the first team. Digging of the second trench was completed in 21 days. In how many days could the second team dig the first trench if it is known that the amount of work performed by the first team in one day is greater than the amount of work performed in one day by the second team?

Solution.It is more convenient to solve this problem if the work being performed is brought to the same scale. If both teams worked together to dig the first trench in 2 days, then obviously they would have dug the second trench (five times longer) in 10 days. Let the first brigade dig this trench in x days, and the second in y, i.e. in 1 day the first one would have dug part of the trench, the second - for 1/y , and together -1/x+1/ y part of the trench.

Then we have . The teams worked separately when digging the second trench. If the second team has completed the amount of workm, then (according to the conditions of the problem) - the first brigade . Becausem + m = m is equal to the volume of all work taken as a unit, thenm = . Consequently, the second brigade dug trenches and spent on it at days. The first brigade dug trenches and spent X days. From here we have orX = 35- . Substituting x into the first equation, we arrive at the quadratic equation2у 2 - 95у +1050 = 0, whose roots will be y 1 = And at 2 = 30. Then accordinglyX 1 = And X 2 =15. From the problem statement choose the one you need: y = 30. Since the found value refers to the second trench, the second team would have dug the first trench (five times shorter) in 6 days.

Task 12. Three excavators participated in digging a pit with a volume of 340 m 3 . In an hour, the first excavator removes 40 m 3 pound, the second - per s m 3 less than the first, and the third is 2c more than the first. First, the first and second excavators worked simultaneously and dug 140 m 3 soil. Then the rest of the pit was dug, working simultaneously, by the first and third excavators. Determine values with(0<с<15), in which the pit was dug in 4 hours if the work was carried out without interruption.

Solution. Since the first excavator takes out 40 m 3 soil per hour, then the second - (40-s) m 3 , and the third - (40+2s) m 3 pounds per hour. Let the first and second excavators work together for x hours. Then from the problem conditions it follows (40+40-с)х = 140 or (80-с)х = 140. If the first and third excavators worked together at the clock, then we have (40+40+2с)у = 340-140 or (80+2c)y - 200. Since the total operating time is 4 hours, we obtain the following equation to determine c: x + y = 4 or

This equation is equivalent to the quadratic equationWith 2 -30s+ 200 =0, whose decisions will be with 1 = 10 m 3 and with 2 = 20m 3 . According to the conditions of the problem, onlyco

s = 10 m 3 .

Task 10. Each of the two workers was assigned to process the same number of parts. The first one started the job right away and completed it in 8 hours. The second one first spent more than 2 hours setting up the device, and then, with its help, finished the job 3 hours earlier than the first one. It is known that the second worker, an hour after the start of his work, processed the same number of parts as the first had processed by that moment. How many times does the device increase the productivity of the machine (i.e., the number of parts processed per hour of operation)?

Solution. This is an example of a problem in which not all unknowns need to be found.

Let us denote the time for setting up the machine by the second worker as x (by condition x>2). Suppose it was necessary to process eachndetails.

Then the first worker per hour processes details, and the second details. Both workers processed the same number of parts an hour after the second one started working. This means that From here we get the equation for determining x: X 2 -4x + 3-0 whose roots are x 1 = 1 andX 2 = 3. Because

x > 2, then required value- this is x = 3. Therefore, the second worker processes per hour details. Because the first worker per hour processes

parts, then we find that the device increases labor productivity in = 4 times.

Task 1 3. Three workers must produce a certain number of parts. At first, only one worker started work, and after a while a second one joined him. When 1/6 of all the parts were made, the third worker began work. They finished the work at the same time, and each made the same number of parts. How long did the third worker work if it is known that he worked two hours less than the second and that the first and second, working together, could produce all the required number of parts 9 hours earlier than the third could have done, working separately?

Solution. Let the first worker work x hours, and the third worker work x hours. Then the second worker worked 2 hours more, i.e. y+2 hours. Each of them produced an equal number of parts, i.e. 1/3 of all parts. Consequently, the first one would produce all the parts in 3 hours, the second one in 3(y+2) hours, and the third one in 3y hours. Therefore, the first one produces in an hour part of all the details, the second - and the third - .

Since all three, during their collaboration, produced all the details, then we get the first equation (all three worked together at the clock)

. (1)

The first and second worker, working together, would have made all the parts together 9 hours earlier than the third worker would have done, working alone. From here we get the second equation

. (2)

These two equations can easily be reduced to the equivalent system

Expressing x from the second equation and substituting it into the first equation, we get y 3 -5у 2 - 32у - 36 = 0. This equation is factorized(y- 9)(y +2) 2 = 0.

Since y > 0, the equation has only one required root, y = 9.Answer:y = 9.

Task 14. Water flows evenly into the pit; 10 identical pumps, operating simultaneously, can pump water out of a filled pit in 12 hours, and 15 such pumps - in 6h.How long can 25 of these pumps pump water out of a filled pit when working together?

Solution.Let the volume of the pitVm 3 , and the productivity of each pump is x m 3 per hour Water flows into the pit continuously.T.since the quantity of its receipt is unknown, we denote it by y m 3 per hour - the volume of water entering the pit. Ten pumps will pump out in 12 hours X= 120x water. This amount of water is equal to the total volume of the pit and the volume of water that enters the pit in 12 hours. This entire volume is equal toV+12 y. Equating these volumes, we create the first equation 120x =V + 12 y .

The equation for 15 such pumps is constructed similarly:15-6 x = V + 6 yor 90x = V + 6 y. From the first equation we have V = 120x - 12y. Substituting V into the second equation, we get y = 5x.

The length of time that 25 of these pumps will operate is unknown. Let's denote it byt. Then, taking into account the conditions of the problem, we construct the last equation by analogy. We have 25tx=V+ty. Substituting y and V into this equation we find 25tx= 120x -12 5x +t 5x or 20tx= 60x. From here we gett= 3 hours.Answer: in 3 hours.

Task 15. The two teams worked together for 15 days and then a third team joined them and 5 days after that the whole job was completed. It is known that the second brigade produces 20% more per day than the first. The second and third brigades together could complete all the work in the time required to complete all the work by the first and third teams when they work together. In what time could all three teams, working together, complete the entire job?

Solution. Let the first, second and third teams perform all the work, working separately, in x, y and respectivelyzdays. Then on the day they perform part of the work. Transforming the first condition of the problem into an equation, assuming that the entire amount of work equal to one, we get

15 or

(1)

20 .

Since the second team produces 120% of what the first does (20% more), we have or . (2)

The second and third teams would have completed all the work in 1/ days, and the first and third - for 1/ days. By condition, the first quantity is equal to

(3)

Second, that is 1/ . From here we get the third equation .

The problem requires determining the time it takes to complete the entire job in three teams working together, that is, the size1/ .

Obviously, it is more convenient to solve the system of equations (1)-(3) if you introduce new variables: , We need to find the value

l/(u + v+ w) .Then we have an equivalent system

Solving this linear system, we easily findu= Then the required value is 1/ SoThus, working together, all three teams will complete all the work in 16 days.

Answer: in 16 days. If the productivity of the second factory doubled, it would be equal to almost all types of productivity tasks encountered.

Tasks

Two workers together can complete some work in 10 days. After 7 days of working together, one of them fell ill, and the other left work after working for another 9 days. What time in days?Can each worker alone do all the work?

A number of workers completed the work in a few days. If the number of workers increasesIf the number of workers increases by 3, then the work will be done 2 days sooner, and if the number of workers increases by 12, then 5 days sooner. Determine the number of workers and the time required to complete this work.

Two pumps of different power, working together, fill a pool in 4 hours. To fill half of the pool, the first pump requires 4 hours more time than the second to fill three quarters of the pool. How long does it take to fill the pool with each pump individually?

10. The ship is loaded with cranes. First, four cranes of equal power worked for 2 hours, then they were joined by two more cranes, but of lower power, and 3 hours after that, loading was completed. If all the cranes started working at the same time, the loading would be remaining work. The productivity of the third brigade is equal to half the sum of the productivity of the first and second brigades. How many times is the productivity of the second team greater than the productivity of the third team?

15. Two teams of plasterers, working together, plastered a residential building in 6 days. Another time they plastered a club and did three times the amount of work they would have done on plastering a residential building. The first team worked at the club at first, and then the second team replaced it and completed the work, and the first team completed the amount of work twice as large as the second. They plastered the club in 35 days. In how many days would the first brigade be able toto tour a residential building if it is known that the second team would spend more than 14 days on it?

Two teams began work at 8 o'clock. Having made 72 parts together, they began to work separately. At 15:00 it turned out that during the separate work, the first team made 8 more parts than the second. The next day, the first team made one more part in 1 hour, and the second team made one less part in 1 hour than on the first day. The teams began working together at 8 o'clock and, having completed 72 parts, began working separately again. Now, during the separate work, the first team made 8 more parts than the second, by 13:00. How many parts did each team make per hour?

Three workers must make 80 identical parts. It is known that all three together make 20 parts in an hour. The first one started work firstworking He made 20 parts, spending more than 3 hours on their production. The rest of the work was done together by the second and third workers. The entire job took 8 hours. How many hours would it take the first worker to make all 80 parts?

The pool fills with water through the first pipe 5 hours faster than through the second pipe, and 30 hours faster than through the third pipe. It is known thatthe carrying capacity of the third pipe is 2.5 times less than the capacity of the first pipe and 24 m 3 /h is less than the capacity of the second pipe. Find the capacity of the first and third pipes.

Two excavators, of which the first has less productivity, dug withjoint work, a pit with a volume of 240 m 3 . Then the first began to dig the second pit, and the second continued to dig the first. 7 hours after the start of their work, the volume of the first pit was 480 m 3 greater than the volume of the second pit. The next day, the second excavator increased its productivity by 10 m 3 /h, and the first one decreased by 10 m 3 /h. First, they dug a pit together at 240 m 3 , after which the first began to dig another pit, and the second continued to dig the first. Now the volume of the first pit has become 480 m 3 greater than the volume of the second pit already 5 hours after the excavators started working. How much soil per hour did the excavators remove on the first day of work?

Three vehicles transport grain, fully loaded on each trip. During one flight, the first and second cars are transported together6 tons of grain, and the first and third together transport in 2 flights the same amount of grain as the second in 3 flights. How much grain does the second vehicle transport in one trip, if it is known that the second and third vehicles transport a certain amount of grain together, withmaking 3 times fewer trips than would be required for a third vehicle to transport the same amount of grain?

Two excavators of different designs must dig two trenches of the same widthnarrow section length 960mi180 m. The entire work lasted 22 days, during which the first excavator laid a large trench. The second excavator started working 6 days later than the first, dug a smaller trench, was repaired for 3 days and then helped the first. If there was no need to waste time on repairs, the work would be completed in 21 days. How many meters of trench can each excavator dig per day?

Three brigades plowed two fields total area 120 hectares. The first field was plowed in 3 days, with all three crews working together. The second field was plowed in 6 days of the first and second brigadami. If all three teams worked on the second field for 1 day, then the first team could plow the rest of the second field in 8 days. How many hectares per day did the second team plow?

Two pipes of equal diameter are connected to two pools(Toeach pool has its own pipe). A certain volume of water was poured into the first pool through the first pipe, and immediately after that the same volume of water was poured into the second pool through the second pipe, and all this took 16 hours. If water flowed through the first pipe as much time as through the second, and through the second - as much time as through the first, then water would be poured through the first pipe for 320 m 3 less than the second one. If through the first one it would pass 10 m 3 less, and after the second - by 10 m 3 more water, then it would take 20 hours to pour the initial volumes of water into the pool (first into the first, and then into the second). How long did the water flow through each of the pipes?

Two convoys consisting of the same number cars transporting cargo. In each of the carsThe vehicles have the same carrying capacity and are fully loaded during flights. Loading capacity of machines in different columns is different, and during 1 trip the first convoy transports 40 tons more cargo than the second convoy. If we reduce the number of vehicles in the first convoy by 2, and in the second convoy by 10, then the first convoy will transport 90 tons of cargo in 1 trip, and the second convoy will transport 90 tons of cargo in 3 trips. What is the carrying capacity of the vehicles in the second convoy?

One worker can produce a batch of parts in 12 hours. One worker began the work, an hour later another one joined him, another hour later a third, etc., until the work was completed. How long did the first worker work? (The labor productivity of all workers is the same.)

A team of workers with the same qualifications had to produce a batch of parts. SnachFirst, one worker began work, an hour later a second one joined him, an hour later a third one, etc., until the whole team began work. If all members of the team had worked from the very beginning, the work would have been completed 2 hours faster. How many workers are in the team?

Three workers were digging a ditch. At first the first worker worked half the time, noit took the other two to dig the entire ditch, then the second worker worked half the time it took the other two to dig the entire ditch, and finally the third worker worked half the time it took the other two to dig the entire ditch. As a result, the ditch was dug. How many times faster would the ditch be dug if all three workers had worked simultaneously from the very beginning?

Tasks to work with solutions

Two workers are doing some work. After 45 minutes of joint work, the first worker was transferred to another job, and the second worker completed the rest of the work in 2 hours 15 minutes. In what time could each worker individually complete all the work, if it is known that the second one will need 1 hour more to do this than the first?
Two teams, working simultaneously, cultivate a plot of land in 12 hours. In what time could this plot be processed by the first brigade separately, if the speed of work performed by the first and second brigades is in the ratio of 3: 2?
One crew can clear a field in 12 days, while another does the same job in 75% of the time it takes the first crew. After the first team worked for 5 days, the second team joined it and they completed the work together. How many days did the teams work together?
Two masters, of whom the second starts working 1.5 days later than the first, can complete the task in 7 days. If everyone did this task separately, then the first one would need 3 days more than the second one. How many days would it take each master individually to complete this task?
The pool can be filled with water from two taps. If the first tap is opened for 10 minutes and the second for 20 minutes, the pool will be filled. If the first tap is opened for 5 minutes, and the second for 15 minutes, then 3/5 of the pool will be filled. How long does it take to fill the entire pool from each tap separately?
Two typists were assigned to perform some task. The second started work 1 hour later than the first. 3 hours after the first one started working, they still had only one task left to complete. At the end of the work, it turned out that each typist had completed half of the entire task. In how many hours could each of them separately complete the entire task?
There are two engines of the same power. One of them, while working, consumed 600 g of gasoline, and the second, who worked 2 hours less, consumed 384 g of gasoline. If the first engine consumed as much gasoline per hour as the second, and the second, on the contrary, as much as the first, then during the same operating time the gasoline consumption in both engines would be the same. How much gasoline does each engine use per hour?
Two people do the work. At first, the first one worked for the time during which the second one does all the work. Then the second one worked for the time during which the first one would have completed the rest of the work. They just completed all the work. How much time does it take for each person to complete this job if it is known that if they work together they will complete it in 3 hours 36 minutes?
Two teams working together must repair a certain section of the road in 18 days. In reality, it turned out that at first only the first brigade worked, and the second brigade, whose productivity was higher than that of the first brigade, completed the repair of the site. As a result, the repair of the site lasted 40 days, and the first team in its working hours completed all the work. How many days would it take to repair a section of the road by each team separately?
Three masons (of different qualifications) laid out a brick wall, and the first one worked for 6 hours, the second one for 4 hours, and the third one for 7 hours. If the first mason had worked for 4 hours, the second one for 2 hours, and the third one for 5 hours, then the job would have been completed just all the work. How many hours would it take the masons to finish the masonry if they worked together for the same amount of time?
Water flows evenly into the pit. Ten identical pumps, operating simultaneously, can pump out water from a filled pit in 12 hours, and 15 such pumps in 6 hours. How long can 25 of these pumps pump water out of a filled pit when working together?
Water enters the reservoir from two pipes of different diameters. On the first day, both pipes, working simultaneously, supplied 14 m 3 of water. On the second day, only the small pipe worked and also supplied 14 m 3 of water, since it worked 5 hours longer than on the previous day. On the third day, both pipes first supplied 21 m 3 of water, and then only the large pipe worked, supplying another 20 m 3 of water, and total duration time, the water supply was the same as on the second day. Determine the productivity of each pipe.

Problems to solve independently

A pool is filled with two pipes in 6 hours. One first pipe fills it 5 hours faster than one second pipe. How long will it take each pipe, acting separately, to fill the pool? Answer: 10 hours; 15 h
The student read a book of 480 pages. Every day he read the same number of pages. If he read 16 more pages every day, he would read the book in 5 days. How many days did the student read the book? Answer: 6 days
Two teams have been assigned to unload the ship. If you add up the time intervals during which the first and second teams can unload the steamer independently, you get 12 hours. Determine these intervals if their difference is 45% of the time during which both teams can unload the steamer together. Answer: h; h
A team of installers could finish the wiring at 4 p.m., laying 8 m of cable per hour. After completing half of the entire task, one worker left the team. In this regard, the team began laying 6 meters of cable per hour and completed the work planned for the day at 18:00. How many meters of cable were laid and in how many hours? Answer: 96 minutes, 14 hours
There are two pipes connected to the pool. Through the first, the pool is filled, and through the second, water flows out of the pool. Half an hour after the simultaneous start of operation of the pipes, the first of them was also switched to draining water from the pool. How long after switching the first pipe does the water level in the pool return to its original level if the capacity of the first pipe is twice the capacity of the second? Answer: in 10 minutes
A number of workers completed the work in a few days. If the number of workers increases by 3, then the work will be done 2 days faster, and if the number of workers increases by 12, then 5 days faster. Determine the number of workers and the time required to complete this work. Answer: 12 working days, 10 days
The pool can be filled with water using two pumps of different capacities. If half of the pool is filled by turning on only the first pump, and then, turning it off, continue filling using the second pump, then the entire pool will be filled in 2 hours 30 minutes. When both pumps operate simultaneously, the pool will fill in 1 hour 12 minutes. What part of the pool is filled in 20 minutes of operation by a pump with a lower capacity? Answer: 1/9
Five people perform some task. The first three of them, working together, will complete the entire task in 7.5 hours; first, third and fifth - in 5 hours; the first, third and fourth - in 6 hours; the second, fourth and fifth - in 4 hours. How long will it take all five people working together to complete this task? Answer: in 3 hours
Two workers completed some work together in 12 hours. If first the first worker did half of this work, and then the other did the rest, then all the work would be completed in 25 hours. How long would it take each worker individually to complete this work? Answer: 16 hours, 16/3 hours
Masters A and B worked the same number of days. If A worked one less day, and B worked 7 days less, then A would earn RUB 7,200 and B would earn RUB 6,480. If, on the contrary, A worked 7 days less, and B worked one day less, then B would earn 3240 rubles. more A. How much did each master actually earn? Answer: 7500 rub.; 9000 rub.
To fill the reservoir, two pipes were opened, through which water was supplied for 20 minutes, then the third pipe was opened, and 5 minutes after that the reservoir was filled and all pipes were closed. The productivity of the second pipe is 1.2 times greater than the productivity of the first. Through the second and third pipes, open simultaneously, the tank is filled in 0.9 times the time required to fill it through the first and third pipes when they work together. How long will it take to fill the tank if all three pipes are opened at the same time? Answer: 16 min
Three automatic lines produce the same products, but have different productivity. The productivity of all three simultaneously operating lines is 1.5 times higher than the productivity of the first and second lines operating simultaneously. The second and third lines, working simultaneously, can complete the shift task of the first line 4 hours 48 minutes faster than the first line performs it; The second line performs the same task 2 hours faster compared to the first. Find the time it takes for the first line to complete its shift task. Answer: 8 hours

CHAPTER 8

ALGEBRAIC AND ARITHMETIC PROBLEMS

793. The typist calculated that if she printed 2 sheets more daily than the norm established for her, she would finish the work 3 days earlier than scheduled; if he prints 4 sheets in excess of the norm, he will finish the work 5 days ahead of schedule. How many sheets should she reprint and in what time frame? . Solution

794. The worker produced a certain number of identical parts within the deadline assigned to him. If he made 10 more of them every day, he would complete this work in 4 1/2 days ahead of schedule, and if he did 5 less details a day, he would be 3 days late against the appointed deadline. How many parts did he complete and in what time frame? . Solution

795. The typist had to complete the job within a certain time frame, printing a certain number of sheets every day. She calculated that if she prints 2 sheets more than the established norm every day, she will finish the job 2 days ahead of schedule, but if she prints at 60% more than normal, then having finished the work 4 days ahead of schedule, he will print 8 sheets more than the intended work. How many sheets should she print per day and in what time should she finish the work? . Solution

796. Two workers working together complete some work in 8 hours. The first of them, working separately, can complete all the work for 12 hours. rather than the second worker, if the latter works separately. In how many hours can each of them, working separately, complete the job? Solution

797. The pool is filled with two pipes in 6 hours. One first pipe fills it for 5 hours. rather than one second. How long will it take each pipe, acting separately, to fill the pool? Solution

798. Two workers were tasked with producing a batch of identical parts. After the first one worked for 7 hours, and the second one for 4 hours, it turned out that they had completed 5/9 of the entire work. After working together for another 4 hours, they determined that they had 1/18 of the total work left to complete. In how many hours could each of them, working separately, complete all the work? Solution

799. The ship is loaded by cranes. First, 4 cranes of equal power began to load. After they worked for 2 hours, 2 more cranes of lower power were attached to them, and after that the loading was completed after 3 hours. If all the cranes started working at the same time, the loading would be completed in 4.5 hours. Determine in what hours one crane of greater and one crane of lesser power could finish loading. . Solution

800. Construction required 8 hours. transport construction material from the station. At first, 30 three-ton vehicles were sent for transportation. After two hours of work of these machines, 9 more five-ton vehicles were sent to help them, together with which the transportation was completed on time. If five-ton vehicles had been sent first, and three-ton vehicles 2 hours later, then only 13/15 of the total cargo would have been removed during the specified period. Determine how many hours it would take one three-ton truck, one five-ton truck to transport this entire cargo, and in what time it would take 30 five-ton trucks to transport the entire cargo. Solution

801. Two typists were assigned to do some work. The second of them started work 1 hour later than the first. 3 hours after the first one started work, they still had 9/20 of the work left to complete. At the end of the work, it turned out that each typist had completed half of the entire work. In how many hours could each of them separately complete all the work? Solution

802. Two trains left station A and B towards each other, and the second of them left half an hour later than the first. 2 hours after the first train left, the distance between the trains was 19/30 of the entire distance between A and B. Continuing on, they met halfway between A and B. How long would it take each train to travel the entire distance between the terminal stations? Solution

803. To wash photographic negatives, use a bath shaped like rectangular parallelepiped, dimensions 20 cm x 90 cm x 25 cm. To constantly mix the water in the bathtub, water flows into it through one tap and simultaneously flows out through another. It takes 5 minutes to empty a full bath using the second tap. less time than to fill it with the first tap if you close the second. If you open both taps, the full bath will be emptied in 1 hour. Find the amount of water flowing through each tap in 1 minute. . Solution

804. When constructing the building, it was necessary to remove 8000 m 3 of earth within a certain period of time. The work was completed 8 days ahead of schedule due to the fact that the excavation team exceeded the plan by 50 m 3 every day. Determine when the work should be completed and find the daily percentage of over-fulfillment. Solution

805. The track was repaired by two teams. Each of them repaired 10 km, despite the fact that the second team worked one day less than the first. How many kilometers of track did each team repair per day if both together repaired 4.5 km per day? . Solution

806. Two workers completed some work together in 12 hours. If first the first one did half of this work, and then the other did the rest, then the whole work would be completed in 25 hours. How long would it take each individual to complete this work? Solution

807. Two tractors of different power, working together, plowed the field for t days. If at first only one tractor worked and plowed half the field, and then one second one finished the job, then under such conditions the field would be plowed in k days. In how many days can each tractor, working separately, plow the entire field? . Solution

808. To deepen the fairway at the entrance to the harbor, 3 different dredgers were used. If only the first of them had been in effect, the work would have taken 10 days longer; if only the second one worked, then the work would drag on for 20 extra days. With only one third dredger, dredging the fairway would take six times longer than with all three machines operating simultaneously. How long will it take to complete the entire job with each dredger separately? Solution

809. Two workers, the second of whom starts work 1 1/2 days later than the first, can complete the job in 7 days. If everyone did this work separately, the first would need 3 days more than the second. In how many days will each of them separately complete this work? Solution

810. When two tractors of different power worked together, the collective farm field was plowed in 8 days. If half the field were plowed first with one tractor, then further work With two tractors, the entire job would be completed in 10 days. In how many days could it take to plow the entire field with each tractor separately? . Solution

811. Several people undertook to dig a ditch and could have finished the job in 6 hours if they had started it at the same time, but they started work one after another at equal intervals. After the same period of time after going to work last participant the ditch was dug, and each of the participants remained at work until the end. How long did it take them to dig a ditch if the one who started work first worked 5 times longer than the one who started last? . Solution

812. Three workers can work together to complete some work in t hour. The first of them, working alone, can complete this work twice as fast as the third and one hour faster than the second. How much time can each of them, working separately, complete this job? . Solution

813. The pool is filled with water from two taps. First, the first tap was opened for one-third of the time it would take to fill the pool by opening only the second tap. Then, on the contrary, the second tap was opened for one third of the time it took to fill the pool with the first tap alone. After this, 13/18 of the pool turned out to be filled. Calculate how much time it takes to fill the pool with each tap separately if both taps, open together, fill the pool in 3 hours 36 minutes. Solution

814. When building a power plant, a team of masons had to lay 120 thousand bricks within a certain period of time. The team completed the work 4 days ahead of schedule. Determine what the norm for daily brick laying was and how many bricks were actually laid daily, if it is known that the team laid 5,000 more bricks in 3 days than was supposed to be laid in 4 days according to the norm. . Solution

815. Three vessels are filled with water. If 1/3 of the water from the first vessel is poured into the second, then 1/4 of the water in the second is poured into the third, and finally 1/10 of the water in the third is poured into the first, then each vessel will contain 9 liters . How much water was in each container? Solution

816. Some of the alcohol was poured out of a tank filled with pure alcohol and the same amount of water was added; then the same number of liters of mixture were poured from the tank; then 49 liters of pure alcohol remained in the tank. Tank capacity 64 l. How much alcohol was poured the first time and how much the second time? The problem is written under the assumption that the volume of the mixture equal to the sum volumes of alcohol and water. In fact, it is somewhat smaller.Solution

817. A 20 liter vessel is filled with alcohol. A certain amount of alcohol is poured from it into another, equal to it, and, having filled the rest of the second vessel with water, the first vessel is supplemented with this mixture. Then 6 2/3 liters are poured from the first into the second, after which both vessels contain the same amount of alcohol. How much alcohol was initially poured from the first vessel into the second? . Solution

818. A vessel with a capacity of 8 liters is filled with air containing 16% oxygen. A certain amount of air is released from this vessel and the same amount of nitrogen is introduced, after which the same amount of mixture is again released as the first time and again supplemented with the same amount of nitrogen. The new mixture contained 9% oxygen. Determine how many liters were released from the vessel each time. . Solution

819. Two collective farmers brought 100 eggs to the market together. Selling eggs for different prices, both earned the same amount. If the first sold as many eggs as the second, she would earn 9 rubles; if the second sold as many eggs as the first, she would earn 4 rubles. How many eggs did each have? . Solution

820. .Two collective farmers having together A l of milk, received the same amounts when selling it, selling milk at different prices. If the first one sold as much as the second one, it would receive T rubles, and if the second one sold as much as the first one, it would receive n , rub. ( t>p ). How many liters of milk did each collective farmer have? Solution

821. When testing the efficiency of two engines internal combustion of the same power, it was found that one of them consumed 600 g of gasoline, and the second, which worked 2 hours less, 384 g. If the first engine consumed as much gasoline per hour as the second, and the second, on the contrary, as much as the first, then During the same operating time, gasoline consumption in both engines would be the same. How much gasoline does each engine use per hour? . Solution

822. There are two alloys of gold and silver; in one the amount of these metals is in the ratio 2:3, in the other - in the ratio 3:7. How much of each alloy must be taken to obtain 8 kg of a new alloy in which gold and silver would be in the ratio 5:11? . Solution

823. One barrel contains a mixture of alcohol and water in a ratio of 2:3, and the other - in a ratio of 3:7. How many buckets must be taken from each barrel to make 12 buckets of a mixture in which alcohol and water would be in a ratio of 3:5? Solution

824. Some alloy consists of two metals in a 1:2 ratio, while another contains the same metals in a 2:3 ratio. From how many parts of both alloys can a third alloy be obtained containing the same metals in a ratio of 17:27?