Information Technology. Text problems

Tasks for working with solutions

1. 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 and 15 minutes. In what time could each worker individually do all the work, if it is known that the second one will need 1 hour more than the first?
2. Two brigades, working simultaneously, process a piece of land in 12 hours. How long could the first team process this area separately, if the speeds of the work done by the first and second teams are 3: 2?
3. One team can clean a field in 12 days, while another does the same job in 75% of the time it takes for the first team. After the first team worked for 5 days, the second team joined it and together they finished the job. How many days did the teams work together?
4. Two masters, of which the second starts working 1.5 days later than the first, can complete the task in 7 days. If this task were performed by each separately, then the first would need 3 days more than the second. In how many days would each master individually complete this task?
5. 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, then 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 for each faucet to fill the entire pool?
6. Two typists were assigned to perform a certain task. The second started work 1 hour later than the first. 3 hours after the first one started work, they still had to complete the entire task. At the end of the work, it turned out that each typist completed half of the entire task. In how many hours could each of them individually complete the entire task?
7. There are two engines of the same power. One of them, while working, used up 600 g of gasoline, and the second, who worked 2 hours less, used up 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 for the same time of operation, the consumption of gasoline in both engines would be the same. How much gasoline per hour does each engine use?
8. Two are doing the work. First, the first worked for the time in which the second does all the work. Then the second worked for the time in which the first would have completed the rest of the work. They just did all the work. How long does it take each to complete this work if it is known that if they work together they will do it in 3 hours and 36 minutes?
9. Two teams, working together, must repair a section of the road in 18 days. In reality, it turned out that at first only the first brigade worked, and only the second brigade completed the repair of the site, the productivity of which is higher than that of the first brigade. As a result, the repair of the site lasted 40 days, and the first team completed all the work during their working hours. How many days would it take each team to repair the road section separately?
10. Three masons (of different qualifications) laid out a brick wall, and the first worked 6 hours, the second - 4 hours, and the third - 7 hours. If the first bricklayer worked 4 hours, the second - 2 hours, the third - 5 hours, then just all the work. How many hours would it take the masons to finish the brickwork if they worked together at the same time?
11. Water flows evenly into the pit. Ten identical pumps, acting simultaneously, can pump water out of a filled pit in 12 hours, and 15 such pumps in 6 hours. How long can 25 such pumps work together to pump water out of a filled pit?
12. Water enters the tank 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 a 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 the total duration of the water supply was the same as on the second day. Determine the capacity of each pipe.

Tasks for independent solution

1. The pool is filled with two pipes in 6 hours. One first pipe fills it 5 hours faster than one second pipe. How long does it take each pipe, acting separately, to fill the pool? Answer: 10 hours; 15 h
2. 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 have finished the book in 5 days. How many days did the student read the book? Answer: 6 days
3. Two brigades were assigned to unload the ship. If we add the time intervals during which the first and second brigades can unload the steamer independently, we get 12 hours. Determine these intervals if their difference is 45% of the time during which both brigades can unload the steamer together. Answer: h; h
4. A team of fitters could finish the wiring at 4 p.m., laying 8 meters of cable per hour. After completing half of the entire task, one worker left the team. In this regard, the team began to lay 6 meters of cable per hour and finished 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
5. Two pipes are connected to the pool. Through the first pool is filled, and through the second water flows out of the pool. Half an hour after the simultaneous start of the pipes, the first of them was also switched to drain water from the pool. How long after switching the first pipe will the water level in the pool return to its original state if the capacity of the first pipe is twice that of the second? Answer: after 10 minutes
6. 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 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. Answer: 12 working, 10 days
7. 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 with the second pump, then the entire pool will be filled in 2 hours and 30 minutes. With simultaneous operation of both pumps, the pool will be filled in 1 hour and 12 minutes. What part of the pool is filled in 20 minutes by a pump of lesser productivity? Answer: 1/9
8. Five people are doing a task. The first three of them, working together, will complete the entire task in 7.5 hours; the first, third and fifth - in 5 hours; the first, third and fourth - for 6 hours; the second, fourth and fifth - in 4 hours. How long will it take all five people to complete this task, working together? Answer: 3 hours
9. 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 for each worker to do this work individually? Answer: 16h, 16/3h
10. Masters A and B worked the same number of days. If A worked one day less and B worked 7 days less, then A would earn 7200 r and B 6480 r. If, on the contrary, A worked 7 days less, and B - one day less, then B would earn 3240 rubles. more A. How much did each artist actually earn? Answer: 7500 rubles; 9000 r.
11. To fill the tank, two pipes were opened, through which water was supplied for 20 minutes, then a third pipe was opened, and 5 minutes after that, the tank was filled, and all pipes were closed. The performance of the second pipe is 1.2 times greater than the performance of the first. Through the second and third pipes, open at the same time, the tank is filled in 0.9 of 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
12. Three automatic lines produce the same products, but have different capacities. 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 and 48 minutes faster than the first line does it; the second line performs the same task 2 hours faster than the first line. Find the time for the first line to complete their shift task. Answer: 8 hours

Working time - the time during which the employee, in accordance with the internal labor regulations and the terms of the employment contract, must perform labor duties, as well as other periods of time that, in accordance with this Code, other federal laws and other regulatory legal acts Russian Federation refer to working time.

Normal working hours may not exceed 40 hours per week.

The procedure for calculating the norm of working time for certain calendar periods (month, quarter, year), depending on the established length of working time per week, is determined by the federal executive body that performs the functions of generating public policy and legal regulation in the sphere of labor.

(Part three was introduced by Federal Law No. 157-FZ of July 22, 2008)

The employer is obliged to keep records of the time actually worked by each employee.

Article 92. Reduced hours of work

Reduced working hours are set:

for employees under the age of sixteen - no more than 24 hours a week;

for employees aged sixteen to eighteen years - no more than 35 hours per week;

for employees who are disabled people of group I or II - no more than 35 hours a week;

for workers employed in jobs with harmful and (or) dangerous working conditions - no more than 36 hours a week in the manner established by the Government of the Russian Federation, taking into account the opinion of the Russian tripartite commission for the regulation of social and labor relations.

(part one as amended by Federal Law No. 90-FZ of 30.06.2006)

Students working hours educational institutions under the age of eighteen, working during the academic year in their free time, may not exceed half of the norms established by the first part of this article for persons of the corresponding age.

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

This Code and other federal laws may establish reduced working hours for other categories of employees (pedagogical, medical and other employees).

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

Article 93. Part-time work

By agreement between the employee and the employer, part-time work (shift) or part-time work week can be established both at the time of employment and subsequently. The employer is obliged to establish a part-time working day (shift) or part-time working week at the request of a pregnant woman, one of the parents (guardian, custodian) who has a child under the age of fourteen (a disabled child under the age of eighteen), as well as a person exercising caring for a sick family member in accordance with a medical report issued in accordance with the procedure established by federal laws and other regulatory legal acts of the Russian Federation.

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

When working on a part-time basis, the employee is paid in proportion to the time worked by him or depending on the amount of work performed by him.

Work on a part-time basis does not entail any restrictions for employees on the duration of the annual basic paid leave, the calculation of seniority and other labor rights.

Article 94. Duration of daily work (shift)

The duration of daily work (shift) cannot exceed:

for employees aged fifteen to sixteen years old - 5 hours, for those aged sixteen to eighteen years old - 7 hours;

for students educational institutions, educational institutions of primary and secondary vocational education those who combine study with work during the academic year, at the age of fourteen to sixteen years - 2.5 hours, at the age of sixteen to eighteen years - 4 hours;

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

for the disabled - in accordance with a medical certificate issued in accordance with the procedure established by federal laws and other regulatory legal acts of the Russian Federation.

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

For workers employed in jobs with harmful and (or) dangerous working conditions, where reduced working hours are established, the maximum allowable duration of daily work (shift) cannot exceed:

with a 36-hour work week - 8 hours;

with a 30-hour work week or less - 6 hours.

The collective agreement may provide for an increase in the duration of daily work (shift) in comparison with the duration of daily work (shift) established by part two of this article for employees employed in work with harmful and (or) dangerous working conditions, subject to the maximum weekly duration of the worker. time (part one of Article 92 of this Code) and hygienic standards of working conditions established by federal laws and other regulatory legal acts of the Russian Federation.

(Part three as amended by Federal Law No. 90-FZ of June 30, 2006)

The duration of the daily work (shift) of creative workers of the media, cinematography organizations, television and video crews, theaters, theater and concert organizations, circuses and other persons involved in the creation and (or) performance (exhibition) of works, in accordance with the lists jobs, professions, positions of these workers, approved by the Government of the Russian Federation, taking into account the opinion of the Russian tripartite commission for the regulation of social and labor relations, may be established by a collective agreement, a local normative act, an employment contract.

(Part four was introduced by Federal Law No. 90-FZ of 30.06.2006, as amended by Federal Law No. 13-FZ of 28.02.2008)

Article 95

The duration of the working day or shift immediately preceding a non-working holiday is reduced by one hour.

In continuously operating organizations and certain types work, where it is impossible to reduce the duration of work (shift) on the holiday day, processing is compensated by providing the employee with additional rest time or, with the consent of the employee, payment according to the norms established for overtime work.

On the eve of the weekend, the duration of work with a six-day working week cannot exceed five hours.

Article 96. Night work

Night time is from 22:00 to 06:00.

The duration of work (shift) at night is reduced by one hour without subsequent working off.

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

The duration of work (shift) at night is not reduced for employees who have a reduced working time, as well as for employees hired specifically for work at night, unless otherwise provided by the collective agreement.

The duration of work at night is equalized with the duration of work in the daytime in cases where it is necessary for working conditions, as well as in shift work with a six-day working week with one day off. The list of the specified works can be determined by the collective agreement, the local normative act.

To work at night are not allowed: pregnant women; employees under the age of eighteen, with the exception of persons involved in the creation and (or) execution works of art, and other categories of employees in accordance with this Code and other federal laws. Women with children under the age of three, disabled people, employees with disabled children, as well as employees caring for sick members of their families in accordance with a medical certificate issued in accordance with the procedure established by federal laws and other regulatory legal acts of the Russian Federation , mothers and fathers raising children under the age of five without a spouse, as well as guardians of children of this age, may be involved in night work only with their written consent and provided that such work is not prohibited to them for health reasons in accordance with medical advice. At the same time, these employees must be informed in writing of their right to refuse to work at night.

(as amended by Federal Laws No. 97-FZ of 24.07.2002, No. 90-FZ of 30.06.2006)

The procedure for work at night of creative workers of the media, cinematography organizations, television and video crews, theaters, theater and concert organizations, circuses and other persons involved in the creation and (or) performance (exhibition) of works, in accordance with the lists of works professions, positions of these workers, approved by the Government of the Russian Federation, taking into account the opinion of the Russian tripartite commission for the regulation of social and labor relations, may be established by a collective agreement, a local normative act, an employment contract.

(as amended by Federal Laws No. 90-FZ of 30.06.2006, No. 13-FZ of 28.02.2008)

Article 97. Work outside the established duration of working hours

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

The employer has the right, in accordance with the procedure established by this Code, to involve an employee in work outside the working hours established for this employee in accordance with this Code, other federal laws and other regulatory legal acts of the Russian Federation, a collective agreement, agreements, local regulations, employment contract (hereinafter referred to as the length of working time established for the employee):

for overtime work (Article 99 of this Code);

if the employee works on irregular working hours (Article 101 of this Code).

Article 98 - Federal Law of June 30, 2006 N 90-FZ.

Article 99. Overtime work

(As amended by Federal Law No. 90-FZ dated June 30, 2006)

Overtime work is work performed by an employee at the initiative of the employer outside the working hours established for the employee: daily work (shift), and in the case of summarized accounting of working time - in excess of the normal number of working hours for the accounting period.

Involving an employee in overtime work by an employer is allowed with his written consent in the following cases:

1) if necessary, perform (finish) the work that has been started, which, due to an unforeseen delay due to the technical conditions of production, could not be performed (completed) within the working hours established for the employee, if failure to perform (non-completion) of this work may lead to damage or loss of property the employer (including the property of third parties held by the employer, if the employer is responsible for the safety of this property), state or municipal property, or endanger the life and health of people;

2) during the production of temporary work on the repair and restoration of mechanisms or structures in cases where their failure may cause a significant number of employees to stop working;

3) to continue work in the absence of a replacement employee, if the work does not allow a break. In these cases, the employer is obliged to immediately take measures to replace the shift with another employee.

Engaging an employer of an employee to work overtime without his consent is allowed in the following cases:

1) in the performance of work necessary to prevent a catastrophe, industrial accident or eliminate the consequences of a catastrophe, industrial accident or natural disaster;

In accordance with Federal Law No. 417-FZ of December 7, 2011, from January 1, 2013, in clause 2 of part three of this article, the words "water supply systems, gas supply, heating, lighting, sewerage," will be replaced by the words "centralized hot water supply systems, cold water supply and (or) water disposal, gas supply systems, heat supply, lighting, ".

2) when performing socially necessary work to eliminate unforeseen circumstances that disrupt the normal functioning of water supply, gas supply, heating, lighting, sewerage, transport, communications;

3) in the performance of work, the need for which is due to the introduction of a state of emergency or martial law, as well as urgent work in emergency situations, that is, in the event of a disaster or threat of disaster (fires, floods, famine, earthquakes, epidemics or epizootics) and in other cases, endangering the life or normal living conditions of the entire population or part of it.

In other cases, involvement in overtime work is allowed with the written consent of the employee and taking into account the opinion of the elected body of the primary trade union organization.

It is not allowed to involve pregnant women, workers under the age of eighteen, other categories of workers in overtime work in accordance with this Code and other federal laws. Involvement in overtime work of disabled people, women with children under the age of three years is allowed only with their written consent and provided that this is not prohibited by them for health reasons in accordance with a medical certificate issued in accordance with the procedure established by federal laws and other regulatory legal acts of the Russian Federation. At the same time, disabled people, women with children under the age of three, must be familiarized with their right to refuse overtime work against signature.

The duration of overtime work should not exceed for each employee 4 hours for two consecutive days and 120 hours per year.

The employer is obliged to ensure that the overtime hours of each employee are accurately recorded.

The assignment problem is a special case of the transportation problem. It has the same structure, but has some features. In the simplest case, the assignment problem can be formulated as follows. There are several various kinds jobs and the same number of employees. One employee is assigned to each type of work. Each of the employees can perform any work, but the salaries of employees and the time they complete the work differ. It is necessary to distribute work among employees in such a way as to minimize cash costs or time to complete work. Just like the transportation problem, the assignment problem can be either balanced or unbalanced.

Example 1. Balanced Assignment Problem

Four workers can do four types of work. Costs Сij fulfillment i-th worker j th work are given in cells A1:D4.

In this table, rows correspond to workers, and columns correspond to jobs. It is necessary to draw up a work plan so that all work is completed, each worker is loaded on only one job, and the total cost of all work is minimal. Note that this problem is balanced, i.e. the number of jobs is the same as the number of workers.

To solve this problem, we construct mathematical model. Let the variable xij= 1 if i-th worker performs j-th work, and xij= 0 if i-th worker is not performed j th work. Then the model looks like this:

Let's create the following table:

Cells A1:D4 - cost of work, must be >= 0.
Cells F2:I5 will be assigned to unknowns. If the employee does work in cell 1, otherwise 0. Unknowns must be integer, binary.
In cell J2, the objective function is the total cost of the work.
The left parts of the constraints are given by the sums. For example:

SUM(F2:I2) should be equal to 1. This means that employee #1 does only one job.
SUM(F2:F5) must be equal to 1. This indicates that only one worker performs job #1.

Find the optimal solution with the Find Solution tool.

Answer: Cost of work = 18.

Example 2 Unbalanced Assignment Problem

Four workers can do five jobs. This task is not balanced, because the number of workers is not equal to the number of types of work. If the problem is not balanced, then before starting the solution, it must be balanced by introducing the missing number of fictitious rows or columns with sufficiently large penalty costs for the work. In our task, a fictitious "Worker 5" has been added. Find out on your own with the solution below.

Answer

Minimum cost of work = 7.

It can be seen from the solution that fictitious worker No. 5 is assigned to job No. 4, that is, no one performs this work.