I will solve the exam 17. There are more complex rules for highlighting calls. The general concept of introductory words and the basic rule for their selection
financial mathematics
For the correct completed task without errors you will receive 3 points.
Approximately 35 minutes.
To solve task 17 in mathematics of a profile level, you need to know:
- The task is divided into several types:
- tasks related to banks, deposits and loans;
- tasks for the optimal choice.
- The formula for calculating the monthly payment: S credit = S/12t
- Formula for calculating simple interest: S=α (1 + tp/m)
- Formula for calculating compound interest: C \u003d x (1 + a%)n
Percent - is one hundredth of a value.
- x*(1 + p/100) - value x increased by p%
- x*(1 - k/100) - value x decreased by k%
- x*(1 + p/100) k - value x increased by p% k once
- x*(1 + p/100)*(1 - k/100) – value X first increased by p%, and then decreased by k%
Tasks for repaying the loan in equal installments:
The loan amount is taken as X. Bank interest - but. Loan repayment - S.
One year after the accrual of interest and payment of the amount S debt - x * (1 + a/100), p = 1 + a/100
- Debt after 2 years: (xp-S)p-S
- Debt after 3 years: ((xp - S)p - S)p - S
- The amount of debt through n years: xp n - S(p n-1 + ... + p 3 + p 2 + p + 1)
On April 15, it is planned to take a loan in the amount of 900 thousand rubles from the bank for 11 months.
The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by p% compared to the end of the previous month;
- on the 15th day of each from the 1st to the 10th month, the debt must be the same amount less than the debt on the 15th day of the previous month;
- on the 15th day of the 10th month, the debt amounted to 200 thousand rubles;
- By the 15th day of the 11th month, the debt must be repaid in full.
Find p if the total amount paid to the bank was 1021 thousand rubles.
On April 15, it is planned to take a bank loan for 700 thousand rubles for (n + 1) month.
The conditions for its return are as follows:
- from the 2nd to the 14th day of each month, it is necessary to pay a part of the debt in one payment;
- On the 15th day of each from the 1st to the nth month, the debt must be the same amount less than the debt on the 15th day of the previous month;
- on the 15th day of the n-th month, the debt amounted to 300 thousand rubles;
- by the 15th day of the (n + 1)th month, the debt must be repaid in full.
Find n if the total amount paid to the bank was 755 thousand rubles.
On August 15, it is planned to take a loan in the amount of 1,100 thousand rubles from a bank for 31 months.
The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by 2% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, it is necessary to pay a part of the debt in one payment;
- on the 15th day of each from the 1st to the 30th month, the debt must be the same amount less than the debt on the 15th day of the previous month;
- by the 15th day of the 31st month, the debt must be repaid in full.
How many thousand rubles is the debt on the 15th day of the 30th month, if the total amount paid to the bank is 1503 thousand rubles?
On March 15, it is planned to take a loan from a bank for a certain amount for 11 months.
The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by 1% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, it is necessary to pay a part of the debt in one payment;
- On the 15th day of each month from the 1st to the 10th, the debt must be the same amount less than the debt on the 15th day of the previous month;
- on the 15th day of the 10th month, the debt will amount to 300 thousand rubles;
What amount is planned to be borrowed if the total amount of payments after its full repayment is 1388 thousand rubles?
On December 15, it is planned to take a loan from the bank for 11 months.
The conditions for its return are as follows:
- On the 15th day of each month from the 1st to the 10th, the debt must be 80 thousand rubles less than the debt on the 15th of the previous month;
- by the 15th day of the 11th month, the loan must be fully repaid.
What debt will be on the 15th day of the 10th month if the total amount of payments after the full repayment of the loan is 1198 thousand rubles?
On December 15, it is planned to take a bank loan in the amount of 300 thousand rubles for 21 months. The return conditions are:
- from the 2nd to the 14th day of each month, part of the debt must be paid;
- On the 15th of each month from the 1st to the 20th, the debt must be the same amount less than the debt on the 15th of the previous month;
- on the 15th day of the 20th month, the debt will amount to 100 thousand rubles;
Find the total amount of payments after the full repayment of the loan.
On December 15, it is planned to take a bank loan for 1,000,000 rubles for (n+1) month. The conditions for its return are as follows:
- on the 1st day of each month, the debt increases by r% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, part of the debt must be paid;
- On the 15th day of each month from the 1st to the nth, the debt must be 40 thousand rubles less than the debt on the 15th day of the previous month;
- On the 15th day of the nth month, the debt will amount to 200 thousand rubles;
- by the 15th day of the (n + 1)th month, the loan must be fully repaid.
Find r if it is known that the total amount of payments after the full repayment of the loan will be 1378 thousand rubles.
On December 15, it is planned to take a loan from the bank for 21 months. The return conditions are:
- On the 1st day of each month, the debt increases by 3% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, part of the debt must be paid;
- On the 15th day of each month from the 1st to the 20th, the debt must be 30 thousand rubles less than the debt on the 15th day of the previous month;
- by the 15th day of the 21st month, the loan must be fully repaid.
What amount is planned to be borrowed if the total amount of payments after its full repayment is 1604 thousand rubles?
On May 25, it is planned to take a loan from a bank for 1.5 years. The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by 7% compared to the end of the previous month;
- from the 1st to the 10th day of each month it is necessary to pay a part of the debt;
- On the 25th day of each month, the debt must be the same amount less than the debt on the 25th day of the previous month.
What amount should be paid to the bank if the average monthly payment for the entire loan term is 18,500 rubles?
The furniture factory produces bookcases and sideboards. One bookcase requires 4/3 m^2 of chipboard, 4/3 m^2 of pine board and 2/3 man-hours of labor time. For the manufacture of one sideboard, 2 m^2 of chipboard, 1.5 m^2 of pine board and 2 man-hours of working time are spent. Profit from the sale of one bookcase is 500 rubles, and a sideboard - 1200 rubles. Within one month, the factory has at its disposal: 180 m^2 of chipboard, 165 m^2 of pine boards and 160 man-hours of working time. What is the maximum expected monthly profit?
Some enterprise manufactures products of two types - A and B, using three types of resources: M, N and K. The rates of use of resources and their reserves are given in the table.
It is required to determine the maximum possible revenue of the enterprise when selling products, if the prices for products A and B are 1500 and 900 rubles per unit of the corresponding product, respectively. Give your answer in thousands of rubles.
Boys of three eleventh grades bought flowers for the girls for the holiday on March 8. If every girl in the first class is given 3 flowers, every girl in the second class is given 5 flowers, and every girl in the third class is given 7 flowers, then at least 40 and at most 50 flowers will be required.
If every girl in the first class is given 5 flowers, every girl in the second class is given 7 flowers, and every girl in the third class is given 3 flowers, then it will take the same number of flowers that is needed to give each girl in the first class 7 flowers, to give each girl in the second class 3 flowers, and give each third grade girl 5 flowers. Find the total number of girls in the 11th grade if it is known that there are more girls in the third grade than in the second.
The amount of the deposit increased on the first day of each month by 2% in relation to the amount on the first day of the previous month. Similarly, the price of a brick has increased by 36% monthly. Postponing the purchase of bricks, on May 1, a certain amount was deposited in the bank. How many percent less in this case can you buy a brick on July 1 of the same year for the entire amount received from the bank, together with interest?
In preparation for the New Year, it was decided to buy several Christmas tree decorations of two types, provided that the cost of decorations of different types should not differ by more than 2 rubles. If you buy 7 decorations of the first type and 8 of the second, you will have to pay more than 165 rubles. If you buy 8 decorations of the first type and 7 of the second, you will have to pay less than 165 rubles. Find the cost of each type of decoration.
The boys of the two eleventh grades bought flowers for the girls for the holiday on March 8. If every first class girl is given 3 flowers, and every second class girl is given 7 flowers, then less than 70 flowers will be needed. If every girl in the first class is given 7 flowers, and every girl in the second class is given 3 flowers, then more than 70 flowers will be needed. Find the number of girls in 11th grade if the number of girls in the classes differ by less than three.
The plant has three types of assembly lines: A, B, C. Two types of products are produced on each of them. The number of products of each type produced by each line is presented in the table.
Under the contract, 1030 products of the first type and 181 products of the second type should be produced. What is the smallest number of assembly lines that can be used?
Three types of aircraft fly between cities A and B, for which the possibilities of transporting passengers and cargo containers are presented in the table
Under the terms of the contract, 1,790 passengers and 195 cargo containers are to be transported. Find the least number of aircraft required.
Ore is mined in two mines: 100 tons per day in the first mine, 220 tons per day in the second mine. The mined ore is processed at two plants. The first is capable of processing no more than 200 tons of ore per day, and the second - no more than 250 tons of ore per day. The cost of transporting one ton of ore from the mine to the plant is presented in the table.
Find the lowest shipping cost.
The depositor decided to place 1000 thousand rubles in the bank for a period of 1 year. The bank offers two strategies: the first is to accrue 7% per annum if the deposit is placed in its entirety. Or it is proposed to divide the contribution into three parts. Then 15% per annum will be charged on the smaller part, 10% on the middle part and 5% per annum on the larger part. What is the maximum profit an investor can receive if the larger part must differ from the smaller part by at least 100,000 rubles, but not more than 300,000 rubles?
The borrower took an amount equal to 691,000 rubles from the bank for 3 years, at 10% per annum, on the condition that the second payment would be twice the first, and the third - three times the first, and payments are made after accruing interest on the balance of the loan. What was the amount of the first payment?
On November 16, Nikita borrowed 1 million rubles from a bank. for six months. The loan repayment terms are as follows:
On the 28th of each month, the debt increases by 10% compared to the 16th of the current month;
From the 1st to the 10th of each month, part of the debt must be paid;
In case of delay in payments (from 1 to 5 days), penalties are additionally charged: for each overdue day, 1% of the amount that had to be paid in the current month;
On the 16th day of each month, the debt must amount to a certain amount in accordance with the table:
Determine how many thousand rubles Nikita will pay the bank in excess of the loan taken, if it is known that he made payments on December 7, January 12, February 10, March 9, April 1 and May 15.
Larin 17) Ivan Petrovich received a loan from a bank at a certain percentage per annum. A year later, in repayment of the loan, he returned to the bank 1/6 of the total amount that he owes the bank by that time. And a year later, on account of the full repayment of the loan, Ivan Petrovich contributed to the bank an amount that was 20% higher than the amount of the loan received. What is the percentage per annum on a loan in this bank?
Two boxes contain pencils: the first is red, the second is blue, moreover, there were fewer reds than blues. First, 40% of the pencils from the first box were transferred to the second. Then, 20% of the pencils that ended up in the second box were transferred to the first, and half of the transferred pencils were blue. After that, there were 26 more red pencils in the first box than in the second, and the total number of pencils in the second box increased by more than 5% compared to the original one. Find the total number of blue pencils.
In July, Viktor plans to take out a loan of 2.5 million rubles. The conditions for its return are as follows:
Each January, the debt increases by 20% compared to the condom of the previous year;
From February to June of each year, Victor must pay off some of the debt.
What is the minimum number of years that Victor can take out a loan for, so that annual payments are no more than 760 thousand rubles?
After how many full years will Sergei have at least 950,000 rubles in his account if he intends to deposit 260,000 rubles into the account every year, provided that the bank once a year on December 31 accrues 10% of the available amount.
Mitrofan wants to borrow 1.7 million rubles. The loan is repaid once a year in equal amounts (except, perhaps, the last one) after interest is charged. The interest rate is 10% per annum. What is the minimum number of years Mitrofan can take out a loan for, so that annual payments are no more than 300 thousand rubles?
On December 31, 2016, Vasily borrowed 5,460,000 rubles from a bank at 20% per annum. The loan repayment scheme is as follows - on December 31 of each next year, the bank accrues interest on the remaining amount of the debt (that is, it increases the debt by 20%), then Vasily transfers x rubles to the bank. What should be the amount for Vasily to pay off the debt in three equal payments (that is, for three years)?
In August, it is planned to take a loan from a bank for a certain amount. The conditions for its return are as follows:
From February to July of each year, it is necessary to pay a part of the debt, equal to 1080 thousand rubles. How many thousand rubles were taken from the bank if it is known that the loan was fully repaid in three equal payments (that is, for 3 years)?
The pension fund owns securities worth [b]10t thousand rubles at the end of year t (t = 1;2;3;...). At the end of any year, the pension fund can sell securities and deposit money into a bank account, while at the end of each next year the amount in the account will increase by 1 + r times. The pension fund wants to sell securities at the end of such a year so that at the end of the twenty-fifth year the amount in its account will be the largest. Calculations have shown that for this the securities must be sold strictly at the end of the eleventh year. For what positive values of r is this possible?
Vadim is the owner of two factories in different cities. The factories produce exactly the same goods using the same technologies. If workers at one of the factories work a total of t^2 hours per week, then they produce t units of goods in that week. For every hour of work at a factory located in the first city, Vadim pays a worker 500 rubles, and at a factory located in the second city, 300 rubles. Vadim is ready to allocate 1,200,000 rubles a week to pay workers. What is the maximum number of units of goods that can be produced in a week at these two factories?
In July 2016, Inga plans to take out a loan for six years in the amount of 4.2 million rubles. The conditions for its return are as follows:
In July 2017, 2018, 2019 and 2020, the debt remains equal to 4.2 million rubles;
Payments in 2021 and 2022 are equal;
By July 2022, the debt will be paid in full.
How many million rubles will the last payment be more than the first?
In July 2016, Timur plans to take out a bank loan for four years in the amount of S million rubles, where S is an integer. The return conditions are as follows:
Each January, the debt increases by 15% compared to the end of the previous year;
The payment must be made once a year from February to June;
In July of each year, the debt must be part of the loan in accordance with the following table:
Find the largest value of S for which the total amount of Timur's payments will be less than 30 million rubles.
In July 2020, it is planned to take a loan from a bank in the amount of 400,000 rubles. The conditions for its return are as follows:
Each January, the debt increases by r% compared to the end of the previous year;
Find the number r if it is known that the loan was fully repaid in two years, and in the first year 330,000 rubles were transferred, and in the second year - 121,000 rubles.
In July 2020, it is planned to take a loan from a bank for a certain amount. The conditions for its return are as follows:
Every January the debt increases by 20% compared to the end of the previous year;
From February to June of each year, it is necessary to pay part of the debt in one payment
How many rubles were taken from the bank if it is known that the loan was fully repaid in three equal payments (that is, for 3 years) and the amount of payments exceeds the amount taken from the bank by 77,200 rubles?
In July, it is planned to take a loan from a bank for a certain amount. The conditions for its return are as follows:
Every January the debt increases by r% compared to the end of the previous year;
From February to June of each year, part of the debt must be paid
Find r if it is known that if you pay 777,600 rubles each, then the loan will be repaid in 4 years, and if you pay 1,317,600 rubles each year, then the loan will be fully repaid in 2 years?
In July, it is planned to take a bank loan in the amount of 18 million rubles for a certain period (an integer number of years). The conditions for its return are as follows:
- each January the debt increases by 10% compared to the end of the previous year;
For how many years was the loan taken if it is known that the total amount of payments after its repayment was 27 million rubles?
In July, it is planned to take a bank loan in the amount of 9 million rubles for a certain period (an integer number of years). The conditions for its return are as follows:
Every January the debt increases by 20% compared to the end of the previous year;
- from February to June of each year, part of the debt must be paid;
- in July of each year, the debt must be the same amount less than the debt in July of the previous year.
What will be the total amount of payments after the full repayment of the loan, if the largest annual payment is 3.6 million rubles?
In July 2026, it is planned to take a loan from a bank for three years in the amount of S million rubles, where S is an integer. The conditions for its return are as follows:
Every January the debt increases by 20% compared to the end of the previous year
From February to June of each year, part of the debt must be paid in one payment
In July of each year, the debt must be part of the loan in accordance with the following table
Find the largest value of S, at which each of the payments will be less than 5 million rubles.
The pension fund owns securities worth t^2 rubles. at the end of each year t(t=1;2...) at the end of any year, the pension fund can sell securities and deposit money into a bank account, while at the end of each next year the amount on the account will increase by (1+ r) once. The pension fund wants to sell securities at the end of a year so that at the end of the twenty-fifth year the amount in its account is the largest. Calculations showed that for this the securities must be sold strictly at the end of the twenty-first year. For what positive r is this possible?
The zoo distributes 111 kg. meat between foxes, leopards and lions. Each fox is entitled to 2 kg. meat, leopard - 14 kg., Lion 21 kg. It is known that each lion has 230 visitors daily, each leopard has 160, each fox has 20. How many foxes, leopards and lions should there be in the zoo so that the number of visitors for these animals would be the largest daily?
At the meeting of shareholders, it was decided to increase the company's profit by expanding the range of products. Economic analysis showed that
1) additional income attributable to each new type of product will be equal to 70 million rubles. in year;
2) additional costs for the development of one new type will amount to 11 million rubles. per year, and the development of each subsequent type will require 7 million rubles. per year more expenses than the development of the previous one. Find the value of the maximum possible increase in profit.
A citizen put 1 million rubles in a bank for 4 years. At the end of each year, 10% is charged on the underlying amount. He decided at the end of each of the first 3 years (after interest) to withdraw the same amount of money. This amount should be such that after 4 years after the accrual of interest for the 4th year, he would have at least 1,200 thousand rubles in his account. What is the maximum amount a citizen can withdraw? Round your answer down to the nearest thousand.
Sasha and Pasha put 100 thousand rubles each. to the bank at 10% per annum for a period of three years. At the same time, Pasha withdrew n thousand rubles a year later. (n is an integer), and a year later he again reported n thousand rubles. to your account. For what is the smallest value of n in three years, the difference between the amounts in the account of Sasha and Pasha will be at least 3 thousand rubles.
It is planned to issue a loan for an integer number of million rubles for 5 years. In the middle of each year of the loan, the borrower's debt increases by 10% compared to the beginning of the year. At the end of the 1st, 2nd and 3rd years, the borrower pays only the interest on the loan, leaving the debt equal to the original. At the end of the 4th and 5th years, the borrower pays the same amount, repaying the entire debt in full. Find the largest loan amount at which the total amount of the borrower's payments will be less than 6 million rubles.
The farmer has two fields, each with an area of 10 hectares. Potatoes and beets can be grown in each field, and the fields can be divided between these crops in any proportion. The yield of potatoes in the first field is 300 c/ha, and in the second 200 c/ha. The beet yield in the first field is 200 c/ha, and in the second - 300 c/ha.
A farmer can sell potatoes at a price of 10,000 rubles. per centner, and beets - at a price of 18,000 rubles. per centner. What is the maximum income a farmer can earn?
On the eve of the New Year, Santa Clauses laid out equal amounts of sweets in gift bags, and these bags were put in bags, 2 bags in one bag. They could arrange the same sweets into bags so that each of them would have 5 less sweets than before, but then each bag would contain 3 bags, and 2 less bags would be required. What is the largest number of sweets that Santa Claus could lay out?
The first car drove from point A to point B at a speed of 80 km/h, and after some time at a constant speed - the second. After stopping for 20 minutes at point B, the second car drove back at the same speed. After 48 km, he met the first car coming towards him, and was at a distance of 120 km from B at the moment when the first car arrived at point B. Find the distance from A to the place of the first meeting if the distance between points A and B is 480 km.
The store received goods of I and II grades for a total of 4.5 million rubles. If all the goods are sold at the price of the second grade, then the losses will amount to 0.5 million rubles, and if all the goods are sold at the price of the first grade, then a profit of 0.3 million rubles will be received. For what amount was the goods of I and II grades purchased separately?
Two mines produce aluminum and nickel. In the first mine there are 80 workers, each of whom is ready to work 5 hours a day. At the same time, one worker produces 1 kg of aluminum or 2 kg of nickel per hour. In the second mine there are 200 workers, each of whom is ready to work 5 hours a day. At the same time, one worker produces 2 kg of aluminum or 1 kg of nickel per hour.
Both mines supply the mined metal to the plant, where an alloy of aluminum and nickel is produced for the needs of industry, in which 2 kg of aluminum accounts for 1 kg of nickel. At the same time, the mines agree among themselves to mine metals so that the plant can produce the largest amount of alloy. How many kilograms of alloy under such conditions can the plant produce daily?
Some enterprise brings losses amounting to 300 million rubles. in year. To turn it into a profitable one, it was proposed to increase the range of products. Calculations have shown that the additional income attributable to each new type of product will amount to 84 million rubles. per year, and additional costs will be equal to 5 million rubles. per year with the development of one new species, but the development of each subsequent one will require 5 million rubles. per year more expenses than the development of the previous one. What is the minimum number of types of new products that must be mastered in order for the enterprise to become profitable? What is the largest annual profit the company can achieve by increasing the range of products?
The cost of developing an electronic version of the textbook
some edition is equal to 800 thousand rubles. Expenses
for the production of x thousand of such electronic textbooks
in this publishing house are (x^2+6x+22100) thousand rubles
in year. If textbooks are sold at a price of Rs. for a unit,
then the publisher's profit for one year will be ax-(x^2+6x+22100).
The publishing house will publish textbooks in such quantity,
to maximize profits. What is the smallest value of a
The development of the textbook will pay off in no more than 2 years?
On November 16, the twins Sasha and Pasha took a bank loan of 500 thousand rubles. each for four months. The loan repayment terms are as follows:
On the 28th of each month, the debt increases by 10% compared to the 16th of the current month;
From the 1st to the 15th day of each month, part of the debt must be paid; On the 16th day of each month, the debt must amount to a certain amount in accordance with the table proposed for each of them:
Which of the brothers will pay the bank the smaller amount in four months? How many rubles?
On March 1, 2016, Valery deposited 100 thousand rubles in the bank. at 10% per annum for a period of 4 years. In two years he plans to withdraw n thousand rubles from his account. (n is an integer) so that by March 1, 2020 he has at least 130 thousand rubles in his account. What is the maximum amount n that Valery can withdraw from his account on March 1, 2018?
Two pedestrians walk towards each other: one from A to B, and the other from B to A. They left at the same time, and when the first walked half the way, the second had another 1.5 hours to go, and when the second walked half the way, then the first There was still 45 minutes to go. How many minutes earlier will the first pedestrian finish his journey than the second?
At the beginning of January 2017, it is planned to take a loan from a bank for S million rubles, where S is an integer, for 4 years. The conditions for its return are as follows:
Every July, the debt increases by 10% compared to the beginning of the current year;
- from August to December of each year, part of the debt must be paid;
- in January of each year, the debt must be part of the loan in accordance with the following table:
Find the largest value of S, at which the difference between the largest and smallest payments will not exceed 2 million rubles.
For the Classic deposit, the bank plans to accrue 12% per annum at the end of each year, and for the Bonus deposit - to increase the deposit amount by 7% in the first year and by the same integer n percent in subsequent years.
Find the smallest value of n at which, over 4 years of storage, the "Bonus" deposit will be more profitable than the "Classic" deposit with equal amounts of initial contributions.
In May 2017, it is planned to take a loan from a bank for six years in the amount of S million rubles. The conditions for its return are as follows:
Every December of every year, the debt increases by 10%;
- from January to April of each year, part of the debt must be paid;
- in May 2018, 2019 and 2020, the debt remains equal to S million rubles;
- payments in 2021, 2022 and 2023 are equal;
- by May 2023, the debt will be paid in full.
Find the smallest integer S for which the total amount of payments does not exceed 13 million rubles.
46 people entered the first course for the specialty "Equipment and Machinery": 34 boys and 12 girls. They are divided into two groups of 22 and 24, with at least one girl in each group. What should be the distribution by groups so that the sum of the numbers equal to the percentage of girls in the first and second groups is the largest?
Leo took out a bank loan for a period of 40 months. According to the agreement, Leo must repay the loan in monthly installments. At the end of each month, p% of this amount is added to the remaining amount of the debt, followed by Leo's payment.
Monthly payments are selected in such a way that the debt decreases evenly.
It is known that Leo's largest payment was 25 times less than the original amount of the debt. Find p.
On December 18, 2015 Andrey borrowed 85,400 rubles from the bank at 13.5% per annum. The loan repayment scheme is as follows: on December 18 of each next year, the bank accrues interest on the remaining amount of the debt, then Andrey transfers X rubles to the bank. What should be the amount X for Andrey to pay off the debt in full in two equal payments?
Ivan wants to borrow 1 million rubles. The loan is repaid once a year in equal amounts (except, perhaps, the last one) after interest is charged. Interest rate 10% per annum. What is the minimum number of years Ivan can take out a loan so that annual payments do not exceed 250 thousand rubles?
On February 1, 2016, Andrei Petrovich took a loan of 1.6 million rubles from the bank. The loan repayment scheme is as follows: on the 1st day of each following month, the bank charges 1% on the remaining amount of the debt, then Andrey Petrovich transfers the payment to the bank. What is the minimum number of months Andrei Petrovich needs to take out a loan so that monthly payments do not exceed 350 thousand rubles?
On November 12, 2015, Dmitry borrowed 1,803,050 rubles from a bank at 19% per annum. The loan repayment scheme is as follows: on November 12 of each next year, the bank accrues interest on the remaining amount of the debt, then Dmitry transfers X rubles to the bank. What should be the amount X for Dmitry to pay off the debt in full in three equal installments?
On two mutually perpendicular highways in the direction of their intersection, two cars simultaneously start moving: one at a speed of 80 km/h, the other at 60 km/h. At the initial moment of time, each car is at a distance of 100 km from the intersection. Determine the time after the start of the movement, after which the distance between the cars will be the smallest. What is this distance?
Arkady, Semyon, Efim and Boris established a company with an authorized capital of 200,000 rubles. Arkady contributed 14% of the authorized capital, Semyon - 42,000 rubles, Efim - 12% of the authorized capital, and Boris contributed the rest of the capital. The founders agreed to share the annual profit in proportion to the contribution made to the authorized capital. What amount of the profit of 500,000 rubles is due to Boris? Give your answer in rubles.
In two regions there are 250 workers each, each of whom is ready to work 5 hours a day in the extraction of aluminum or nickel. In the first region, one worker produces 0.2 kg of aluminum or 0.1 kg of nickel per hour. In the second region, it takes x^2 man-hours of labor to mine x kg of aluminum per day, and y^2 man-hours of labor to mine y kg of nickel per day.
In two regions there are 50 workers each, each of whom is ready to work 10 hours a day in the extraction of aluminum or nickel. In the first region, one worker produces 0.2 kg of aluminum or 0.1 kg of nickel per hour. In the second region, it takes x^2 man-hours of labor to mine x kg of aluminum per day, and y^2 man-hours of labor to mine a kg of nickel per day.
Both regions supply the mined metal to the plant, where an alloy of aluminum and nickel is produced for the needs of industry, in which 1 kg of aluminum accounts for 2 kg of nickel. At the same time, the regions agree among themselves to mine metals so that the plant can produce the largest amount of alloy. How many kilograms of alloy under such conditions can the plant produce daily?
Timofey wants to borrow 1.1 million rubles. The loan is repaid once a year in equal amounts (except, perhaps, the last one) after interest is charged. The interest rate is 10% per annum. What is the minimum number of years Timofey can take a loan for, so that annual payments are no more than 270 thousand rubles?
Galina took a loan of 12 million rubles for a period of 24 months. According to the agreement, Galina must return part of the money to the bank at the end of each month. Every month, the total amount of debt increases by 3%, and then decreases by the amount paid by Galina to the bank at the end of the month. The amounts paid by Galina are selected so that the amount of debt decreases evenly, that is, by the same amount every month. How many more rubles will Galina return to the bank during the first year of lending compared to the second year?
On January 15, it is planned to take a loan from the bank for 15 months. The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by 3% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, part of the debt must be paid;
It is known that the eighth payment amounted to 99.2 thousand rubles. How much must be repaid to the bank during the entire loan term?
On December 31, 2014, Oleg borrowed a certain amount from the bank at a certain percentage per annum. The loan repayment scheme is as follows - on December 31 of each next year, the bank accrues interest on the remaining amount of the debt (that is, it increases the debt by a%), then Oleg transfers the next tranche. If he pays 328,050 rubles every year, he will pay off the debt in 4 years. If for 587,250 rubles, then for 2 years. Find a.
Two identical pools simultaneously began to fill with water. The first pool receives 30 m ^ 3 more water per hour than the second. At some point in the two pools together there was as much water as the volume of each of them. After that, after 2 hours and 40 minutes, the first pool was filled, and after another 3 hours and 20 minutes, the second one. How much water was supplied per hour to the second pool? How long did it take for the second pool to fill up?
On the 1st of each month, the debt increases by 2% compared to the end of the previous month;
From the 2nd to the 14th of each month, part of the debt must be paid;
On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
What percentage of the loan amount is the total amount of money that needs to be paid to the bank for the entire loan period?
On December 20, Valery took a loan from a bank in the amount of 500 thousand rubles. for a period of five months. The loan repayment terms are as follows:
On the 5th day of each month, the debt increases by an integer n percent compared to the previous month;
From the 6th to the 19th of each month, part of the debt must be paid;
On the 20th of each month, the debt must amount to a certain amount in accordance with the table:
Find the smallest n at which the amount of payments in excess of the loan taken (interest payments) will be more than 200 thousand rubles.
Three automatic machines of different power must produce 800 parts each. First, the first machine was launched, after 20 minutes - the second, and after another 35 minutes - the third. Each of them worked without failures and stops, and in the course of work there was a moment when each machine completed the same part of the task. How many minutes before the second machine finished the work of the third, if the first completed the task 1 hour 28 minutes after the third?
In two regions there are 90 workers each, each of whom is ready to work 5 hours a day in the extraction of aluminum or nickel. In the first region, one worker produces 0.3 kg of aluminum or 0.1 kg of nickel per hour. In the second area, it takes x^2 man-hours of labor to mine x kg of aluminum per day, and y^2 man-hours of labor to mine y kg of nickel per day.
For the needs of industry, either aluminum or nickel can be used, and 1 kg of aluminum can be replaced by 1 kg of nickel. What is the largest mass of metals that can be mined in the two regions in total for the needs of industry?
In July 2016, it is planned to take a loan from a bank for three years in the amount of S million rubles, where S is an integer. The conditions for its return are as follows:
Every January the debt increases by 25% compared to the end of the previous year;
- from February to June of each year, it is necessary to pay a part of the debt in one payment;
- in July of each year, the debt must be part of the loan in accordance with the following table.
Find the smallest value of S, at which each of the payments will be more than 5 million rubles
On August 1, 2016, Valery opened a “Refill” account with a bank for four years at 10% per annum, having invested 100 thousand rubles.
On August 1, 2017 and August 1, 2019, he plans to report n thousand rubles to the account. Find the smallest integer n such that by August 1, 2020, Valery will have at least 200,000 rubles in his account.
On January 15, it is planned to take a loan from a bank for six months in the amount of 1 million rubles. The conditions for its return are as follows:
On the 1st day of each month, the debt increases by r percent compared to the end of the previous month, where r is an integer;
From the 2nd to the 14th of each month, part of the debt must be paid;
On the 15th of each month, the debt must be a certain amount in accordance with the following table
Find the largest value of r at which the total amount of payments will be less than 1.2 million rubles.
In July, a loan of 8.8 million rubles was taken for several years. At the beginning of each next year, the balance of the debt increases by 25% compared to the end of the previous year. By July 1 of each year, the client must repay part of the debt in such a way that, as of July 1, the debt is reduced by the same amount every year. The last payment is 1 million rubles. Find the total amount paid to the bank.
In July, it is planned to take a loan from a bank in the amount of 14 million rubles for a certain period (an integer number of years). The conditions for its return are as follows:
Every January the debt increases by 10% compared to the end of the previous year;
- from February to June of each year, part of the debt must be paid;
- in July of each year, the debt must be the same amount less than the debt in July of the previous year
What will be the total amount of payments after the full repayment of the loan, if the smallest annual payment is 3.85 million rubles?
At the beginning of the year, the Zhilstroyservice company chooses a bank to receive a loan from among several banks lending at different interest rates. The company plans to dispose of the received loan as follows: 75% of the loan will be directed to the construction of cottages, and the remaining 25% to the provision of real estate services to the population. The first project can bring profit in the amount of 36% to 44% per annum, and the second - from 20% to 24% per annum. At the end of the year, the company must return the loan to the bank with interest and at the same time expects a net profit from these activities from at least 13%, but not more than 21% per annum of the total loan received. What should be the lowest and highest interest rates of the selected banks in order for the firm to guarantee itself the above level of profit?
On January 15, 2012, the bank issued a loan in the amount of 1 million rubles. The conditions for his return were as follows:
- On January 1st of each year, the debt increases by a% compared to the end of the previous year;
- the payment of part of the debt occurs in January of each year after interest is accrued.
The loan was repaid in two years, and at the same time, the amount of 600 thousand rubles was transferred in the first year, and 550 thousand rubles in the second time.
Find a.
The construction of a new plant costs 78 million rubles. Production costs x thousand units. products at such a plant are equal to 0.5x² + 2x + 6 million rubles per year. If the plant's products are sold at a price of r thousand rubles per unit, then the company's profit (in million rubles) for one year will be (px - (0.5x² + 2x + 6)). When the factory is built, the firm will produce products in such quantities that profits are the greatest. At what minimum value of p will the construction of the plant pay off in no more than 3 years?
At the beginning of 2001, Alexey purchased a security for 25,000 rubles. At the end of each year, the price of paper increases by 3,000 rubles. At the beginning of any year, Alexey can sell the paper and put the proceeds into a bank account. Every year the amount on the account will increase by 10%. At the beginning of what year should Alexei sell the security so that fifteen years after the purchase of this security, the amount in the bank account would be the largest?
Each of the two plants employs 1,800 people. At the first plant, one worker produces 1 part A or 2 parts B per shift. At the second plant, t^2 man-shifts are required to manufacture t parts (both A and B).
Each of the two plants employs 200 people. At the first plant, one worker produces 1 part A or 3 parts B per shift. At the second plant, t^2 man-shifts are required to manufacture t parts (both A and B).
Both of these plants supply parts to the plant, from which they assemble a product, for the manufacture of which 1 part A and 1 part B are needed. At the same time, the plants agree among themselves to produce parts so that the largest number of products can be assembled. How many products under such conditions can the plant assemble per shift?
Parts A and B are made in each of the two plants. The first plant has 40 people and one worker makes 15 A parts or 5 B parts per shift. The second plant has 160 people and one worker makes 5 A parts or 15 parts per shift. details B.
Both of these plants supply parts to the plant, from which they assemble a product, for the manufacture of which 2 parts A and 1 part B are needed. At the same time, the plants agree among themselves to produce parts so that the largest number of products can be assembled. How many products under such conditions can the plant assemble per shift?
For the production of some product C containing 40% alcohol, Alexey can purchase raw materials from two suppliers A and B. Supplier A offers a 90% alcohol solution in 1000 l canisters at a price of 100 thousand rubles. for the canister. Supplier B offers an 80% alcohol solution in 2,000-liter canisters at a price of 160,000 rubles. for the canister. The resulting product B is bottled in 0.5 liter bottles. What is the minimum amount that Alexey must spend on raw materials if he plans to produce exactly 60,000 bottles of product B?
On March 1, 2016, Ivan Lvovich put 20,000 rubles on a bank deposit for a period of 1 year with monthly interest and capitalization at 21% per annum. This means that on the first day of each month the deposit amount increases by the same amount of interest, calculated in such a way that it will increase by exactly 21% in 12 months. In how many months will the deposit amount exceed 22,000 rubles for the first time?
On May 15, the businessman planned to take a bank loan in the amount of 12 million rubles for 19 months. The conditions for its return are as follows:
On the 1st day of each month, the debt increases by 2% compared to the end of the previous month;
From the 2nd to the 14th day of each month, part of the debt must be paid;
On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
How many percent more in relation to the loan taken will have to pay a businessman?
For the production of some product C containing 40% alcohol, Alexey can purchase raw materials from two suppliers A and B. Supplier A offers a 90% alcohol solution in 1000 l canisters at a price of 100 thousand rubles. for the canister. Supplier B offers an 80% alcohol solution in 2,000-liter canisters at a price of 160,000 rubles. for the canister. The resulting product B is bottled in 0.5 liter bottles. What is the minimum amount that Alexey must spend on raw materials if he plans to produce exactly 60,000 bottles of product B?
Vladimir owns two factories for the production of refrigerators. The productivity of the first plant does not exceed 950 refrigerators per day. The output of the second plant was initially 95% of that of the first. After commissioning an additional line, the second plant increased the production of refrigerators per day by exactly 23% of the number of refrigerators produced at the first plant, and began to produce more than 1000 of them. How many refrigerators did each plant produce per day before the reconstruction of the second plant?
On the 1st of each month, the debt increases by 2% compared to the end of the previous month;
From the 2nd to the 14th of each month, part of the debt must be paid;
On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
It is known that for the fourth month of lending, you need to pay 54 thousand rubles. How much must be repaid to the bank during the entire loan term?
In July, the client took out a loan in the amount of 8.8 million rubles for several years.
The return conditions are as follows:
At the beginning of each next year, the balance of the debt increases by 25% compared to the end of the previous year.
- before July 1 of each year, the client must return to the bank a part of the debt in such a way that, compared to July 1, the debt is reduced by the same amount every year.
It is known that the last payment will be 1 million rubles. Find the total amount of payments that the client will pay to the bank.
Friends Polina and Christina dream of becoming models. On January 1, they decided to start losing weight. At the same time, Polina's weight turned out to be 10% more than Christina's.
In February, Christina is going to lose another 2%.
A) What is the smallest integer % that Polina needs to lose weight in February so that by March 1 her weight becomes less than Christina's?
B) How much will Christina weigh by the end of February if it is known that on January 1 Polina weighed 55 kg?
The deposit is planned to be opened for four years. The initial contribution is an integer number of millions of rubles. At the end of each year, the deposit increases by 10% compared to its size at the beginning of the year, and, in addition, at the beginning of the third and fourth years, the deposit is replenished annually by 3 million rubles. Find the largest amount of the initial contribution, at which in four years the contribution will be less than 25 million rubles.
Each of the two plants employs 20 people. At the first plant, one worker produces 2 parts A or 2 parts B per shift. At the second plant, t^2 man-shifts are required to manufacture t parts (both A and B).
Both of these plants supply parts to the plant, from which they assemble a product, for the manufacture of which 1 part A and 1 part B are needed. At the same time, the plants agree among themselves to produce parts so that the largest number of products can be assembled. How many products under such conditions can the plant assemble per shift?
On the 1st of each month, the debt increases by 2% compared to the end of the previous month;
From the 2nd to the 14th of each month, part of the debt must be paid;
On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
It is known that for the fifth month (from June 2 to 14) of lending, 44 thousand rubles must be paid to the bank. How much must be paid to the bank during the entire term of the loan?
The farmer has 2 fields, each with an area of 10 hectares. Each field can grow potatoes and beets, the field can be divided between these crops in any proportion. The yield of potatoes in the first field is 300 c/ha, and in the second - 200 c/ha. The beet harvest in the first field is 200 c/ha, and in the second - 300 c/ha.
A farmer can sell potatoes at a price of 4,000 rubles. per centner, and beets - at a price of 5000 rubles. per centner. What is the maximum income a farmer can earn?
The workshop received an order for the manufacture of 2,000 type A parts and 14,000 type B parts. Each of the 146 workers in the shop spends on the manufacture of one type A part the time in which he could make 2 type B parts. How should the workers of the shop be divided into two teams, in order to complete the order in the shortest time, provided that both teams start work at the same time, and each of the teams will be busy manufacturing parts of only one type?
Each of the two plants employs 100 people. At the first plant, one worker produces 3 parts A or 1 part B per shift. At the second plant, t^2 man-shifts are required to manufacture t parts (both A and B).
Both of these plants supply parts to the plant, from which they assemble a product, for the manufacture of which 1 part A and 3 parts B are needed. At the same time, the plants agree among themselves to produce parts so that the largest number of products can be assembled. How many products under such conditions can the plant assemble per shift?
On December 17, 2014, Anna borrowed 232,050 rubles from the bank at 10% per annum. The loan repayment scheme is as follows: on December 17 of each next year, the bank accrues interest on the remaining amount of the debt, and then Anna transfers X rubles to the bank. What amount X must be for Anna to repay the debt in full in four equal installments?
These are:
- On the 1st day of each month, the debt increases by 3% compared to the end of the previous month;
- from the 2nd to the 14th day of each month, part of the debt must be paid;
- On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
What amount should be returned to the bank during the first year (first 12 months) of lending?
According to the business plan, it is planned to invest 10 million rubles in a four-year project. According to the results of each year, it is planned to increase the invested funds by 15% compared to the beginning of the year. Accrued interest remains invested in the project. In addition, immediately after the accrual of interest, additional investments are needed: an integer of n million rubles in the first and second years, as well as an integer of m million rubles in the third and fourth years.
Find the smallest values of n and m that will at least double the initial investment in two years and at least triple in four years
In two regions there are 100 workers each, each of whom is ready to work 10 hours a day in the extraction of aluminum or nickel. In the first region, one worker produces 0.3 kg of aluminum or 0.1 kg of nickel per hour. In the second region, it takes x^2 man-hours of labor to mine x kg of aluminum per day, and y^2 man-hours of labor to mine y kg of nickel per day.
Both regions supply the mined metal to the plant, where an alloy of aluminum and nickel is produced for the needs of industry, in which 2 kg of aluminum accounts for 1 kg of nickel. At the same time, the regions agree among themselves to mine metals so that the plant can produce the largest amount of alloy. How many kilograms of alloy under such conditions can the plant produce daily?
For how many years is it planned to take a loan if it is known that the total amount of payments after its full repayment will be 18 million rubles?
The farmer has two fields, each with an area of 10 hectares. Potatoes and beets can be grown in each field, and the fields can be divided between these crops in any proportion. Potato yield in the first field is 500 c/ha, and in the second - 300 c/ha. The beet yield in the first field is 300 c/ha, and in the second - 500 c/ha.
A farmer can sell potatoes at a price of 2,000 rubles. per centner, and beets - at a price of 3,000 rubles. per centner. What is the maximum income a farmer can earn?
On June 10, the bank took a loan for 15 months. At the same time, on the 3rd day of each month, the debt increases by a% compared to the end of the previous month, from the 4th to the 9th day of each month, part of the debt must be paid, and on the 10th day, the debt must be the same amount less debt on the 10th day of the previous month.
On the 1st of each month, the debt increases by 1% compared to the end of the previous month;
From the 2nd to the 14th of each month, part of the debt must be paid;
On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month. It is known that for the first 12 months it is necessary to pay 177.75 thousand rubles to the bank. How much are you planning to borrow?
The entrepreneur bought the building and is going to open a hotel in it. The hotel can have standard rooms of 21 square meters and deluxe rooms of 49 square meters. The total area that can be allocated for rooms is 1099 square meters. The entrepreneur can divide this area between rooms of different types, as he wants. An ordinary room brings the hotel 2,000 rubles per day, and a deluxe room brings 4,500 rubles per day. What is the maximum amount of money an entrepreneur can earn from his hotel?
- On the 15th day of each month, the debt must be the same amount less than the debt on the 15th day of the previous month.
It is known that over the past 12 months it is necessary to pay 1,597.5 thousand rubles to the bank. How much are you planning to borrow?
On January 15, it is planned to take a loan from the bank for 14 months. The conditions for its return are as follows:
- On the 1st day of each month, the debt increases by r% compared to the end of the previous month;
- from the 2nd to the 14th of each month, part of the debt must be repaid;
- On the 15th day of each month, the debt should be the same amount less than the debt on the 15th day of the previous month.
It is known that the total amount of payments after the full repayment of the loan is 15% more than the amount taken on credit. Find r.
At the beginning of 2001, Alexey purchased a security for 7,000 rubles. At the end of each year, the price of paper increases by 2000 rubles. At the beginning of any year, Alexey can sell the paper and deposit the proceeds into a bank account. Every year the amount on the account will increase by 10%. At the beginning of what year should Alexei sell the security so that fifteen years after the purchase of this security, the amount in the bank account would be the largest?
Gregory is the owner of two factories in different cities. The factories produce exactly the same goods, but the factory located in the second city uses more advanced equipment.
As a result, if the workers at the factory located in the first city work a total of t^2 hours per week, then in that week they produce 3t units of goods; if the workers at the factory located in the second city work a total of t^2 hours per week, then in that week they produce 4t units of goods.
- each January the debt increases by 10% compared to the end of the previous year;
- from February to June of each year, part of the debt must be paid;
- in July of each year, the debt should be the same amount less than the debt in July of the previous year.
How many million rubles was the total amount of payments after repayment of the loan?
In July, it is planned to take a loan from a bank in the amount of 6 million rubles for a certain period. The conditions for its return are as follows:
- each January, the debt increases by 20% compared to the end of the previous year;
- from February to June of each year, part of the debt must be paid;
- in July of each year, the debt should be the same amount less than the debt in July of the previous year.
For what minimum term should a loan be taken so that the largest annual loan payment does not exceed 1.8 million rubles?
In July, it is planned to take a loan in the amount of 4,026,000 rubles. The conditions for its return are as follows:
- Every January, the debt increases by 20% compared to the end of last year.
- From February to June of each year, it is necessary to pay some part of the debt.
How many more rubles will have to be paid if the loan is fully repaid in four equal installments (that is, over 4 years) compared to the case if the loan is fully repaid in two equal installments (that is, over 2 years)?
In July, it is planned to take a loan from a bank in the amount of 100,000 rubles. The conditions for its return are as follows:
- each January, the debt increases by a% compared to the end of the previous year;
- From February to June of each year, part of the debt must be paid.
Find the number a if it is known that the loan was fully repaid in two years, and 55,000 rubles were transferred in the first year, and 69,000 rubles in the second.
The bank placed the amount of 3900 thousand rubles at 50% per annum. At the end of each of the first four years of storage, after the calculation of interest, the depositor additionally deposited the same fixed amount into the account. By the end of the fifth year after the accrual of interest, it turned out that the amount of the deposit had increased by 725% compared to the original one. How much did the contributor annually add to the deposit?
The entrepreneur took a bank loan in the amount of 9,930,000 rubles at 10% per annum. Loan repayment scheme: once a year, the client must pay the bank the same amount, which consists of two parts. The first part is 10% of the remaining debt, and the second part is aimed at repaying the remaining debt. Each subsequent year, interest is charged only on the remaining amount of the debt. What should be the annual payment amount (in rubles) for the entrepreneur to fully repay the loan in three equal installments?
17 task of the profile level of the exam in mathematics is a task related to finance, namely, this task can be for interest, part of debts, etc. The difficulty lies in the fact that it is necessary to calculate the interest or part over a long period, so this task is not direct analogy of standard problems with percentages. In order not to talk about the general, let's go directly to the analysis of a typical task.
Analysis of typical options for assignments No. 17 USE in mathematics at the profile level
The first version of the task (demo version 2018)
The conditions for its return are as follows:
- On the 1st of each month, the debt increases by r percent compared to the end of the previous month, where r is an integer;
- from the 2nd to the 14th of each month, part of the debt must be paid;
- On the 15th day of each month, the debt must amount to a certain amount in accordance with the following table.
Find the largest value of r for which the total amount of payments will be less than 1.2 million rubles.
Solution algorithm:
- We consider what the amount of payments on the loan is monthly.
- We determine the debt for each month.
- Find the required percentage.
- We determine the amount of payments for the entire period.
- We calculate the percentage r of the amount of debt payments.
- We write down the answer.
Solution:
1. According to the condition, the debt to the bank should decrease monthly in the following order:
1; 0,6; 0,4; 0,3; 0,2; 0,1; 0.
2. Let k = 1 + r / 100, then the debt each month is:
k; 0.6k; 0.4k; 0.3k; 0.2k; 0.1k.
3. So, payments from the 2nd to the 14th monthly are:
k - 0.6; 0.6k - 0.4; 0.4k - 0.3; 0.3k - 0.2; 0.2k - 0.1; 0.1k
4. The total amount of payments is equal to:
By condition, the entire amount of payments is less than 1.2 million rubles, therefore,
The largest integer solution of the resulting inequality is 7. Then it is the required one - 7.
The second option (from Yaschenko, No. 1)
In July 2020, it is planned to take a loan from a bank in the amount of 300,000 rubles. The conditions for its return are as follows:
- each January, the debt increases by r% compared to the end of the previous year;
- From February to June of each year, part of the debt must be paid in one payment.
Find r if it is known that the loan will be fully repaid in two years, and in the first year 160,000 rubles will be paid, and in the second year - 240,000 rubles.
Algorithm for solving the problem:
- Determine the amount of debt.
- We calculate the amount of debt after the first installment.
- Finding the amount of debt after the second installment
- Find the required percentage.
- We write down the answer.
Solution:
1. 300,000 rubles were borrowed. According to the condition, the amount of debt to be repaid increases by r%, which means times. To pay off the debt, you need to give the bank 300,000∙k.
2. After making a payment equal to 160,000 rubles. The balance of the debt is
Fill in all the missing punctuation marks: indicate the number(s) that should be replaced by a comma(s) in the sentence.
Farewell (1) my homeland! North (2) goodbye, -
Forever (3) I will remain your son!
Farewell (4) peaks under the roof of snows,
Farewell (5) valleys and slopes of meadows,
Farewell (6) forests drooping into the abyss,
(S. Marshak)
Explanation (see also Rule below).
Here is the correct spelling.
Goodbye, My motherland! North, Goodbye, -
Fatherland of glory and valor.
We are driven by fate through the white world,
I will always be your son!
Farewell, peaks covered with snow,
Farewell, valleys and slopes of meadows,
Farewell, abyssal forests,
(S. Marshak)
There are 6 references in this poem, all of them are separated by commas.
Answer: 124567
Answer: 124567
Relevance: Current academic year
Difficulty: advanced
Codifier section: Punctuation marks in sentences with words and constructions not grammatically related to sentence members
Rule: Introductory words and appeal. Task 18 USE., Introductory words and appeal. Task 18 USE.
Put punctuation marks: indicate the number (s) in the place of which (s) in the sentence should (s) be a comma (s).
(M.Yu. Lermontov)
For example: Obviously; Fortunately.
introductory sentences. For example: evening, Do you remember, the blizzard was angry ... (Pushkin).
Adjoining the input units insert structures (two) I ordered to send to Yalta ; about Mozart.
GROUPS OF INTRODUCTORY WORDS.
…
Maybe may be, he is ill. You, should be seems, I saw him somewhere.
You, obviously maybe I'm going to rest. You, it is seen
probably (=must be) right exactly naturally
He is a good sportsman. By the way He also studies well.
Her parents, friends and, by the way, best friend against the trip.
First of all
From this hill really Undoubtedly undoubtedly proper and the whole story.
AND, Then Further, we will talk about our findings. In this way finally eventually
Rain, but but!
but, it's incredibly difficult.
His works, at all
Your child, to my mind, caught a cold. This, In your
We, certainly ready to help you with everything.
I, anyway anyway
I anyway
Indeed, are you making a smart guy out of yourself?
in its turn in its turn
The message is complex means means means she feels right.
He didn't mean to hurt her, uh vice versa vice versa sitting at home all day.
Misha, at least
From my grandmother's point of view from the point of view of the examiners
in particular in particular
mainly mainly
Many Russian people mainly
for example
In many big cities, for example for example
for example
On the one side with another On the one side with another On the one side with another.
Tatiana, dear Tatiana! With you now I shed tears Use life living Noise, noise obedient sail ; Don't make noise rye, ripe ear.
Personal pronouns you And you, as a rule, act not in the role of appeal, and as a subject: Sorry, peaceful valleys, and you , familiar mountain peaks, and you , familiar woods!
Old man! Forget about the past; A young native of Naples!
Think master of culture; hello to you people of peaceful labor!; Are you here, cute?; You are a pig brother…
; Vaska! Vaska! Vaska! Great!
And or Yes, do not separate with a comma: sing along people, cities and rivers! sing along mountains, steppes and fields!; Hi, sunshine and happy morning!
Ivan Ilyich, dispose, brother, about snacks; ... I therefore Thomas, isn't it better brother, breake down?
Stronger equine, bey, hoof, chasing a step! ; we see you forty one year.
Task 18 tests the ability to punctuate words that are not grammatically related to the sentence. These include introductory words (constructions, phrases, sentences), plug-in constructions and appeals.
In the USE 2016-2017, one part of tasks 18 will be presented in the form of a narrative sentence with introductory words
Put punctuation marks: indicate the number (s) in the place of which (s) in the sentence should (s) be a comma (s).
Dacha (1) can be (2) called the cradle from which for each of us began to comprehend the world, at first limited to a garden, then a huge street, then plots and (3) finally (4) the entire country side.
The other part (judging by the demo and the book by I.P. Tsybulko Model Exam Materials 2017) will look like this:
Put punctuation marks: indicate the number (s) in the place of which (s) in the sentence should (s) be a comma (s).
Listen (1) maybe (2) when we leave
Forever this world, where the soul is so cold,
Perhaps (3) in a country where they do not know deceit,
You (4) will be an angel, I will become a demon!
Swear then to forget (5) dear (6)
For a former friend, all the happiness of paradise!
May (7) the gloomy exile, condemned by fate,
You will be paradise, and you will be the universe to me!
(M.Yu. Lermontov)
Consider the rules and concepts necessary to perform this type of task.
17.1 The general concept of introductory words and the basic rule for their selection.
Introductory words are words (or phrases) that are not grammatically related to the sentence and introduce additional semantic shades. For example: Obviously communication with children develops many good qualities in a person; Fortunately the secret remains a secret.
These meanings are conveyed not only by introductory words, but also introductory sentences. For example: evening, Do you remember, the blizzard was angry ... (Pushkin).
Adjoining the input units insert structures which contain various additional remarks, amendments and clarifications. Plug-in constructions, like introductory ones, are not connected with other words in the sentence. They abruptly tear up the offer. For example: Journals of foreign literature (two) I ordered to send to Yalta ; Masha talked to him about Rossini (Rossini was just coming into fashion) about Mozart.
The main mistake of most writers is associated with inaccurate knowledge of the list of introductory words. Therefore, first of all, you should learn which words can be introductory, which groups of introductory words can be distinguished and which words are never introductory.
GROUPS OF INTRODUCTORY WORDS.
1. introductory words expressing the speaker's feelings in connection with what was said: fortunately, unfortunately, unfortunately, to annoyance, to horror, to misfortune, what good ...
2. introductory words expressing the speaker's assessment of the degree of reliability of what he said: of course, undoubtedly, of course, indisputably, obviously, certainly, probably, probably, probably, probably, probably, probably, apparently, apparently, in essence, in fact, I think ... This group of introductory words is the most numerous.
3. introductory words indicating the sequence of thoughts presented and their connection with each other: firstly, so, therefore, in general, means, by the way, further, however, finally, on the one hand… This group is also quite large and treacherous.
4. introductory words indicating the techniques and ways of formulating thoughts: in a word, in other words, in other words, rather, more precisely, so to speak ...
5. introductory words indicating the source of the message: they say, in my opinion, according to ..., according to rumors, according to information ..., according to ..., in my opinion, I remember ...
6. introductory words, which are the speaker's appeal to the interlocutor: you see (whether), you know, understand, forgive, please, agree ...
7. introductory words indicating an assessment of the measure of what is being said: at the most, at least...
8. introductory words showing the degree of commonness of what was said: happens, happens, as usual...
9. introductory words expressing the expressiveness of the statement: joking aside, it's funny to say, to be honest, between us...
17.1. 1 ARE NOT INTRODUCTORY WORDS and therefore the following words are not separated by commas in the letter:
literally, as if, in addition, suddenly, after all, here, there, hardly, after all, ultimately, hardly, even, precisely, exclusively, as if, as if, just, meanwhile, almost, therefore, therefore, approximately, approximately, moreover, moreover, simply, decisively, as if ... - this group includes particles and adverbs, which most often turn out to be erroneously isolated as introductory.
according to tradition, according to the advice ..., according to the instructions ..., according to the demand ..., according to the order ..., according to the plan ... - these combinations act as non-separated (not separated by commas) members of the sentence:
On the advice of her older sister, she decided to enter Moscow State University.
By order of the doctor, the patient was put on a strict diet.
17.1. 2 Depending on the context, the same words can act either as introductory words or as members of a sentence.
MAY and MAY BE, SHOULD BE, SEEMS (seemed) act as introductory if they indicate the degree of reliability of the reported:
Maybe, I will come tomorrow? Our teacher has been gone for two days; may be, he is ill. You, should be, for the first time you meet with such a phenomenon. I, seems, I saw him somewhere.
The same words can be used as predicates:
What can a meeting with you bring me? How can a person be so optional! This should be your own decision. All this seems very suspicious to me. Note: you can never throw out its predicate from a sentence, but the introductory word can.
OBVIOUSLY, POSSIBLY, VISIBLY turn out to be introductory if they indicate the degree of reliability of the statement:
You, obviously Do you want to apologize for what you did? Next month I maybe I'm going to rest. You, it is seen Would you like to tell us the whole truth?
The same words can be included in the predicates:
It became obvious to everyone that another way to solve the problem had to be found. This was made possible thanks to the coordinated actions of the fire brigade. The sun is not visible because of the clouds.
PROBABLY, TRUE, EXACTLY, NATURALLY turn out to be introductory when indicating the degree of reliability of the reported (in this case they are interchangeable or can be replaced by words of this group that are close in meaning) - You, probably (=must be) and you don't understand how important it is to do it on time. You, right, and there is the same Sidorov? She, exactly, was a beauty. All these arguments naturally So far, only our guesses.
The same words turn out to be members of the sentence (circumstances) - He correctly (=correctly, the circumstance of the mode of action) translated the text. I don't know for sure (=probably a modus operandi), but he must have done it to spite me. The student accurately (=correctly) solved the problem. This naturally (=in a natural way) led us to the only correct answer.
BTW is an introductory word if it indicates a connection of thoughts:
He is a good sportsman. By the way He also studies well.
The same word does not act as an introductory word in the meaning of "at the same time":
I'll go for a walk, by the way I'll buy some bread.
BY THE WAY turns out to be an introductory word, indicating the connection of thoughts:
Her parents, friends and, by the way, best friend against the trip.
This word can be used as a non-introductory word in the context:
He made a long speech, in which, among other things, he noted that he would soon become our boss.
FIRST of all, as an introductory word, it indicates the connection of thoughts:
First of all(= firstly), is it even necessary to raise such a sensitive topic?
The same word can act as a circumstance of time (=first):
First of all, I want to say hello from your parents.
It must be said that in the same phrase "first of all" can be considered as an introductory, or not, depending on the will of the author.
REALLY, DEFINITELY, DEFINITELY, ACTUALLY will be introductory if they indicate the degree of reliability of the reported:
From this hill really(=exactly, in fact, without any doubt), the view was the best. Undoubtedly(=really, really), your child is capable of music. He, undoubtedly read this novel. - or at the reception of the formulation of thoughts - Here, proper and the whole story.
The same words are not introductory if they appear in other meanings:
I really am what you imagined me to be (=really, actually). He was undoubtedly a talented composer (= no doubt, actually). She is certainly right in offering us such a simple way to solve the problem (=very, quite right). I didn't really have anything against the school, but I didn't want to go to this one (= in general, exactly). The words "really" and "unconditionally", depending on the intonation proposed by the speaker, may in the same context be either introductory or not.
AND, Then she turned out to be a celebrity. Further, we will talk about our findings. In this way(=so), our results do not contradict those obtained by other scientists. She is smart, beautiful and, finally she is very kind to me. What, eventually you want from me? Usually sentences containing the above words complete a series of enumerations, the words themselves have the meaning "and more". In the context above, the words "firstly", "secondly", "on the one hand", etc. may occur. "Thus" in the meaning of the introductory word turns out to be not only the completion of the enumeration, but also the conclusion.
The same words are not distinguished as introductory in the meanings: "in this way" = "in this way":
Thus he was able to move the heavy cabinet.
Usually in the previous context there are circumstances of time, for example "at first". "then" = "then, after that":
And then he became a famous scientist.
"Finally" = "in the end, finally, after all, as a result of everything":
Finally, all cases were successfully completed. Usually, in this sense, the particle "-something" can be added to the word "finally", which cannot be done if "finally" is an introductory word. In the same meanings as indicated above for "finally", the combination "in the end" is not an introductory combination:
In the end (=as a result) an agreement was reached.
HOWEVER is introductory if it is in the middle or at the end of a sentence:
Rain, but, has been going on for the second week, despite the forecasts of weather forecasters. How I deftly do it, but!
"However" does not turn out to be introductory at the beginning of a sentence and at the beginning of a part of a complex sentence, when it acts as an adversative conjunction (= but): However, people did not want to believe in his good intentions. We did not hope to meet, but we were lucky.
We draw attention to the fact that sometimes the word “however” can also be at the beginning of a sentence, but does not perform the function of a union: but, it's incredibly difficult.
IN GENERAL is introductory in the sense of "generally speaking" when it indicates the way thoughts are framed:
His works, at all, is of interest only to a narrow circle of specialists. In other senses, the word "in general" is an adverb in the sense of "in general, completely, in all respects, under all conditions, always":
Ostrovsky is to the Russian theater what Pushkin is to literature in general. Under the new law, smoking in the workplace is generally prohibited.
MY, YOUR, OUR, YOUR are introductory, indicating the source of the message:
Your child, to my mind, caught a cold. This, In your, proves something? The word "in his own way" is not introductory: He is right in his own way.
OF COURSE is most often introductory, indicating the degree of reliability of the statement:
We, certainly ready to help you with everything.
Sometimes this word is not isolated if intonation is distinguished by a tone of confidence, conviction. In this case, the word "of course" is considered an amplifying particle: I certainly would agree if you warned me in advance.
In any case, it is more often introductory and is used to evaluate:
I, anyway I don't want to be reminded of it. These words, anyway testify to the seriousness of his attitude to life.
In the meaning of "always, under any circumstances" this combination is not introductory:
I anyway was supposed to meet him today and talk to him.
IN REALLY, it is NOT introductory more often, speaking in the meaning of "really" - Petya is really well versed in computers. I really don't belong here. Less often, this phrase turns out to be introductory if it serves to express bewilderment, indignation - What are you, Indeed, are you making a smart guy out of yourself?
In turn, it can be introductory when it indicates the connection of thoughts or the way the thought is formed:
Among many modern writers, Vladimir Sorokin is of interest, and among his books, in its turn, you can highlight the "Roman". Asking me to help him with his work, he, in its turn, also did not mess around. The same phrase can be non-introductory in the meanings "in response", "on my part" (= when the turn comes) - Masha, in turn, told about how she spent the summer.
MEAN is introductory if it can be replaced by the words "therefore", "therefore":
The message is complex means, it must be submitted today. The rain has already stopped means we can go for a walk. If she fights us so hard means she feels right.
This word may turn out to be a predicate, close in meaning to "means":
The dog means more to him than the wife. When you are truly friends with a person, it means that you trust him in everything. "So" can be between the subject and the predicate, especially when they are expressed in infinitives. In this case, the "mean" is preceded by a dash:
To be offended means to recognize oneself as weak. To be friends means to trust your friend.
ON the contrary, it is introductory if it indicates a connection of thoughts:
He didn't mean to hurt her, uh vice versa tried to ask her forgiveness. Instead of playing sports, she, vice versa sitting at home all day.
The combination "and vice versa" is not an introductory combination, which can act as a homogeneous member of a sentence, it is used as a word that replaces the whole sentence or part of it:
In the spring, girls change: brunettes become blondes and vice versa (i.e. blondes become brunettes). The more you study, the higher marks you get, and vice versa (i.e. if you study a little, the marks will be bad; the comma before "and" appears at the end of the sentence part - it turns out, as it were, a compound sentence, where "on the contrary" replaces its second part). I know that he will fulfill my request and vice versa (i.e. I will fulfill it, there is no comma before "and", since "vice versa" replaces a homogeneous clause).
It is AT LEAST introductory if the score matters:
Misha, at least, knows how to behave, and does not pick his teeth with a fork.
This phrase can be used in the meanings "not less than", "the least", then it is not isolated:
At least she would know that her father did not live in vain. At least five of the class must take part in cross-country skiing.
FROM THE POINT OF VIEW is introductory in the sense of "according to":
From my grandmother's point of view, the girl should not wear pants. her answer, from the point of view of the examiners worthy of the highest praise.
The same turnover can have the meaning "in relation to" and then it is not introductory:
Work is progressing according to plan in terms of timelines. If we evaluate the behavior of the heroes of some literary works from the point of view of modern morality, then it should be considered immoral.
IN PARTICULAR, it stands out as introductory if it indicates the connection of thoughts in the statement: She is interested, in particular, the question of the contribution of this scientist to the development of the theory of relativity. The firm is actively involved in charitable activities and, in particular, helps orphanage No. 187.
If the combination IN PARTICULAR turned out to be at the beginning or at the end of the connecting structure, then it is not separated from this structure (this will be discussed in more detail in the next section):
I love books about animals, especially about dogs. My friends, in particular Masha and Vadim, vacationed this summer in Spain. The indicated combination is not distinguished as an introductory one if it is connected by the union "and" with the word "generally":
The conversation turned to politics in general and the latest government decisions in particular.
MAINLY it is introductory, when it serves to evaluate some fact, highlight it in the statement: The textbook should be rewritten and, mainly, add such chapters to it ... The room was used on special occasions and, mainly for the organization of ceremonial dinners.
This combination may be part of the connecting construction, in which case, if it is at its beginning or end, it is not separated from the construction itself by a comma:
Many Russian people mainly intellectuals did not believe the promises of the government.
In the meaning of "first of all", "most of all", this combination is not introductory and is not isolated:
He was afraid of writing mainly because of his illiteracy. What I like most about him is his relationship with his parents.
FOR EXAMPLE will always be introductory, but is formatted differently. It can be separated by commas on both sides:
Pavel Petrovich is a person who is extremely attentive to his appearance, for example He takes good care of his nails. If "for example" appears at the beginning or at the end of an already isolated member, then it is not separated from this turnover by a comma:
In many big cities, for example in Moscow, there is an unfavorable ecological situation. Some works of Russian writers, for example"Eugene Onegin" or "War and Peace" served as the basis for the creation of feature films not only in Russia, but also in other countries. In addition, after "for example" there can be a colon, if "for example" is after the generalizing word before a number of homogeneous members:
Some fruits can cause allergies, for example: oranges, tangerines, pineapple, red berries.
17.1.3 There are special cases of punctuation in introductory words.
To highlight introductory words and sentences, not only commas, but also dashes, as well as combinations of dashes and commas, can be used.
These cases are not included in the secondary school course and are not used in the USE assignments. But some turns, often used, need to be remembered. Here are some examples from Rosenthal's Punctuation Guide.
So, if the introductory combination forms an incomplete construction (any word restored from the context is missing), then it is highlighted with a comma and a dash: Makarenko repeatedly emphasized that pedagogy is based On the one side, on boundless trust in a person, and with another- on high requirements to it; Chichikov ordered to stop for two reasons: On the one side to give the horses a rest, with another- to relax and refresh yourself(the comma before the subordinate clause is "absorbed" by the dash); On the one side, it was important to make an urgent decision, but caution was required - with another.
17.2 The general concept of treatment and the basic rule for its selection.
For the first time included in the tasks of the exam in 2016-2017. Students will have to look for appeals in poetic works, which greatly complicates the task.
Addresses are words that name the person to whom the speech is addressed. The appeal has the form of the nominative case and is pronounced with a special intonation: Tatiana, dear Tatiana! With you now I shed tears. Addresses are usually expressed by animate nouns, as well as adjectives and participles in the meaning of nouns. For example: Use life living . In artistic speech, inanimate nouns can also be addresses. For example: Noise, noise obedient sail ; Don't make noise rye, ripe ear.
Personal pronouns you And you, as a rule, act not in the role of appeal, and as a subject: Sorry, peaceful valleys, and you , familiar mountain peaks, and you , familiar woods!
17.1.2. There are also more complex rules for selecting hits.
1. If the appeal at the beginning of the sentence is pronounced with an exclamatory intonation, then an exclamation mark is placed after it (the word following the appeal is capitalized): Old man! Forget about the past; A young native of Naples! What did you leave on the field in Russia?
2. If the appeal is at the end of the sentence, then a comma is placed before it, and after it - the punctuation mark that is required by the content and intonation of the sentence: Think master of culture; hello to you people of peaceful labor!; Are you here, cute?; You are a pig brother…
3.Duplicate calls are separated by a comma or an exclamation point: The steppe is wide, the steppe is deserted Why are you looking so cloudy?; Hi, wind, formidable wind, tailwind of world history!; Vaska! Vaska! Vaska! Great!
4. Homogeneous appeals connected by a union And or Yes, do not separate with a comma: sing along people, cities and rivers! sing along mountains, steppes and fields!; Hi, sunshine and happy morning!
5. If there are several appeals to one person, located in different places of the sentence, each of them is separated by commas: Ivan Ilyich, dispose, brother, about snacks; ... I therefore Thomas, isn't it better brother, breake down?
6. If the common appeal is “broken” by other words - members of the sentence, then each part of the appeal is separated by commas according to the general rule: Stronger equine, bey, hoof, chasing a step! ; For blood and tears, thirsting for retribution we see you forty one year.
Secondary general education
Line UMK G.K. Muravina. Algebra and the beginnings of mathematical analysis (10-11) (deep)
Line UMK Merzlyak. Algebra and the Beginnings of Analysis (10-11) (U)
Maths
Preparation for the exam in mathematics (profile level): tasks, solutions and explanations
We analyze tasks and solve examples with the teacherThe profile-level examination paper lasts 3 hours 55 minutes (235 minutes).
Minimum Threshold- 27 points.
The examination paper consists of two parts, which differ in content, complexity and number of tasks.
The defining feature of each part of the work is the form of tasks:
- part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of an integer or a final decimal fraction;
- part 2 contains 4 tasks (tasks 9-12) with a short answer in the form of an integer or a final decimal fraction and 7 tasks (tasks 13-19) with a detailed answer (full record of the decision with the rationale for the actions performed).
Panova Svetlana Anatolievna, teacher of mathematics of the highest category of the school, work experience of 20 years:
“In order to get a school certificate, a graduate must pass two mandatory exams in the form of the Unified State Examination, one of which is mathematics. In accordance with the Concept for the Development of Mathematical Education in the Russian Federation, the Unified State Exam in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.
Task number 1- checks the ability of USE participants to apply the skills acquired in the course of 5-9 grades in elementary mathematics in practical activities. The participant must have computational skills, be able to work with rational numbers, be able to round decimal fractions, be able to convert one unit of measure to another.
Example 1 In the apartment where Petr lives, a cold water meter (meter) was installed. On the first of May, the meter showed an consumption of 172 cubic meters. m of water, and on the first of June - 177 cubic meters. m. What amount should Peter pay for cold water for May, if the price of 1 cu. m of cold water is 34 rubles 17 kopecks? Give your answer in rubles.
Solution:
1) Find the amount of water spent per month:
177 - 172 = 5 (cu m)
2) Find how much money will be paid for the spent water:
34.17 5 = 170.85 (rub)
Answer: 170,85.
Task number 2- is one of the simplest tasks of the exam. The majority of graduates successfully cope with it, which indicates the possession of the definition of the concept of function. Task type No. 2 according to the requirements codifier is a task for using the acquired knowledge and skills in practical activities and everyday life. Task No. 2 consists of describing, using functions, various real relationships between quantities and interpreting their graphs. Task number 2 tests the ability to extract information presented in tables, diagrams, graphs. Graduates need to be able to determine the value of a function by the value of the argument with various ways of specifying the function and describe the behavior and properties of the function according to its graph. It is also necessary to be able to find the largest or smallest value from the function graph and build graphs of the studied functions. The mistakes made are of a random nature in reading the conditions of the problem, reading the diagram.
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Example 2 The figure shows the change in the exchange value of one share of a mining company in the first half of April 2017. On April 7, the businessman purchased 1,000 shares of this company. On April 10, he sold three-quarters of the purchased shares, and on April 13 he sold all the remaining ones. How much did the businessman lose as a result of these operations?
Solution:
2) 1000 3/4 = 750 (shares) - make up 3/4 of all purchased shares.
6) 247500 + 77500 = 325000 (rubles) - the businessman received after the sale of 1000 shares.
7) 340,000 - 325,000 = 15,000 (rubles) - the businessman lost as a result of all operations.
Answer: 15000.
Task number 3- is a task of the basic level of the first part, it checks the ability to perform actions with geometric shapes according to the content of the course "Planimetry". Task 3 tests the ability to calculate the area of a figure on checkered paper, the ability to calculate degree measures of angles, calculate perimeters, etc.
Example 3 Find the area of a rectangle drawn on checkered paper with a cell size of 1 cm by 1 cm (see figure). Give your answer in square centimeters.
Solution: To calculate the area of this figure, you can use the Peak formula:
To calculate the area of this rectangle, we use the Peak formula:
|
S= B + |
G | |
| 2 |
|
S = 18 + |
6 | |
| 2 |
See also: Unified State Examination in Physics: solving vibration problems
Task number 4- the task of the course "Probability Theory and Statistics". The ability to calculate the probability of an event in the simplest situation is tested.
Example 4 There are 5 red and 1 blue dots on the circle. Determine which polygons are larger: those with all red vertices, or those with one of the blue vertices. In your answer, indicate how many more of one than the other.
Solution: 1) We use the formula for the number of combinations from n elements by k:
all of whose vertices are red.
3) One pentagon with all red vertices.
4) 10 + 5 + 1 = 16 polygons with all red vertices.
whose vertices are red or with one blue vertex.
whose vertices are red or with one blue vertex.
8) One hexagon whose vertices are red with one blue vertex.
9) 20 + 15 + 6 + 1 = 42 polygons that have all red vertices or one blue vertex.
10) 42 - 16 = 26 polygons that use the blue dot.
11) 26 - 16 = 10 polygons - how many polygons, in which one of the vertices is a blue dot, are more than polygons, in which all vertices are only red.
Answer: 10.
Task number 5- the basic level of the first part tests the ability to solve the simplest equations (irrational, exponential, trigonometric, logarithmic).
Example 5 Solve Equation 2 3 + x= 0.4 5 3 + x .
Solution. Divide both sides of this equation by 5 3 + X≠ 0, we get
| 2 3 + x | = 0.4 or | 2 | 3 + X | = | 2 | , | ||
| 5 3 + X | 5 | 5 |
whence it follows that 3 + x = 1, x = –2.
Answer: –2.
Task number 6 in planimetry for finding geometric quantities (lengths, angles, areas), modeling real situations in the language of geometry. The study of the constructed models using geometric concepts and theorems. The source of difficulties is, as a rule, ignorance or incorrect application of the necessary theorems of planimetry.
Area of a triangle ABC equals 129. DE- median line parallel to side AB. Find the area of the trapezoid ABED.
Solution. Triangle CDE similar to a triangle CAB at two corners, since the corner at the vertex C general, angle CDE equal to the angle CAB as the corresponding angles at DE || AB secant AC. Because DE is the middle line of the triangle by the condition, then by the property of the middle line | DE = (1/2)AB. So the similarity coefficient is 0.5. The areas of similar figures are related as the square of the similarity coefficient, so
Consequently, S ABED = S Δ ABC – S Δ CDE = 129 – 32,25 = 96,75.
Task number 7- checks the application of the derivative to the study of the function. For successful implementation, a meaningful, non-formal possession of the concept of a derivative is necessary.
Example 7 To the graph of the function y = f(x) at the point with the abscissa x 0 a tangent is drawn, which is perpendicular to the straight line passing through the points (4; 3) and (3; -1) of this graph. Find f′( x 0).
Solution. 1) Let's use the equation of a straight line passing through two given points and find the equation of a straight line passing through points (4; 3) and (3; -1).
(y – y 1)(x 2 – x 1) = (x – x 1)(y 2 – y 1)
(y – 3)(3 – 4) = (x – 4)(–1 – 3)
(y – 3)(–1) = (x – 4)(–4)
–y + 3 = –4x+ 16| · (-one)
y – 3 = 4x – 16
y = 4x– 13, where k 1 = 4.
2) Find the slope of the tangent k 2 which is perpendicular to the line y = 4x– 13, where k 1 = 4, according to the formula:
3) The slope of the tangent is the derivative of the function at the point of contact. Means, f′( x 0) = k 2 = –0,25.
Answer: –0,25.
Task number 8- checks the knowledge of elementary stereometry among the participants of the exam, the ability to apply formulas for finding surface areas and volumes of figures, dihedral angles, compare the volumes of similar figures, be able to perform actions with geometric figures, coordinates and vectors, etc.
The volume of a cube circumscribed around a sphere is 216. Find the radius of the sphere.
Solution. 1) V cube = a 3 (where but is the length of the edge of the cube), so
but 3 = 216
but = 3 √216
2) Since the sphere is inscribed in a cube, it means that the length of the diameter of the sphere is equal to the length of the edge of the cube, therefore d = a, d = 6, d = 2R, R = 6: 2 = 3.
Task number 9- requires the graduate to transform and simplify algebraic expressions. Task No. 9 of an increased level of complexity with a short answer. Tasks from the section "Calculations and transformations" in the USE are divided into several types:
- conversion of numeric/letter trigonometric expressions.
transformations of numerical rational expressions;
transformations of algebraic expressions and fractions;
transformations of numerical/letter irrational expressions;
actions with degrees;
transformation of logarithmic expressions;
Example 9 Calculate tgα if it is known that cos2α = 0.6 and
| 3π | < α < π. |
| 4 |
Solution. 1) Let's use the double argument formula: cos2α = 2 cos 2 α - 1 and find
| tan 2 α = | 1 | – 1 = | 1 | – 1 = | 10 | – 1 = | 5 | – 1 = 1 | 1 | – 1 = | 1 | = 0,25. |
| cos 2 α | 0,8 | 8 | 4 | 4 | 4 |
Hence, tan 2 α = ± 0.5.
3) By condition
| 3π | < α < π, |
| 4 |
hence α is the angle of the second quarter and tgα< 0, поэтому tgα = –0,5.
Answer: –0,5.
#ADVERTISING_INSERT# Task number 10- checks the ability of students to use the acquired early knowledge and skills in practical activities and everyday life. We can say that these are problems in physics, and not in mathematics, but all the necessary formulas and quantities are given in the condition. The tasks are reduced to solving a linear or quadratic equation, or a linear or quadratic inequality. Therefore, it is necessary to be able to solve such equations and inequalities, and determine the answer. The answer must be in the form of a whole number or a final decimal fraction.
Two bodies of mass m= 2 kg each, moving at the same speed v= 10 m/s at an angle of 2α to each other. The energy (in joules) released during their absolutely inelastic collision is determined by the expression Q = mv 2 sin 2 α. At what smallest angle 2α (in degrees) must the bodies move so that at least 50 joules are released as a result of the collision?
Solution. To solve the problem, we need to solve the inequality Q ≥ 50, on the interval 2α ∈ (0°; 180°).
mv 2 sin 2 α ≥ 50
2 10 2 sin 2 α ≥ 50
200 sin2α ≥ 50
Since α ∈ (0°; 90°), we will only solve
We represent the solution of the inequality graphically:
Since by assumption α ∈ (0°; 90°), it means that 30° ≤ α< 90°. Получили, что наименьший угол α равен 30°, тогда наименьший угол 2α = 60°.
Task number 11- is typical, but it turns out to be difficult for students. The main source of difficulties is the construction of a mathematical model (drawing up an equation). Task number 11 tests the ability to solve word problems.
Example 11. During spring break, 11-grader Vasya had to solve 560 training problems to prepare for the exam. On March 18, on the last day of school, Vasya solved 5 problems. Then every day he solved the same number of problems more than the previous day. Determine how many problems Vasya solved on April 2 on the last day of vacation.
Solution: Denote a 1 = 5 - the number of tasks that Vasya solved on March 18, d– daily number of tasks solved by Vasya, n= 16 - the number of days from March 18 to April 2 inclusive, S 16 = 560 - the total number of tasks, a 16 - the number of tasks that Vasya solved on April 2. Knowing that every day Vasya solved the same number of tasks more than the previous day, then you can use the formulas for finding the sum of an arithmetic progression:560 = (5 + a 16) 8,
5 + a 16 = 560: 8,
5 + a 16 = 70,
a 16 = 70 – 5
a 16 = 65.
Answer: 65.
Task number 12- check students' ability to perform actions with functions, be able to apply the derivative to the study of the function.
Find the maximum point of a function y= 10ln( x + 9) – 10x + 1.
Solution: 1) Find the domain of the function: x + 9 > 0, x> –9, that is, x ∈ (–9; ∞).
2) Find the derivative of the function:
4) The found point belongs to the interval (–9; ∞). We define the signs of the derivative of the function and depict the behavior of the function in the figure:
The desired maximum point x = –8.
Download for free the work program in mathematics to the line of UMK G.K. Muravina, K.S. Muravina, O.V. Muravina 10-11 Download free algebra manualsTask number 13- an increased level of complexity with a detailed answer, which tests the ability to solve equations, the most successfully solved among tasks with a detailed answer of an increased level of complexity.
a) Solve the equation 2log 3 2 (2cos x) – 5log 3 (2cos x) + 2 = 0
b) Find all the roots of this equation that belong to the segment.
Solution: a) Let log 3 (2cos x) = t, then 2 t 2 – 5t + 2 = 0,
|
|
log3(2cos x) = | 2 | ⇔ |
|
2cos x = 9 | ⇔ |
|
cos x = | 4,5 | ⇔ because |cos x| ≤ 1, |
| log3(2cos x) = | 1 | 2cos x = √3 | cos x = | √3 | ||||||
| 2 | 2 |
| then cos x = | √3 |
| 2 |
|
|
x = | π | + 2π k |
| 6 | |||
| x = – | π | + 2π k, k ∈ Z | |
| 6 |
b) Find the roots lying on the segment .
It can be seen from the figure that the given segment has roots
| 11π | And | 13π | . |
| 6 | 6 |
| Answer: but) | π | + 2π k; – | π | + 2π k, k ∈ Z; b) | 11π | ; | 13π | . |
| 6 | 6 | 6 | 6 |
The circumference diameter of the base of the cylinder is 20, the generatrix of the cylinder is 28. The plane intersects its bases along chords of length 12 and 16. The distance between the chords is 2√197.
a) Prove that the centers of the bases of the cylinder lie on the same side of this plane.
b) Find the angle between this plane and the plane of the base of the cylinder.
Solution: a) A chord of length 12 is at a distance = 8 from the center of the base circle, and a chord of length 16, similarly, is at a distance of 6. Therefore, the distance between their projections on a plane parallel to the bases of the cylinders is either 8 + 6 = 14, or 8 − 6 = 2.
Then the distance between chords is either
= = √980 = = 2√245
= = √788 = = 2√197.
According to the condition, the second case was realized, in which the projections of the chords lie on one side of the axis of the cylinder. This means that the axis does not intersect this plane within the cylinder, that is, the bases lie on one side of it. What needed to be proven.
b) Let's denote the centers of the bases as O 1 and O 2. Let us draw from the center of the base with a chord of length 12 the perpendicular bisector to this chord (it has a length of 8, as already noted) and from the center of the other base to another chord. They lie in the same plane β perpendicular to these chords. Let's call the midpoint of the smaller chord B, greater than A, and the projection of A onto the second base H (H ∈ β). Then AB,AH ∈ β and, therefore, AB,AH are perpendicular to the chord, that is, the line of intersection of the base with the given plane.
So the required angle is
| ∠ABH = arctan | AH | = arctg | 28 | = arctg14. |
| BH | 8 – 6 |
Task number 15- an increased level of complexity with a detailed answer, checks the ability to solve inequalities, the most successfully solved among tasks with a detailed answer of an increased level of complexity.
Example 15 Solve the inequality | x 2 – 3x| log 2 ( x + 1) ≤ 3x – x 2 .
Solution: The domain of definition of this inequality is the interval (–1; +∞). Consider three cases separately:
1) Let x 2 – 3x= 0, i.e. X= 0 or X= 3. In this case, this inequality becomes true, therefore, these values are included in the solution.
2) Let now x 2 – 3x> 0, i.e. x∈ (–1; 0) ∪ (3; +∞). In this case, this inequality can be rewritten in the form ( x 2 – 3x) log 2 ( x + 1) ≤ 3x – x 2 and divide by a positive expression x 2 – 3x. We get log 2 ( x + 1) ≤ –1, x + 1 ≤ 2 –1 , x≤ 0.5 -1 or x≤ -0.5. Taking into account the domain of definition, we have x ∈ (–1; –0,5].
3) Finally, consider x 2 – 3x < 0, при этом x∈ (0; 3). In this case, the original inequality will be rewritten in the form (3 x – x 2) log 2 ( x + 1) ≤ 3x – x 2. After dividing by a positive expression 3 x – x 2 , we get log 2 ( x + 1) ≤ 1, x + 1 ≤ 2, x≤ 1. Taking into account the area, we have x ∈ (0; 1].
Combining the obtained solutions, we obtain x ∈ (–1; –0.5] ∪ ∪ {3}.
Answer: (–1; –0.5] ∪ ∪ {3}.
Task number 16- advanced level refers to the tasks of the second part with a detailed answer. The task tests the ability to perform actions with geometric shapes, coordinates and vectors. The task contains two items. In the first paragraph, the task must be proved, and in the second paragraph, it must be calculated.
In an isosceles triangle ABC with an angle of 120° at the vertex A, a bisector BD is drawn. Rectangle DEFH is inscribed in triangle ABC so that side FH lies on segment BC and vertex E lies on segment AB. a) Prove that FH = 2DH. b) Find the area of the rectangle DEFH if AB = 4.
Solution: but)
1) ΔBEF - rectangular, EF⊥BC, ∠B = (180° - 120°) : 2 = 30°, then EF = BE due to the property of the leg opposite the angle of 30°.
2) Let EF = DH = x, then BE = 2 x, BF = x√3 by the Pythagorean theorem.
3) Since ΔABC is isosceles, then ∠B = ∠C = 30˚.
BD is the bisector of ∠B, so ∠ABD = ∠DBC = 15˚.
4) Consider ΔDBH - rectangular, because DH⊥BC.
| 2x | = | 4 – 2x |
| 2x(√3 + 1) | 4 |
| 1 | = | 2 – x |
| √3 + 1 | 2 |
√3 – 1 = 2 – x
x = 3 – √3
EF = 3 - √3
2) S DEFH = ED EF = (3 - √3 ) 2(3 - √3 )
S DEFH = 24 - 12√3.
Answer: 24 – 12√3.
Task number 17- a task with a detailed answer, this task tests the application of knowledge and skills in practical activities and everyday life, the ability to build and explore mathematical models. This task is a text task with economic content.
Example 17. The deposit in the amount of 20 million rubles is planned to be opened for four years. At the end of each year, the bank increases the deposit by 10% compared to its size at the beginning of the year. In addition, at the beginning of the third and fourth years, the depositor annually replenishes the deposit by X million rubles, where X - whole number. Find the highest value X, at which the bank will add less than 17 million rubles to the deposit in four years.
Solution: At the end of the first year, the contribution will be 20 + 20 · 0.1 = 22 million rubles, and at the end of the second - 22 + 22 · 0.1 = 24.2 million rubles. At the beginning of the third year, the contribution (in million rubles) will be (24.2 + X), and at the end - (24.2 + X) + (24,2 + X) 0.1 = (26.62 + 1.1 X). At the beginning of the fourth year, the contribution will be (26.62 + 2.1 X), and at the end - (26.62 + 2.1 X) + (26,62 + 2,1X) 0.1 = (29.282 + 2.31 X). By condition, you need to find the largest integer x for which the inequality
(29,282 + 2,31x) – 20 – 2x < 17
29,282 + 2,31x – 20 – 2x < 17
0,31x < 17 + 20 – 29,282
0,31x < 7,718
| x < | 7718 |
| 310 |
| x < | 3859 |
| 155 |
| x < 24 | 139 |
| 155 |
The largest integer solution to this inequality is the number 24.
Answer: 24.
Task number 18- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. A task of a high level of complexity is not a task for applying one solution method, but for a combination of different methods. For the successful completion of task 18, in addition to solid mathematical knowledge, a high level of mathematical culture is also required.
At what a system of inequalities
| x 2 + y 2 ≤ 2ay – a 2 + 1 | |
| y + a ≤ |x| – a |
has exactly two solutions?
Solution: This system can be rewritten as
| x 2 + (y– a) 2 ≤ 1 | |
| y ≤ |x| – a |
If we draw on the plane the set of solutions to the first inequality, we get the interior of a circle (with a boundary) of radius 1 centered at the point (0, but). The set of solutions of the second inequality is the part of the plane that lies under the graph of the function y = |
x| –
a,
and the latter is the graph of the function
y = |
x|
, shifted down by but. The solution of this system is the intersection of the solution sets of each of the inequalities.
Consequently, this system will have two solutions only in the case shown in Fig. one.
The points of contact between the circle and the lines will be the two solutions of the system. Each of the straight lines is inclined to the axes at an angle of 45°. So the triangle PQR- rectangular isosceles. Dot Q has coordinates (0, but), and the point R– coordinates (0, – but). In addition, cuts PR And PQ are equal to the circle radius equal to 1. Hence,
| QR= 2a = √2, a = | √2 | . |
| 2 |
| Answer: a = | √2 | . |
| 2 |
Task number 19- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. A task of a high level of complexity is not a task for applying one solution method, but for a combination of different methods. For the successful completion of task 19, it is necessary to be able to search for a solution, choosing various approaches from among the known ones, modifying the studied methods.
Let be sn sum P members of an arithmetic progression ( a p). It is known that S n + 1 = 2n 2 – 21n – 23.
a) Give the formula P th member of this progression.
b) Find the smallest modulo sum S n.
c) Find the smallest P, at which S n will be the square of an integer.
Solution: a) Obviously, a n = S n – S n- one . Using this formula, we get:
S n = S (n – 1) + 1 = 2(n – 1) 2 – 21(n – 1) – 23 = 2n 2 – 25n,
S n – 1 = S (n – 2) + 1 = 2(n – 1) 2 – 21(n – 2) – 23 = 2n 2 – 25n+ 27
means, a n = 2n 2 – 25n – (2n 2 – 29n + 27) = 4n – 27.
B) because S n = 2n 2 – 25n, then consider the function S(x) = | 2x 2 – 25x|. Her graph can be seen in the figure.
It is obvious that the smallest value is reached at the integer points located closest to the zeros of the function. Obviously these are points. X= 1, X= 12 and X= 13. Since, S(1) = |S 1 | = |2 – 25| = 23, S(12) = |S 12 | = |2 144 – 25 12| = 12, S(13) = |S 13 | = |2 169 – 25 13| = 13, then the smallest value is 12.
c) It follows from the previous paragraph that sn positive since n= 13. Since S n = 2n 2 – 25n = n(2n– 25), then the obvious case when this expression is a perfect square is realized when n = 2n- 25, that is, with P= 25.
It remains to check the values from 13 to 25:
S 13 = 13 1, S 14 = 14 3, S 15 = 15 5, S 16 = 16 7, S 17 = 17 9, S 18 = 18 11, S 19 = 19 13 S 20 = 20 13, S 21 = 21 17, S 22 = 22 19, S 23 = 23 21, S 24 = 24 23.
It turns out that for smaller values P full square is not achieved.
Answer: but) a n = 4n- 27; b) 12; c) 25.
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