In this article, we will consider solving problems from Task 17, in which it is required to optimally distribute production in order to obtain maximum profit.

Task 1. The cannery produces fruit compotes in two types of containers - glass and tin. The production capacity of the plant allows to produce 90 centners of compotes in glass containers or 80 centners in tin containers per day. To meet the conditions of assortment, which are presented by retail chains, products in each type of container must be produced at least 20 centners. The table shows the cost and selling price of the plant for 1 centner of products for both types of containers.

Assuming that all the products of the plant are in demand (sold without a trace), find the maximum possible profit of the plant in one day (profit is the difference between the selling price of all products and its cost).

The amount of profit depends on how the production capacities will be distributed at the plant, that is, what part of the capacities will be directed to the production of compotes in glass containers, and which one - in a tin. The value on which the profit depends will be taken as an unknown.

Let the value be a part of the plant's capacity aimed at the production of compotes in glass containers. Then the remaining capacities, that is, are aimed at the production of compotes in tin containers.

In this case, the plant will produce centners of compote in glass containers, and centners in tin.

The profit from one centner of production is equal to the difference between the selling price and the cost price. Thus

1 centner of compotes in glass containers makes a profit rub

1 centner of compotes in tin containers makes a profit rub

As a result, the resulting profit, depending on will be

Simplify the expression for the function

The coefficient at is greater than zero, therefore, this is an increasing function, and the larger the value, the greater the profit. But according to the condition of the problem, it is impossible to give all the capacities for the production of compotes in glass containers: to meet the conditions of assortment, which are presented by retail chains, products in each type of container must be produced at least 20 centners.

Let's find what part of the capacities should be given to the production of compotes in tin containers:

For the production of compotes in tin containers, it is necessary to give part of all the capacities of the plant, therefore, for the production of compotes in glass containers, you can give the maximum of all capacities.

Answer: .

Task 2. The farmer has two fields, each with an area of ​​10 hectares. Each field can grow potatoes and beets, 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 5,000 rubles. per centner, and beets - at a price of 8000 rubles. per centner. What is the maximum income a farmer can earn?

(from the collection Typical test tasks in mathematics, edited by I. V. Yashchenko. 2016)

The amount of the farmer's income depends on how the area of ​​each field will be distributed between potato and beet plantings.

Let the farmer take a hectare in the first field for potatoes. Then ha remains under the beets.

The yield of potatoes in the first field is 500 c/ha, and beets 300 c/ha.

In this case, the profit from the first field will be - we have an increasing function that takes the largest value at the maximum possible . Since there are no restrictions on the distribution of planting areas between potatoes and beets, it is beneficial for the farmer to give the entire first field for potatoes, then he will make a profit:

Rub.

We will do the same with the second field.

Let the farmer take a hectare in the second field for potatoes. Then ha remains under the beets.

The yield of potatoes in the second field is 300 c/ha, and that of beets is 500 c/ha.

If you think about it, you don’t even need to make a function here, since the yield of beets in the second field is higher than potatoes, and the cost of one centner of beets is also higher. Therefore, it is obvious that it is more profitable for a farmer to grow only beets in the second field. In this case, the profit from the second field will be

Rub.

The total profit of the farmer is RUR.

Answer:

Today we will digress a little from standard logarithms, integrals, trigonometry, etc., and together we will consider a more vital task from the Unified State Examination in mathematics, which is directly related to our backward Russian resource-based economy. And to be precise, we will consider the problem of deposits, interest and loans. Because it is the tasks with percentages that have recently been added to the second part of the unified state exam in mathematics. I’ll make a reservation right away that for solving this problem, according to the specifications of the Unified State Examination, three primary points are offered at once, i.e. examiners consider this task one of the most difficult.

At the same time, to solve any of these tasks from the Unified State Examination in mathematics, you need to know only two formulas, each of which is quite accessible to any school graduate, however, for reasons I do not understand, these formulas are completely ignored by both school teachers and compilers of various tasks for preparation to the exam. Therefore, today I will not just tell you what these formulas are and how to apply them, but I will derive each of these formulas literally before your eyes, taking as a basis the tasks from open bank USE in mathematics.

Therefore, the lesson turned out to be quite voluminous, quite meaningful, so make yourself comfortable, and we begin.

Putting money in the bank

First of all, I would like to make a small lyrical digression related to finance, banks, loans and deposits, on the basis of which we will get the formulas that we will use to solve this problem. So, let's digress a little from the exams, from the upcoming school problems, and look into the future.

Let's say you have grown up and are going to buy an apartment. Let's say you are going to buy not some bad apartment on the outskirts, but a good quality apartment for 20 million rubles. At the same time, let's also assume that you got a more or less normal job and earn 300 thousand rubles a month. In this case, for the year you can save about three million rubles. Of course, earning 300 thousand rubles a month, for the year you will get a slightly larger amount - 3,600,000 - but let these 600,000 be spent on food, clothes and other daily household joys. The total input data is as follows: it is necessary to earn twenty million rubles, while we have at our disposal only three million rubles a year. A natural question arises: how many years do we need to put aside three million in order to get these same twenty million. It is considered elementary:

\[\frac(20)(3)=6,....\to 7\]

However, as we have already noted, you earn 300 thousand rubles a month, which means that you are smart people and will not save money "under the pillow", but take it to the bank. And, therefore, annually on those deposits that you bring to the bank, interest will be charged. Let's say you choose a reliable, but at the same time more or less profitable bank, and therefore your deposits will grow by 15% per annum annually. In other words, we can say that the amount on your accounts will increase by 1.15 times every year. Let me remind you the formula:

Let's calculate how much money will be in your accounts after each year:

In the first year, when you just start saving money, no interest will accumulate, that is, at the end of the year you will save three million rubles:

At the end of the second year, interest will already be accrued on those three million rubles that have remained from the first year, i.e. we need to multiply by 1.15. However, during the second year, you also reported another three million rubles. Of course, these three million had not yet accrued interest, because by the end of the second year, these three million had only appeared in the account:

So, the third year. At the end of the third year, interest will be accrued on this amount, that is, it is necessary to multiply this entire amount by 1.15. And again, throughout the year you worked hard and put aside three million rubles:

\[\left(3m\cdot 1.15+3m \right)\cdot 1.15+3m\]

Let's calculate another fourth year. Again, the entire amount that we had at the end of the third year is multiplied by 1.15, i.e. Interest will be charged on the entire amount. This includes interest on interest. And three million more is added to this amount, because during the fourth year you also worked and also saved money:

\[\left(\left(3m\cdot 1.15+3m \right)\cdot 1.15+3m \right)\cdot 1.15+3m\]

And now let's open the brackets and see what amount we will have by the end of the fourth year of saving money:

\[\begin(align)& \left(\left(3m\cdot 1,15+3m \right)\cdot 1,15+3m \right)\cdot 1,15+3m= \\& =\left( 3m\cdot ((1,15)^(2))+3m\cdot 1,15+3m \right)\cdot 1,15+3m= \\& =3m\cdot ((1,15)^(3 ))+3m\cdot ((1,15)^(2))+3m\cdot 1,15+3m= \\& =3m\left(((1,15)^(3))+((1 ,15)^(2))+1,15+1 \right)= \\& =3m\left(1+1,15+((1,15)^(2))+((1,15) ^(3)) \right) \\\end(align)\]

As you can see, in brackets we have elements of a geometric progression, i.e. we have the sum of the elements of a geometric progression.

Let me remind you that if the geometric progression is given by the element $((b)_(1))$, as well as the denominator $q$, then the sum of the elements will be calculated according to the following formula:

This formula must be known and clearly applied.

Please note: the formula n th element sounds like this:

\[((b)_(n))=((b)_(1))\cdot ((q)^(n-1))\]

Because of this degree, many students are confused. In total, we have just n for the sum n- elements, and n-th element has degree $n-1$. In other words, if we now try to calculate the sum of a geometric progression, then we need to consider the following:

\[\begin(align)& ((b)_(1))=1 \\& q=1,15 \\\end(align)\]

\[((S)_(4))=1\cdot \frac(((1,15)^(4))-1)(1,15-1)\]

Let's calculate the numerator separately:

\[((1,15)^(4))=((\left(((1,15)^(2)) \right))^(2))=((\left(1,3225 \right ))^(2))=1.74900625\approx 1.75\]

In total, returning to the sum of the geometric progression, we get:

\[((S)_(4))=1\cdot \frac(1.75-1)(0.15)=\frac(0.75)(0.15)=\frac(75)(15 )=5\]

As a result, we get that in four years of savings, our initial amount will not increase four times, as if we had not deposited money in the bank, but five times, that is, fifteen million. Let's write it separately:

4 years → 5 times

Looking ahead, I’ll say that if we had been saving not for four years, but for five years, then as a result, our amount of savings would have increased by 6.7 times:

5 years → 6.7 times

In other words, by the end of the fifth year, we would have the following amount in the account:

That is, by the end of the fifth year of savings, taking into account interest on the deposit, we would have already received over twenty million rubles. Thus, the total account of savings due to bank interest would decrease from almost seven years to five years, that is, by almost two years.

Thus, even though the bank accrues enough low interest on our deposits (15%), in five years these same 15% give an increase that significantly exceeds our annual earnings. At the same time, the main multiplier effect occurs in recent years and even, rather, in Last year savings.

Why did I write all this? Of course, not to agitate you to carry money to the bank. Because if you really want to increase your savings, then you need to invest them not in the bank, but in real operating business, where these same percentages, i.e., profitability under conditions Russian economy rarely falls below 30%, i.e. twice as much as bank deposits.

But what is really useful in all this reasoning is a formula that allows us to find the final amount of the deposit through the amount of annual payments, as well as through the interest that the bank charges. So let's write:

\[\text(Vklad)=\text(platezh)\frac(((\text(%))^(n))-1)(\text(%)-1)\]

By itself, % is calculated using the following formula:

This formula also needs to be known, as well as the basic formula for the amount of the contribution. And, in turn, the main formula can significantly reduce calculations in those problems with percentages where it is required to calculate the contribution.

Why use formulas instead of tables?

Many will probably have a question, why all these difficulties at all, is it possible to simply write each year on a plate, as is done in many textbooks, calculate separately each year, and then calculate the total amount of the contribution? Of course, you can generally forget about the sum of a geometric progression and count everything using classic tablets - this is done in most collections to prepare for the exam. However, firstly, the volume of calculations increases sharply, and secondly, as a result, the probability of making an error increases.

In general, using tables instead of this wonderful formula is the same as digging trenches with your hands at a construction site instead of using an excavator standing nearby and fully working.

Well, or the same thing as multiplying five by ten not using the multiplication table, but adding five to itself ten times in a row. However, I have already digressed, so I will repeat the most important idea once again: if there is some way to simplify and shorten the calculations, then this is the way to use.

Interest on loans

We figured out the deposits, so we move on to the next topic, namely, to interest on loans.

So, while you are saving money, carefully planning your budget, thinking about your future apartment, your classmate, and now a simple unemployed person, decided to live for today and just took out a loan. At the same time, he will still tease and laugh at you, they say, he has credit phone and a used car loan, and you still ride the subway and use an old push-button telephone. Of course, for all these cheap "show-offs" your former classmate will have to pay dearly. How expensive - this is what we will calculate right now.

First, a brief introduction. Let's say your former classmate took two million rubles on credit. At the same time, according to the contract, he must pay x rubles per month. Let's say that he took a loan at a rate of 20% per annum, which in the current conditions looks quite decent. Also, assume that the loan term is only three months. Let's try to connect all these quantities in one formula.

So, at the very beginning, as soon as your former classmate left the bank, he has two million in his pocket, and this is his debt. At the same time, not a year has passed, and not a month, but this is only the very beginning:

Then, after one month, interest will accrue on the amount owed. As we already know, to calculate interest, it is enough to multiply the original debt by a coefficient, which is calculated using the following formula:

In our case, we are talking about a rate of 20% per annum, i.e. we can write:

This is the ratio of the amount that will be charged per year. However, our classmate is not very smart and he did not read the contract, and in fact he was given a loan not at 20% per year, but at 20% per month. And by the end of the first month, interest will be accrued on this amount, and it will increase by 1.2 times. Immediately after that, the person will need to pay the agreed amount, i.e. x rubles per month:

\[\left(2m\cdot 1,2-x\right)\cdot 1,2-x\]

And again, our boy makes a payment in the amount of $x$ rubles.

Then, by the end of the third month, the amount of his debt increases again by 20%:

\[\left(\left(2m\cdot 1,2- x\right)\cdot 1,2- x\right)1,2- x\]

And according to the condition for three months, he must pay in full, that is, after making the last third payment, his amount of debt should be equal to zero. We can write this equation:

\[\left(\left(2m\cdot 1,2- x\right)\cdot 1,2- x\right)1,2 - x=0\]

Let's decide:

\[\begin(align)& \left(2m\cdot ((1,2)^(2))- x\cdot 1,2- x\right)\cdot 1,2- x=0 \\& 2m \cdot ((1,2)^(3))- x\cdot ((1,2)^(2))- x\cdot 1,2- x=0 \\& 2m\cdot ((1,2 )^(3))=\cdot ((1,2)^(2))+\cdot 1,2+ \\& 2m\cdot ((1,2)^(3))=\left((( 1,2)^(2))+1,2+1 \right) \\\end(align)\]

Before us is again a geometric progression, or rather, the sum of the three elements of a geometric progression. Let's rewrite it in ascending order of elements:

Now we need to find the sum of the three elements of a geometric progression. Let's write:

\[\begin(align)& ((b)_(1))=1; \\& q=1,2 \\\end(align)\]

Now let's find the sum of the geometric progression:

\[((S)_(3))=1\cdot \frac(((1,2)^(3))-1)(1,2-1)\]

It should be recalled that the sum of a geometric progression with such parameters $\left(((b)_(1));q \right)$ is calculated by the formula:

\[((S)_(n))=((b)_(1))\cdot \frac(((q)^(n))-1)(q-1)\]

This is the formula we just used. Substitute this formula into our expression:

For further calculations, we need to find out what $((1,2)^(3))$ is equal to. Unfortunately, in this case, we can no longer paint as last time in the form of a double square, but we can calculate like this:

\[\begin(align)& ((1,2)^(3))=((1,2)^(2))\cdot 1,2 \\& ((1,2)^(3)) =1,44\cdot 1,2 \\& ((1,2)^(3))=1,728 \\\end(align)\]

We rewrite our expression:

This is a classic linear expression. Let's go back to the next formula:

In fact, if we generalize it, we will get a formula linking interest, loans, payments and terms. The formula goes like this:

Here it is, the most important formula of today's video lesson, with the help of which at least 80% of all economic tasks from the exam in mathematics in the second part.

Most often, in real tasks, you will be asked for a payment, or a little less often for a loan, that is, the total amount of debt that our classmate had at the very beginning of the payments. In more complex tasks, you will be asked to find a percentage, but for very complex ones, which we will analyze in a separate video lesson, you will be asked to find the time frame during which, with the given loan and payment parameters, our unemployed classmate will be able to fully pay off the bank.

Perhaps someone will now think that I am a fierce opponent of loans, finance and in general banking system. So, nothing like that! On the contrary, I believe that credit instruments are very useful and essential for our economy, but only on the condition that the loan is taken for business development. As a last resort, you can take out a loan to buy a home, i.e. a mortgage or an emergency medical treatment- that's it, there are simply no other reasons to take a loan. And all sorts of unemployed people who take loans to buy "show-offs" and at the same time do not think at all about the consequences in the end and become the cause of crises and problems in our economy.

Returning to the topic of today's lesson, I would like to note that it is also necessary to know this formula connecting loans, payments and interest, as well as the amount of a geometric progression. It is with the help of these formulas that real economic problems from the Unified State Examination in mathematics are solved. Well, now that you know all this very well, when you understand what a loan is and why you should not take it, let's move on to solving real economic problems from the Unified State Examination in mathematics.

We solve real problems from the exam in mathematics

Example #1

So the first task is:

On December 31, 2014, Alexei took a loan of 9,282,000 rubles from the bank at 10% 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, increases the debt by 10%), then Alexey transfers X rubles to the bank. What should be the amount X for Alexey to pay off the debt in four equal payments (i.e. for four years)?

So, this is a problem about a loan, so we immediately write down our formula:

We know the loan - 9,282,000 rubles.

We will deal with percentages now. We are talking about 10% of the problem. Therefore, we can translate them:

We can make an equation:

We have obtained an ordinary linear equation with respect to $x$, although with quite formidable coefficients. Let's try to solve it. First, let's find the expression $((1,1)^(4))$:

$\begin(align)& ((1,1)^(4))=((\left(((1,1)^(2)) \right))^(2)) \\& 1,1 \cdot 1,1=1,21 \\& ((1,1)^(4))=1,4641 \\\end(align)$

Now let's rewrite the equation:

\[\begin(align)& 9289000\cdot 1,4641=x\cdot \frac(1,4641-1)(0,1) \\& 9282000\cdot 1,4641=x\cdot \frac(0, 4641)(0,1)|:10000 \\& 9282000\cdot \frac(14641)(10000)=x\cdot \frac(4641)(1000) \\& \frac(9282\cdot 14641)(10) =x\cdot \frac(4641)(1000)|:\frac(4641)(1000) \\& x=\frac(9282\cdot 14641)(10)\cdot \frac(1000)(4641) \\ & x=\frac(2\cdot 14641\cdot 1000)(10) \\& x=200\cdot 14641 \\& x=2928200 \\\end(align)\]\[\]

That's it, our problem with percentages is solved.

Of course, this was only the simplest task with percentages from the Unified State Examination in mathematics. In a real exam, there will most likely not be such a task. And if it does, consider yourself very lucky. Well, for those who like to count and do not like to take risks, let's move on to the next more difficult tasks.

Example #2

On December 31, 2014, Stepan borrowed 4,004,000 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 (i.e., increases the debt by 20%), then Stepan makes a payment to the bank. Stepan paid off the entire debt in 3 equal payments. How many rubles less would he give to the bank if he could pay off the debt in 2 equal payments.

Before us is a problem about loans, so we write down our formula:

\[\]\

What do we know? First, we know the total credit. We also know the percentages. Let's find the ratio:

As for $n$, you need to carefully read the condition of the problem. That is, first we need to calculate how much he paid for three years, i.e. $n=3$, and then perform the same steps again but calculate payments for two years. Let's write an equation for the case where the payment is paid for three years:

Let's solve this equation. But first, let's find the expression $((1,2)^(3))$:

\[\begin(align)& ((1,2)^(3))=1,2\cdot ((1,2)^(2)) \\& ((1,2)^(3)) =1,44\cdot 1,2 \\& ((1,2)^(3))=1,728 \\\end(align)\]

We rewrite our expression:

\[\begin(align)& 4004000\cdot 1,728=x\cdot \frac(1,728-1)(0,2) \\& 4004000\cdot \frac(1728)(1000)=x\cdot \frac(728 )(200)|:\frac(728)(200) \\& x=\frac(4004\cdot 1728\cdot 200)(728) \\& x=\frac(4004\cdot 216\cdot 200)( 91) \\& x=44\cdot 216\cdot 200 \\& x=8800\cdot 216 \\& x=1900800 \\\end(align)\]

In total, our payment will be 1900800 rubles. However, pay attention: in the problem we were required to find not monthly payment, and how much Stepan will pay in total for three equal payments, that is, for the entire time of using the loan. Therefore, the resulting value must be multiplied by three again. Let's count:

In total, Stepan will pay 5,702,400 rubles for three equal payments. That's how much it will cost him to use the loan for three years.

Now consider the second situation, when Stepan pulled himself together, got ready and paid off the entire loan not in three, but in two equal payments. We write down our same formula:

\[\begin(align)& 4004000\cdot ((1,2)^(2))=x\cdot \frac(((1,2)^(2))-1)(1,2-1) \\& 4004000\cdot \frac(144)(100)=x\cdot \frac(11)(5)|\cdot \frac(5)(11) \\& x=\frac(40040\cdot 144\ cdot 5)(11) \\& x=3640\cdot 144\cdot 5=3640\cdot 720 \\& x=2620800 \\\end(align)\]

But that's not all, because now we have calculated only one of the two payments, so in total Stepan will pay exactly twice as much:

Great, now we are close to the final answer. But pay attention: in no case have we yet received a final answer, because for three years of payments Stepan will pay 5,702,400 rubles, and for two years of payments he will pay 5,241,600 rubles, that is, a little less. How much less? To find out, you need to subtract the second payment amount from the first payment amount:

The total final answer is 460,800 rubles. Exactly how much Stepan will save if he pays not three years, but two.

As you can see, the formula linking interest, terms, and payments greatly simplifies calculations compared to classical tables, and, unfortunately, for unknown reasons, tables are still used in most problem collections.

Separately, I would like to draw your attention to the term for which the loan was taken, and the amount of monthly payments. The fact is that this connection is not directly visible from the formulas that we wrote down, but its understanding is necessary for the quick and effective solution of real problems in the exam. In fact, this relationship is very simple: the longer the loan is taken, the smaller the amount will be in monthly payments, but the larger the amount will accumulate over the entire period of using the loan. And vice versa: the shorter the term, the greater the monthly payment, however, the total overpayment is less and less total cost loan.

Of course, all these statements will be equal only on the condition that the amount of the loan and the interest rate in both cases is the same. In general, for now, just remember this fact - it will be used to solve the most difficult problems on this topic, but for now we will analyze a simpler problem, where you just need to find the total amount of the original loan.

Example #3

So, one more task for a loan and, in combination, the last task in today's video tutorial.

On December 31, 2014, Vasily took out a certain amount from the bank on credit at 13% 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 13%), then Vasily transfers 5,107,600 rubles to the bank. What amount did Vasily borrow from the bank if he repaid the debt in two equal installments (for two years)?

So, first of all, this problem is again about loans, so we write down our wonderful formula:

Let's see what we know from the condition of the problem. First, the payment - it is equal to 5,107,600 rubles a year. Secondly, percentages, so we can find the ratio:

In addition, according to the condition of the problem, Vasily took a loan from the bank for two years, i.e. paid in two equal installments, hence $n=2$. Let's substitute everything and also note that the loan is unknown to us, i.e. the amount he took, and let's denote it as $x$. We get:

\[{{1,13}^{2}}=1,2769\]

Let's rewrite our equation with this fact in mind:

\[\begin(align)& x\cdot \frac(12769)(10000)=5107600\cdot \frac(1,2769-1)(0,13) \\& x\cdot \frac(12769)(10000 )=\frac(5107600\cdot 2769)(1300)|:\frac(12769)(10000) \\& x=\frac(51076\cdot 2769)(13)\cdot \frac(10000)(12769) \ \& x=4\cdot 213\cdot 10000 \\& x=8520000 \\\end(align)\]

That's it, this is the final answer. It was this amount that Vasily took on credit at the very beginning.

Now it is clear why in this problem we are asked to take a loan for only two years, because double-digit percentages appear here, namely 13%, which, squared, already gives a rather “brutal” number. But this is not the limit - in the next separate lesson we will consider more complex tasks, where it will be required to find the loan term, and the rate will be one, two or three percent.

In general, learn to solve problems for deposits and loans, prepare for exams and pass them "excellently". And if something is not clear in the materials of today's video lesson, then do not hesitate - write, call, and I will try to help you.

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;
- 15th of each from 1st to 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.
How much debt will be on the 15th of the 10th month if the total amount of payments after full repayment the loan will amount to 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;
- 15th of every month from 1st to nth 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 decorations of two types, provided that the cost of decorations 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 factory has assembly lines three types: A, B, C. Each of them produces two types of products. 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?

In 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)?

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 to 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 per each the new kind products 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 maximum amount can be removed by a citizen. 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 largest size a loan in 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 put the same sweets into bags so that each of them would have 5 less candies 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?

Development cost electronic version 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.

At the end of each year, the bank plans to accrue 12% per annum on the "Classic" deposit, and on 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, after 4 years of storage, the "Bonus" deposit will be more profitable than a deposit"Classic" 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 highest payment Leo 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, Andrey 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 a loan in order to monthly payments did 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 amount contributed to authorized capital contribution. 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 under different percentages. 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 firm must return the loan to the bank with interest, and at the same time counts on net profit from these types of activities from at least 13%, but not more than 21% per annum of the total loan received. What should be the smallest and largest interest rates lending to selected banks so that the firm is guaranteed to secure 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, Alexei acquired a security for 25000 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 at each of the two factories. At the first factory, 40 people work, and one worker makes 15 parts A or 5 parts B per shift. At the second plant, 160 people work, and one worker makes 5 parts A or 15 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 cans 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?

March 1, 2016 Ivan Lvovich put 20,000 rubles on 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-liter 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 contribution 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 contribution is annually replenished 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 manufacturing one part of type A, in which he could produce 2 parts of type B. 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. total area, which can be assigned to rooms is 1099 square meters. The entrepreneur can divide this area between rooms various 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 of bank account was 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.
Which minimum term should you take out a loan so that the largest annual payment on the loan 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, in 4 years) compared to the case if the loan is fully repaid in two equal installments (that is, in 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?

USE in mathematics profile level

The work consists of 19 tasks.
Part 1:
8 tasks with a short answer of the basic level of complexity.
Part 2:
4 tasks with a short answer
7 tasks with a detailed answer of a high level of complexity.

Run time - 3 hours 55 minutes.

Examples of USE assignments

Solving USE tasks in mathematics.

For a standalone solution:

1 kilowatt-hour of electricity costs 1 ruble 80 kopecks.
The electricity meter on November 1 showed 12625 kilowatt-hours, and on December 1 it showed 12802 kilowatt-hours.
How much do you need to pay for electricity in November?
Give your answer in rubles.

Problem with solution:

In a regular triangular pyramid ABCS with a base ABC, the edges are known: AB \u003d 5 roots out of 3, SC \u003d 13.
Find the angle formed by the plane of the base and the straight line passing through the midpoint of the edges AS and BC.

Solution:

1. Since SABC is a regular pyramid, then ABC is an equilateral triangle, and the remaining faces are equal isosceles triangles.
That is, all sides of the base are 5 sqrt(3), and all side edges are 13.

2. Let D be the midpoint of BC, E the midpoint of AS, SH the height from point S to the base of the pyramid, EP the height from point E to the base of the pyramid.

3. Find AD from the right triangle CAD using the Pythagorean theorem. You get 15/2 = 7.5.

4. Since the pyramid is regular, point H is the intersection point of heights / medians / bisectors of triangle ABC, which means it divides AD in a ratio of 2: 1 (AH = 2 AD).

5. Find SH from right triangle ASH. AH = AD 2/3 = 5, AS = 13, by the Pythagorean theorem SH = sqrt(13 2 -5 2) = 12.

6. Triangles AEP and ASH are both right-angled and have a common angle A, hence similar. By assumption, AE = AS/2, hence both AP = AH/2 and EP = SH/2.

7. It remains to consider the right triangle EDP (we are just interested in the angle EDP).
EP = SH/2 = 6;
DP = AD 2/3 = 5;

Angle tangent EDP = EP/DP = 6/5,
Angle EDP = arctg(6/5)

Answer:

USE 2019 in mathematics task 17 with a solution

Demo version of the Unified State Examination 2019 in mathematics

Unified State Examination in Mathematics 2019 in pdf format Basic level | Profile level

Tasks for preparing for the exam in mathematics: basic and profile level with answers and solutions.