Today in the modern world it is impossible to do without interest. Even at school, starting from grade 5, children recognize this concept and solve problems with this magnitude. Interest is found in any area of modern structures. Take, for example, banks: the size of the overpayment of a loan depends on the amount specified in the contract; profitability is also affected by the interest rate. Therefore, it is vital to know what percentage is.
Concept of interest
According to one legend, the percentage came from a silly typo. The typesetter should have set the number 100, but mixed up and set it like this: 010. This caused the first zero to rise slightly and the second to fall. The unit has become a backslash. Such manipulations served as a sign of percent. Of course, there are other legends about the origin of this quantity.
Indians knew about percent in the 5th century. In Europe, decimal fractions, with which our concept is closely interconnected, appeared after a millennium. For the first time in the Old World, the judgment of what percentage was was introduced by the scientist from Belgium, Simon Stevin. In 1584, a table of quantities was first published by the same scientist.
The word "percentage" originates in the Latin language as pro centum. If you translate the phrase, you get "from a hundred." So, the percentage refers to one hundredth of any value, number. This value is indicated by%.
Thanks to the interest, it became possible to compare parts of one whole without much difficulty. The appearance of shares has greatly simplified the calculations, so they have become so common.
Convert fractions to percent
To convert a decimal fraction to percent, you may need the so-called percent formula: the fraction is multiplied by 100,% is assigned to the result.
If you want to convert an ordinary fraction into percent, first you need to make it a decimal, and then use the above formula.
Interest transfer in fractions
As such, the percent formula is rather arbitrary. But you need to know how to translate this value into a fractional expression. To convert fractions (percent) to decimal fractions, you need to remove the% sign and divide the indicator by 100.
The formula for calculating the percentage of the number
For exams in chemistry, 30% of students received an “excellent” mark. There are a total of 40 students in the class. How many students wrote a quiz on "5"? This task clearly shows how to find out the percentage of the number.
Decision:
1) 40 x 30 = 1200.
2) 1200: 100 = 12 (students).
Answer: 12 students wrote the test paper for "5".
You can use the finished table, which shows some fractions and percentages that correspond to them.
It turns out that the formula of percent of the number is as follows: C = (A ∙ B) / 100 , where A is the original number (in the specific example, 40); In - the number of percent (in this task, B = 30%); C is the desired result.
The formula for calculating the number of percent
The next task will demonstrate what percentage is and how to find the number by percentage.
The garment factory made 1200 dresses, where 32% of them are dresses of a new style. How many dresses of the new style did the garment factory make?
Decision:
1. 1200: 100 = 12 (dresses) - 1% of all manufactured products.
2.212 x 32 = 384 (dresses).
Answer: the factory manufactured 384 dresses of a new style.
If you need to find the number by its percentage, you can use the following formula: C = (A ∙ 100) / B , where A is the total number of items (in this case, A = 1200); B - the number of percent (in a specific task, B = 32%); C is the desired value.
Increase, decrease the number by a specified number of percent
Schoolchildren must learn what percentages are, how to count them and solve various problems. To do this, you need to understand how the number increases or decreases by N%.
Often tasks are given, and in life you need to find out what the number will be equal to, increased by a given number of percent. For example, the number X is given. You need to find out what the value of X will be if you increase it, say, by 40%. First you need to convert 40% to a fractional number (40/100). So, the result of increasing the number X will be: X + 40% ∙ X = ( 1 + 40/100) ∙ X = 1.4 ∙ X. If you substitute any number instead of X, take, for example, 100, then the whole expression will be equal : 1.4 ∙ X = 1.4 ∙ 100 = 140.
Approximately the same principle is used when reducing the number by a given number of percent. It is necessary to carry out calculations: X - X ∙ 40% = X ∙ ( 1-40 / 100 ) = 0.6 ∙ X. If the value is 100, then 0.6 ∙ X = 0.6 . 100 = 60.
There are tasks where you need to find out how many percent the number has increased.
For example, the task is given: The driver was driving along one section of the track at a speed of 80 km / h. In another section, the train speed increased to 100 km / h. How many percent did train speed increase?
Decision:
Suppose 80 km / h is 100%. Then we make the calculations: (100% ∙ 100 km / h) / 80 km / h = 1000: 8 = 125%. It turns out that 100 km / h is 125%. To find out how much speed has increased, you need to calculate: 125% - 100% = 25%.
Answer: train speed in the second section increased by 25%.
Proportion
There are frequent cases when it is necessary to solve the problem of interest, using the proportion. In fact, this method of finding the result greatly facilitates the task of students, teachers and more.
So what is proportion? This term refers to the equality of two relations, which can be expressed as follows: A / B = C / D.
In the textbooks of mathematics, the following rule appears: the product of the extreme members equals the product of the average. This is expressed by the following formula: A x D = B x C.
Thanks to this formulation, any number can be calculated if the other three members of the proportion are known. For example, A is an unknown number. To find him, you need
When solving problems by the method of proportion, it is necessary to understand from what number to take percent. There are times when shares need to be taken from different values. Compare:
1. After the sale in the store, the cost of a T-shirt increased by 25% and amounted to 200 rubles. What was the cost at the time of sale.
Decision:
In this case, you need a value of 200 rubles corresponds to 125% of the initial (sale) price of a T-shirt. Then, to find out its cost during the sale, you need (200 x 100): 125. You get 160 rubles.
2. On the planet Vicenza, 200,000 inhabitants: people and representatives of the humanoid race of Navi. Navi account for 80% of the entire population of Vicenza. Of the people, 40% are engaged in the maintenance of the mine, the rest mine tetanium. How many people mine tetanium?
Decision:
First of all, you need to find in numerical form the number of people and the number of Naavi. So, 80% of 200,000 will be equal to 160,000. So many representatives of the humanoid race live on Vicenza. The number of people, respectively, is 40,000. Of these, 40%, that is 16,000, serve the mine. So, 24,000 people are mining tetanium.
Repeatedly change the number by a certain percentage
When it is already clear what percentage is, you need to study the concept of absolute and relative change. Absolute conversion refers to an increase in a number by a specific number. So, X increased by 100. No matter what you substitute for X, this number will still increase by 100: 15 + 100; 99.9 + 100; a + 100, etc.
By relative change is meant an increase in the value by a certain number of percent. Let's say X increased by 20%. This means that X will be equal to: X + X ∙ 20%. A relative change is implied every time it comes to an increase of half or a third, a decrease of a quarter, an increase of 15%, etc.
There is another important point: if the value of X is increased by 20%, and then another 20%, then as a result the total increase will be 44%, but not 40%. This can be seen from the following calculations:
1.X + 20% ∙ X = 1.2 ∙ X
2.2.2 ∙ X + 20% ∙ 1.2 ∙ X = 1.2 ∙ X + 0.24 ∙ X = 1.44 ∙ X
This shows that X increased by 44%.
Interest task examples
1. How many percent of number 36 is number 9?
Decision:
According to the formula for finding the percentage of the number, you need 9 times 100 and divide by 36.
Answer: the number 9 is 25% of 36.
2. Calculate the number C, which is 10% of 40.
Decision:
According to the formula for finding the number by its percentage, you need to multiply 40 by 10 and divide the result by 100.
Answer: the number 4 is 10% of 40.
3. The first partner invested 4,500 rubles in the business, the second - 3,500 rubles, the third - 2,000 rubles. They made a profit of 2400 rubles. They divided the profit equally. How much in rubles did the first partner lose, compared to how much he would have gained if they had divided the income according to the percentage of invested funds?
Decision:
So, together they invested 10,000 rubles. The income for each amounted to an equal share of 800 rubles. To find out how much the first partner should have received and how much he, accordingly, lost, you need to know the percentage of invested funds. Then you need to find out how much profit in rubles is this contribution. And the last thing is to subtract 800 rubles from the result.
Answer: the first partner lost 280 rubles during the profit sharing.
A bit of economics
Today, a fairly popular question is how to get a loan for a certain period. But how to choose a profitable loan so as not to overpay? First, you need to look at the interest rate. It is desirable that this indicator be as low as possible. Then apply the formula for calculating interest on the loan.
As a rule, the amount of the overpayment is affected by the amount of the debt, interest rate and method of repayment. Distinguish between annuity and differentiated payments. In the first case, the loan is repaid in equal installments every month. Immediately, the amount that covers the main loan grows, and the cost of interest gradually decreases. In the second case, the borrower pays constant amounts to repay the loan, to which interest on the balance of the principal debt is added. Monthly, the total amount of payments will be reduced.
Now you need to consider both ways of repaying the loan. So, with the annuity option, the amount of the overpayment will be higher, and with the differential one, the amount of the first payments. Naturally, the loan conditions are the same for both cases.
Conclusion
So, interest. How to count them? Simple enough. However, sometimes they can cause difficulties. This topic is beginning to be studied at school, but it catches up with everyone in the field of loans, deposits, taxes, etc. Therefore, it is advisable to understand the essence of this issue. If you still can’t make the calculations, there are lots of online calculators that will help you cope with the task.