What is the percentage? Interest formula. Interest - how to count?

Today in the modern world, no interest can not be dispensed with. Even in school, starting from the 5th grade, children learn this concept and solve problems with this magnitude. Interest is found in any area of modern structures. Take, for example, banks: the amount of overpayment of a loan depends on the amount specified in the contract; The profit rate is also affected by the interest rate. Therefore, it is vital to know what percentage is.

The concept of interest

According to one legend, the percentage appeared due to a stupid typing error. The setter had to set the number 100, but confused and put it like this: 010. This caused the first zero to rise slightly, and the second dropped. The unit turned into a backslash. Such manipulations served to the fact that a percent sign appeared. Of course, there are other legends about the origin of this magnitude.

About the Indians knew as far back as the 5th century. In Europe, the same decimals, with which our concept is closely interrelated, appeared after a millennium. For the first time in the Old World, a scientist from Belgium Simon Stevin introduced a judgment on what percentage is. In 1584 the table of values was published for the first time by the same scientist.

The word "percent" originates in Latin as a pro centum. If you translate the phrase, you get "one hundred." So, by interest is meant one hundredth part of any quantity, number. This quantity is denoted by%.

Thanks to interest, it became possible to compare parts of one whole without much difficulty. The appearance of shares greatly simplified the calculations, so they became so common.

Fraction conversion as a percentage

To convert a decimal into percentages, you may need the so-called interest formula: the fraction is multiplied by 100, the result is assigned to%.

If you want to convert to a percentage of ordinary fraction, it must first be made decimal, and then use the above formula.

Transfer of interest in fractions

As such, the interest formula is relatively arbitrary. But you need to know how to translate this value into a fractional expression. To convert percentages to decimals, you need to remove the% sign and divide the indicator by 100.

The formula for calculating the percentage of the number

For the control work on chemistry, an "excellent" score was received by 30% of the students. There are 40 pupils in the class. How many students wrote a test paper for "5"? This task clearly shows how to know the percentage of a number.

Decision:

1) 40 x 30 = 1200.

2) 1200: 100 = 12 (students).

Answer: 12 students wrote the test work for "5".

You can use the finished table, which indicates some fractions and percentages that correspond to them.

It turns out that the percent formula of the number looks like this: C = ^{(A ∙ B)} / ₁₀₀ , where A - the original number (in the specific example, equal to 40); B is the number of percentages (in this task, B = 30%); C is the desired result.

The formula for calculating the number of percentages

The next task will show what percentage is and how to find the number by percentage.

The garment factory produced 1200 dresses, of which 32% are dresses of a new style. How many dresses of a new style did the garment factory produce?

Decision:

1. 1200: 100 = 12 (dresses) - 1% of all manufactured items.

2. 12 x 32 = 384 (dresses).

The answer: the factory has made 384 dresses of a new style.

If you want 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 is the number of percentages (in the specific task B = 32%); C is the desired quantity.

Increase, decrease the number by a specified number of percentages

Schoolchildren must learn what percentages are, how to count them and solve various tasks. To do this, you need to understand how the number increases or decreases by N%.

Often given tasks, and in life you need to know what the number, increased by a specified number of percent, will be equal to. For example, the number X is given. It is necessary to find out what the value of X will be if it is increased, say, by 40%. First you need to translate 40% into a fractional number (40/100). So, the result of increasing the number of X becomes: X + 40% ∙ X = ( ^{1 +} 40/100) ∙ X = 1,4 ∙ X. If instead of X substitute any number, take, for example, 100, then the whole expression will be : 1.4 ∙ X = 1.4 ∙ 100 = 140.

Approximately the same principle is used and with decreasing the number by a specified number of percentages. It is necessary to make the 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 know how many percent the number has increased.

For example, given the task: The driver was traveling on one section of the road at a speed of 80 km / h. In another section, the speed of the train increased to 100 km / h. How many percent increased the speed of the train?

Decision:

Suppose, 80 km / h - 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: The speed of the train increased by 25% in the second section.

Proportion

It is not uncommon for cases when it is necessary to solve the problem of interest using a proportion. In fact, this method of finding the result greatly facilitates the task for students, teachers and not only.

So, what is the proportion? By this term we mean the equality of two relations, which can be expressed as follows: ^A / _B = ^C / _D.

In the textbooks of mathematics there is a rule: the product of extreme members is equal to the product of means. This is expressed by the following formula: A x D = B x C.

Due to this formulation, it is possible to calculate any number if the other three terms of the proportion are known. For example, A is an unknown number. To find it, you need

When solving problems by the method of proportion, it is necessary to understand from what number to take interest. There are cases when the shares need to be taken from different values. Compare:

1. After the end of the sale in the store, the cost of a T-shirt increased by 25% and amounted to 200 rubles. What was the cost during the sale.

Decision:

In this case you need a value of 200 rubles that corresponds to 125% of the original (sale) price of the T-shirt. Then, to find out its value during the sale, you need (200 x 100): 125. It will be 160 rubles.

2. On the planet Vicence 200 000 inhabitants: people and representatives of the humanoid race Naavi. Naavi accounts for 80% of the total population of Vicenza. Of the people, 40% are engaged in servicing the mine, others are mining tetanium. How many people get 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 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 engaged in the production of tetanium.

Multiple number change by a certain number of percentages

When it is already clear that such a percentage, you need to study the concept of absolute and relative change. An absolute transformation is an increase in the number by a specific number. So, X increased by 100. What would be substituted instead of X, all the same this number will increase by 100: 15 + 100; 99.9 + 100; A + 100, and so on.

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: X + X ∙ 20%. A relative change is implied every time it comes to increasing by half or a third, decreasing by a quarter, increasing by 15%, and so on.

There is one more important point: if the X value is increased by 20%, and then by another 20%, then the total increase will be 44%, but not 40%. This can be seen from the following calculations:

1. Х + 20% ∙ Х = 1,2 ∙ Х

2. 1.2 ∙ X + 20% ∙ 1.2 ∙ X = 1.2 ∙ X + 0.24 ∙ X = 1.44 ∙ X

This shows that X increased by 44%.

Examples of tasks for interest

1. How many percent of the number 36 is the number 9?

Decision:

By the formula for finding a percentage of the number, you need to multiply 9 by 100 and divide by 36.

Answer: the number 9 is 25% of 36.

2. Calculate the number C, which is 10% of 40.

Decision:

By the formula for finding the number by its percentage, you need to 40 multiply 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 partner - 3,500 rubles, the third - 2000 rubles. They received a profit of 2400 rubles. They shared the profit equally. How much did the first partner lose in rubles, compared to how much he would receive if they divided the income according to the percentage of the invested funds?

Decision:

So, together they invested 10 000 rubles. The income for each accounted for an equal share of 800 rubles. To find out how much the first partner should have received and how much he, respectively, lost, you need to find out the percentage of funds invested. Then you need to find out how much this contribution makes in rubles. And the last - subtract 800 rubles from the result.

Answer: the first partner lost 280 rubles when dividing the profit.

A bit of economics

Today, a rather popular issue is the issue of 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 should be as low as possible. Then you should apply the formula for calculating interest on the loan.

Typically, the amount of overpayment affects the amount of debt, interest rate and method of repayment. There are 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 increases, and the cost of interest gradually decreases. In the second case, the lending borrower pays permanent amounts to repay the loan, to which interest is added to the balance of the principal. Monthly total payments will be reduced.

Now we need to consider both ways of repaying the loan. So, with an annuity option, the amount of overpayment will be higher, and for a differential option, the amount of the first payments. Naturally, the terms of the loan 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 in school, but it overtakes everyone in the field of loans, deposits, taxes, etc. Therefore, it is advisable to get to the heart of this issue. If you still can not make the calculations, there are a lot of online calculators that will help to cope with the task.