Today, in the modern world, it is impossible to do without interest. Even at school, starting from the 5th grade, children learn this concept and solve problems with this value. Interest is found in every area of modern structures. Take, for example, banks: the amount of overpayment of the loan depends on the amount specified in the contract; The interest rate also affects the size of the profit. Therefore, it is vital to know what a percentage is.
The concept of interest
According to one legend, the percentage appeared due to a silly typo. The compositor was supposed to set the number 100, but mixed it up and put 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 led to the appearance of the percent sign. Of course, there are other legends about the origin of this value.
Indians knew about percentages in the 5th century. In Europe, decimal fractions, withwhich our concept is closely interconnected, appeared after a millennium. For the first time in the Old World, the judgment of what a percentage is was introduced by a scientist from Belgium, Simon Stevin. In 1584, a table of magnitudes was first published by the same scientist.
The word "percentage" originates in Latin as pro centum. If you translate the phrase, you get "from a hundred." So, a percentage is understood as one hundredth of a value, a number. This value is denoted by the sign %.
Thanks to percentages, it became possible to compare parts of one whole without much difficulty. The introduction of shares has greatly simplified the calculations, which is why they have become so common.
Conversion of fractions to percentages
To convert a decimal fraction to a percentage, you may need the so-called percent formula: the fraction is multiplied by 100, %.
If you need to convert a fraction to a percentage, first you need to make it decimal, and then use the above formula.
Conversion of percentages to fractions
As such, the percentage formula is rather conditional. But you need to know how to convert this value into a fractional expression. To convert shares (percentages) to decimal fractions, you need to remove the% sign and divide the indicator by 100.
Formula for calculating the percentage of a number
For the test in chemistry, 30% of students received an "excellent" grade. There are 40 students in the class in total. How muchstudents wrote a test on the "5"? This task clearly shows how to find out the percentage of a number.
Solution:
1) 40 x 30=1200.
2) 1200: 100=12 (students).
Answer: 12 students wrote the test for "5".
You can use the ready-made table, which shows some fractions and percentages that correspond to them.
It turns out that the formula for percent of a number looks like this: C=(A∙B)/100, where A is the original number (in a specific example equal to 40); B - the number of percent (in this problem, B=30%); С – desired result.
Formula for calculating a number from a percentage
The following problem will demonstrate what a percentage is and how to find a number from a percentage.
The garment factory made 1200 dresses, where 32% of them are new-style dresses. How many new-style dresses did the clothing factory make?
Solution:
1. 1200: 100=12 (dresses) - 1% of all items released.
2. 12 x 32=384 (dresses).
Answer: the factory made 384 new style dresses.
If you need to find a number by its percentage, you can use the following formula: C=(A∙100)/B, where A - 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 givenpercentages
Students should 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 percentage. For example, given the number X. You need to find out what the value of X will be if it is increased, 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 we substitute any number instead of X, let's take, for example, 100, then the whole expression will be equal to: 1, 4 ∙ X=1, 4 ∙ 100=140.
Approximately the same principle is used when decreasing a number by a given percentage. 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 by what percentage the number has increased.
For example, given the task: The driver was driving along one section of the track at a speed of 80 km/h. On another section, the speed of the train increased to 100 km/h. By what percent did the speed of the train increase?
Solution:
Assume 80 km/h is 100%. Then we make 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 the speed has increased, you need to calculate: 125% - 100%=25%.
Answer: the speed of the train on the second section has increased by 25%.
Proportion
There are often cases when it is necessary to solve percentage problems using a proportion. In fact, this method of finding the result greatly facilitates the task for students, teachers and not only.
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 there is such a rule: the product of the extreme terms is equal to the product of the average ones. 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 terms of the proportion are known. For example, A is an unknown number. To find him, you need to
When solving problems using the proportion method, you need to understand what number to take percentages from. There are times when shares need to be taken from different values. Compare:
1. After the end of the sale in the store, the cost of the T-shirt increased by 25% and amounted to 200 rubles. What was the price during the sale.
Solution:
In this case, the value of 200 rubles 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. You get 160 rubles.
2. There are 200,000 inhabitants on the planet Vitsencia: people and representatives of the humanoid race Naavi. Naavi make up 80% of the total populationVicencii. Of the people, 40% are employed in the maintenance of the mine, the rest are mined for tetanium. How many people are mining tetanium?
Solution:
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 equal 160,000. So many representatives of the humanoid race live on Vicencia. 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.
Repeated change of a number by a certain percentage
When you already understand what a percentage is, you need to study the concept of absolute and relative change. An absolute transformation is understood as an increase in a number by a specific number. Thus, X has increased by 100. Whatever one substitutes for X, this number will still increase by 100: 15 + 100; 99, 9 + 100; a + 100 etc.
A relative change is understood as an increase in a value by a certain number of percent. Let's say X has increased by 20%. This means that X will be equal to: X + X ∙ 20%. Relative change is implied whenever it comes to increasing by half or a third, decreasing by a quarter, increasing by 15%, etc.
There is another important point: if the value of X 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. X + 20% ∙ X=1, 2 ∙ X
2. 1, 2 ∙ X + 20% ∙ 1, 2 ∙ X=1, 2 ∙ X + 0, 24 ∙ X=1, 44 ∙ X
It showsthat X increased by 44%.
Examples of interest problems
1. What percentage of 36 is 9?
Solution:
According to the formula for finding the percentage of a number, you need to multiply 9 by 100 and divide by 36.
Answer: 9 is 25% of 36.
2. Calculate the number C, which is 10% of 40.
Solution:
According to the formula for finding a number by its percentage, you need to multiply 40 by 10 and divide the result by 100.
Answer: 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 shared the profits equally. How much did the first partner lose in rubles, compared to how much he would have received if they divided the income according to the percentage of invested funds?
Solution:
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 lost, respectively, you need to find out the percentage of invested funds. Then you need to find out how much profit this contribution makes in rubles. And the last thing is to subtract 800 rubles from the result.
Answer: the first partner lost 280 rubles when sharing profits.
A bit of economy
Today, a rather popular question is obtaining a loan for a certain period. But how to choose a profitable loan so as not to overpay? First, you need to lookinterest rate. It is desirable that this indicator be as low as possible. Then you should apply the formula for calculating the interest on the loan.
As a rule, the size of the overpayment is affected by the amount of debt, the interest rate and the 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 grows, and the cost of interest gradually decreases. In the second case, the borrower pays fixed amounts to repay the loan, to which interest is added on the balance of the principal debt. Monthly, the total amount of payments will decrease.
Now we 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 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 be problematic. This topic begins to be studied at school, but it catches up with everyone in the field of loans, deposits, taxes, etc. Therefore, it is advisable to delve into the essence of this issue. If you still cannot make the calculations, there are a lot of online calculators that will help you cope with the task.
