How to solve percentage problems. Examples. All about percentages. Such an understandable theory. Analysis of tasks Increase, decrease the number by a given percentage
1% is a hundredth of a number.1% = 0,01.
Finding percentages of a number.
To find a percentage of a number, you can express the percentage as a decimal fraction and multiply the number by the resulting decimal fraction.
Finding a number by its percentage.
To find a number by its percentage, you can represent the percentage as a decimal fraction and divide this number by the resulting decimal fraction.
To find how many percent one number is from another, you can divide one number by another and multiply the resulting product by 100.
How to solve percentage problems. Examples.
Finding a percentage of a number is related to finding a fraction of a number. Interest is a special way of writing an ordinary fraction, so you should begin to reveal the meaning of the concept of interest from understanding the concept of an ordinary fraction.
Let's take a few common fractions, for example. What is the meaning of each such entry?
These are examples of regular fractions. The denominator of each of them shows how many equal parts a certain real or abstract object needs to be divided into, the numerator shows how many such parts need to be taken. Let's take a regular fraction as an example. For example. The meaning of this expression can be revealed as follows. Some real object was divided into 3 equal parts and 2 parts were taken from them.
As a real object, you can take, for example, a rectangle.
This expression is the quotient of a and b, where b is not equal to 0.
This is the ratio of the numbers a and b, where b is not equal to 0.
This is an ordinary fraction. a is the numerator, b is the denominator (b is not equal to 0).
Example 1 The capacity of the barrel was 200 liters. The barrels were filled with water. What is the meaning of this proposal?
- this fraction means that a certain object was divided into 5 equal parts and 2 parts were taken from them. The object in this problem is the volume of the barrel equal to 200 liters, therefore,
200:5 = 40,
402 = 80.
80 liters of water were poured into a barrel.
The above example is a typical example of finding a fraction of a number.
To find a fraction of a number, you need to multiply the number by that fraction.
Now we can move on to percentages.
The concept of percentage is defined as follows: 1% of a number is a hundredth of a number, i.e. 1% \u003d 0.01.
Then the meaning of the sentence a% of number b can be explained like this. Some object (the value of which is equal to b units) divided into 100 equal parts and taken from them a parts.
Example 2 Masha had 400 rubles. She spent 24% of this amount. What is the meaning of this saying?
Since 24% \u003d 0.24, and 0.24 means that a certain object was divided into 100 equal parts and 24 parts were taken from them. In this case, the object is the amount of money equal to 400 rubles, therefore,
400: 100 =4,
424 = 96.
Masha spent 96 rubles.
The above example is a typical example of finding percentages of a number.
Example 3 Need to find R%
from number b
.
Let x be the number we need to find.
p%
= 0,01p,
x = b
0,01p
To find percentages of a number, you need to represent the number of percents as a decimal fraction and multiply the given number by this decimal fraction.
Another approach to this problem. You can use the concept and properties of proportion. If we recall that proportion is the equality of two ratios, and the ratio of two numbers is an ordinary fraction, then this method is also associated with the concept of an ordinary fraction.
b - 100%,
x - p%,
We have a proportion:
b: 100 = x: p, (b is to 100 as x is to p) whence,
Example 4 Let there be numbers a And b , moreover, a >b Then the number a more number b on %.
Let's approach this problem a little differently. We will consider a simple special case, for example, this: "How many percent is the number 10 greater than the number 2?".
1. Subtract the smaller number from the larger number. 10 - 2 = 8. Then 10 is greater than 2 by 8.
2. Find the ratio of the found number to a smaller number. 8:2=4 is the ratio of two numbers!
3 We express the ratio as a percentage 4100 = 400%.
The number 10 is greater than the number 2 by 400%.
If we divide 8 by 10, we will find a ratio showing how much of 10 2 is less than 10 (here the comparison is with the number 10.
The number 2 is less than the number 10 by 80%.
Example 5 The tractor driver plowed 6 hectares, which is from the entire field. What is the area of the entire field.
This is a typical problem of finding a number by its fraction. Let the area of the entire field be x,
then we have the equation x= 6. Whence x = 6:; x = 26. Field area is 26 ha.
To find a number by its fraction, you need to divide the number corresponding to the given fraction by the fraction.
Example 6 . Given a number b, which is p% from number a. Find a number A.
p%
= 0,01p
b
= 0,01pa
a = b: (0.01p)
Given a number b , which is p% from number a .
Find a number A .
a - 100%
b-p%
a:100 = b:p
Compound interest formula.
If the deposit has an amount a monetary units, and the bank charges R%
per annum, then through n
years, the amount on the deposit will be monetary units, or
a(1+0.01p)n
monetary units.
Example 7 The construction of the house cost 9,800 rubles, of which 35% was paid for the work, and the rest was paid for the material. How much did the materials cost?
Paid for work:
0,359800 = 3430.
Therefore, the materials cost: 9800 - 3430 = 6370.
Answer: 6370 rubles.
Example 8 37.4 tons of gasoline were poured into the tank, after which 6.5% of the tank's capacity remained unfilled. How much gasoline must be added to the tank to fill it?
If the unfilled part of the tank is 6.5% of the capacity, then the filled part is: 100% - 6.5% = 93.5%. Then, if x is the mass of gasoline that remains to be added to the tank, then we have the proportion
where 
.
Answer: 2.6 tons.
Example 9 Find a number knowing that 25% of it is 45% of 640.
Let x be the desired number. We have
0.25x = 0.45640.
Answer: 1152.
Example 10 The number a is 92% of the number b. If the number b is increased by 700, then the new number will be 9% greater than the number a. Find numbers a and b.
From the condition of the problem we have a system of equations:
Solving the resulting system, we find, a = 230000, b = 250000.
Answer: 230000; 250000.
Example 11. The first number is 50% of the second. What percentage of the first is the second?
Let's denote the second number by x, then the first number is equal to 0.5x. To find out what percentage is the number x of the number 0.5x; Let's make a proportion:
from which we find
Answer: 200%.
Example 12. There are 260 students in the lyceum, of which 10% fail. After the expulsion of a certain number of poor performers, their percentage dropped to 6.4%. How many students dropped out?
Before expulsion, the number of underachievers before expulsion was solo
Let x people be expelled. Then, in total, 260 students remained in the lyceum, of which 26 were unsuccessful. We have a proportion
260 - x - 100%,
(260 - x)0.064=(26 - x)100,
Solving the resulting equation, we find x = 10.
Example 13 By what percentage is 250 greater than 200?
Let's do two things.
1) We find out how many percent is the number 250 tons of the number 200:
2) Since the number 200 in this example is 100%, then the number 250 is greater than the number 200 by 125% -100% = 25%.
Answer: 25%.
Example 14 What percentage is 200 less than 250?
1) Find out how many percent is the number 200 of the number 250 (unlike the previous example, here you need to take the number 250 as 100%!):
2) The number 200 is less than the number 250 by 100% - 80% = 20%.
Answer: 20%.
Example 15 The length of the brick was increased by 30%, the width by 20%, and the height was reduced by 40%. Did the volume of bricks increase or decrease from this and by what percentage?
Let the original length of the brick be x, width - y, height - z. Then the initial volume of the brick: V 1 = xyz. New brick sizes: 1.3x; 1.2y; 0.6z and new volume: V 2 \u003d 1.3x1.2y0.6z \u003d 0.936xyz. Since V 2< V 1 , объем кирпича уменьшился. Уменьшение V 2 - V 1 = 0,064xyz и составляет 6,4% от V 1.
Answer: decreased by 6.4%.
Example 16 The price of a commodity went down by 40%, then another 25%. By what percentage has the price of the product decreased from its original price?
Let x be the original price of the product. After the first decrease, the price will be equal to
x - 0, 4x = 0.6x.
The second price decrease is 25% of the new price of 0.6x, so after the second decrease we will have the price
0.6x - 0.250.6x = 0.45x;.
After two declines, the total price change is:
x - 0.45x = 0.55x.
Since the value is 0.55x; is 55% of x, then the price of the good has decreased by 55%.
Answer: 55%.
Example 17. The initial cost of a unit of production was 75 rubles. During the first year of production, it increased by a certain number of percent, and during the second year it decreased (in relation to the increased value) by the same number of percent, as a result of which it became equal to 72 rubles. Determine the percentage increase and decrease in the cost of a unit of production.
Let x% be the percentage increase (and decrease) in the cost of a unit of output. By definition, x% of 75 is 750.01x. Then after the first increase the price will be equal to 75 + 0.75x.
During the second year, the price will decrease by
0.01x(75+0.75x) = 0.75x + 0.0075x2.
Now we can write the equation for the final price
(75 + 0.75x) - (0.75x + 0.0075x 2) = 72;
x 2 \u003d 400; hence x 1 = - 20, x 2 = 20.
Only one root of this equation is suitable: x 2 \u003d 20.
Answer: 20%.
Example 18. 10 thousand rubles were deposited in the bank account. After the money lay for one year, 1 thousand rubles were withdrawn from the account. A year later, the account was 11 thousand rubles. Determine what percentage per annum the bank charges.
Let the bank charge p% per annum.
1) The amount of 10,000 rubles, deposited in a bank account at p% per annum, in a year will increase to the value
10000 + 0.01p10000 = 10000 + 100 rub.
When 1000 rubles are withdrawn from the account, 9000 + 100 rubles will remain there.
2) In another year, the latter value will increase to 9000 + 100r + 0.01p (9000 + 100r) = r 2 + 190r + 9000 rubles due to the accrual of interest.
By condition, this value is equal to 11,000 rubles, so we have a quadratic equation.
p 2 + 190r + 9000 = 11000;
r 2 + 190r - 2000 = 0
, we solve this quadratic equation using Viette's theorem, p 1 \u003d 10, p 2 \u003d -200.
The negative root is not suitable.
Answer: 10%.
Example 19. The city currently has 48,400 inhabitants. It is known that the population of this city increases annually by 10%. How many inhabitants were there in the city two years ago?
Suppose that two years ago the number of inhabitants of the city was x people, then the number of inhabitants is currently expressed through x using the compound interest formula:
x(1+0.1) 2 = 1.21x.
From the problem statement:
Answer: 40,000 people.
Percentage is one of the interesting and frequently used tools in practice. Interest is partially or fully applied in any science, in any job, and even in everyday communication. A person who is well versed in percentages gives the impression of being intelligent and educated. In this lesson, we will learn what percentage is and what actions you can perform with it.
Lesson contentWhat is a percentage?
In everyday life, fractions are the most common. They even got their own names: half, third and quarter, respectively.
But there is another fraction that also occurs frequently. This is a fraction (one hundredth). This fraction is called percent. What does one hundredth mean? This fraction means that something is divided into one hundred parts and one part is taken from there. So a percentage is one hundredth of something.
A percent is one hundredth of something
For example, from one meter is 1 cm. One meter was divided into one hundred parts, and one part was taken (remember that 1 meter is 100 cm). And one part of these hundred parts is 1 cm. So one percent of one meter is 1 cm.
From one meter is already 2 centimeters. This time, one meter was divided into one hundred parts and not one, but two parts were taken from there. And two parts out of a hundred are two centimeters. So two percent of one meter is 2 centimeters.
Another example, from one ruble is one penny. The ruble was divided into one hundred parts, and one part was taken from there. And one part of these hundred parts is one penny. So one percent of one ruble is one penny.
Percentages were so common that people replaced the fraction with a special icon that looks like this:
This entry reads "one percent". It replaces the fraction. It also replaces the decimal 0.01 because if we convert the common fraction to a decimal, we get 0.01. Therefore, between these three expressions, you can put an equal sign:
1% = = 0,01
Two percent in fractional form would be written as , in decimal form as 0.02 and with a special sign, two percent would be written as 2%.
2% = = 0,02
How to find percentage?
The principle of finding a percentage is the same as the usual finding of a fraction of a number. To find the percentage of something, you need to divide it into 100 parts and multiply the resulting number by the desired percentage.
For example, find 2% of 10 cm.
What does 2% mean? The entry 2% replaces the entry . If we translate this task into a more understandable language, then it will look like this:
Find from 10 cm
And we already know how to solve such tasks. This is the usual finding of a fraction of a number. To find a fraction of a number, you need to divide this number by the denominator of the fraction, and multiply the result by the numerator of the fraction.
So, we divide the number 10 by the denominator of the fraction
Got 0.1. Now we multiply 0.1 by the numerator of the fraction
0.1 x 2 = 0.2
We got the answer 0.2. So 2% of 10 cm is 0.2 cm. And if, then we get 2 millimeters:
0.2cm=2mm
So 2% of 10 cm is 2 mm.
Example 2 Find 50% of 300 rubles.
To find 50% of 300 rubles, you need to divide these 300 rubles by 100, and multiply the result by 50.
So, we divide 300 rubles by 100
300: 100 = 3
Now multiply the result by 50
3 × 50 = 150 rubles
So 50% of 300 rubles is 150 rubles.
If at first it is difficult to get used to the notation with the% sign, you can replace this notation with a regular fractional notation.
For example, the same 50% can be replaced with the entry. Then the task will look like this: Find from 300 rubles, and it is still easier for us to solve such problems
300: 100 = 3
3 x 50 = 150
In principle, there is nothing complicated here. If difficulties arise, we advise you to stop and re-examine and.
Example 3 The garment factory produced 1200 suits. Of these, 32% are suits of a new style. How many suits of the new style did the factory produce?
Here you need to find 32% of 1200. The number found will be the answer to the problem. Let's use the percentage rule. Divide 1200 by 100 and multiply the result by the desired percentage, i.e. at 32
1200: 100 = 12
12 x 32 = 384
Answer: 384 suits of the new style were produced by the factory.
The second way to find the percentage
The second way to find the percentage is much simpler and more convenient. It lies in the fact that the number from which the percentage is sought will immediately be multiplied by the desired percentage, expressed as a decimal fraction.
For example, let's solve the previous problem in this way. Find 50% of 300 rubles.
The entry 50% replaces the entry, and if we translate these into a decimal fraction, we get 0.5
Now, to find 50% of 300, it will be enough to multiply the number 300 by the decimal fraction 0.5
300 x 0.5 = 150
By the way, the mechanism for finding the percentage on calculators works on the same principle. To find a percentage using a calculator, you need to enter the number from which the percentage is being searched into the calculator, then press the multiplication key and enter the percentage you are looking for. Then press the percentage key
Finding a number by its percentage
Knowing the percentage of a number, you can find out the whole number. For example, an enterprise paid us 60,000 rubles for work, and this is 2% of the total profit received by the enterprise. Knowing our share, and what percentage it is, we can find out the total profit.
First you need to find out how many rubles is one percent. How to do it? Try to guess by carefully studying the following figure:
If two percent of the total profit is 60 thousand rubles, then it is easy to guess that one percent is 30 thousand rubles. And to get these 30 thousand rubles, you need to divide 60 thousand by 2
60 000: 2 = 30 000
We found one percent of the total profit, i.e. . If one part is 30 thousand, then to determine one hundred parts, you need to multiply 30 thousand by 100
30,000 × 100 = 3,000,000
We found the total profit. It is three million.
Let's try to form a rule for finding a number by its percentage.
To find a number by its percentage, you need to divide the known number by the given percentage, and multiply the result by 100.
Example 2 The number 35 is 7% of some unknown number. Find this unknown number.
Read the first part of the rule:
To find a number by its percentage, you need to divide the known number by the given percentage.
Our known number is 35, and the given percentage is 7. Divide 35 by 7
35: 7 = 5
Read the second part of the rule:
and multiply the result by 100
Our result is the number 5. Multiply 5 by 100
5 x 100 = 500
500 is the unknown number that was required to be found. You can do a check. To do this, we find 7% of 500. If we did everything right, we should get 35
500: 100 = 5
5 x 7 = 35
We got 35. So the problem was solved correctly.
The principle of finding a number by its percentage is the same as the usual finding of an integer by its fraction. If percentages are confusing and confusing at first, then the percentage entry can be replaced with a fractional entry.
For example, the previous problem can be stated as follows: the number 35 is from some unknown number. Find this unknown number. We already know how to solve such problems. This is finding a number from a fraction. To find a number from a fraction, we divide this number by the numerator of the fraction and multiply the result by the denominator of the fraction. In our example, the number 35 must be divided by 7 and the result multiplied by 100
35: 7 = 5
5 x 100 = 500
In the future, we will solve percentage problems, some of which will be difficult. In order not to complicate learning at first, it is enough to be able to find the percentage of the number, and the number by percentage.
Tasks for independent solution
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Anonymous The number A is 56% less than the number B, which is 2.2 times less than the number C. What is the percentage of the number C relative to the number A? NMitra A = B - 0.56 ⋅ B = B ⋅ (1 - 0.56) = 0.44 ⋅ B B = A: 0.44 C = 2.2 ⋅ B = 2.2 ⋅ A: 0.44 = 5 ⋅ A C 5 times more A C 400% more A Anonymous Help. In 2001, revenue increased by 2 percent compared to 2000, although it was planned to double. By what percentage is the plan underfulfilled? NMitra A - 2000 B - 2001 B = A + 0.02A = A ⋅ (1 + 0.02) = 1.02 ⋅ A B = 2 ⋅ A (plan) 2 - 100% 1.02 - x% x = 1.02 ⋅ 100: 2 = 51% (target met) 100 - 51 = 49% (target not met) Anonymous Help answer the question. Watermelon contains 99% moisture, but after drying (put in the sun for a few days), its moisture content is 98%. By what % will the WEIGHT of the watermelon change after drying? If you calculate mathematically, it turns out that my watermelon has completely dried up. For example: with a weight of 20 kg, water is 99% of the mass, that is, the dry weight is 1% \u003d 0.2 kg. Here the watermelon loses liquid, and is already 98%, therefore, the dry weight is 2%. But dry weight cannot change due to water loss, so it is still 0.2 kg. 2%=0.2 => 100%=10 kg. Anonymous Tell me, please, how to calculate the percentage itself in the range of 2 values? Say, what is the percentage of the number 37 in the range of values 22-63? I need a formula for an application, I used to solve such problems in a couple of minutes, but now my brain has shrunken). Help out. NMitra It looks like this for me: percentage = (number - z0) ⋅ 100: (z1-z0) z0 - start value of the range z1 - end value of the range For example, x = (37-22) ⋅ 100: (63-22) = 1500 : 41 = 37% For the example below converges
| 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 35 | 50% | 10 | 45 |
| 16 | 23% | 4,6 | 20,6 |
| 18 | 26% | 5,2 | 23,2 |
| 1 | 1% | 0,2 | 1,2 |
| 70 | 100% | 20 | 90 |
| 35 | 50% | 10 | 45 | 67,5 |
| 16 | 23% | 4,6 | 20,6 | 30,9 |
| 18 | 26% | 5,2 | 23,2 | 34,8 |
| 1 | 1% | 0,2 | 1,2 | 1,8 |
| 70 | 100% | 20 | 90 | 135 |
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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 dimension of profit is also affected. 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.
The Hindus knew about percentages as early as the 5th century. In Europe, with which 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 "percent" 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 appearance of shares greatly simplified the calculations, which is why they have become so common.
Converting fractions to percentages
To convert a decimal fraction to a percentage, you may need the so-called percentage formula: the fraction is multiplied by 100,% is added to the result.
If you need to convert an ordinary fraction to a percentage, first you need to make it a decimal, and then use the above formula.
Converting percentages to fractions
As such, the percentage formula is rather arbitrary. 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.
The formula for calculating the percentage of a number
1) 40 x 30 = 1200.
2) 1200: 100 = 12 (students).
Answer: control work on "5" was written by 12 students.
You can use the ready-made table, which shows some fractions and percentages that correspond to them.
It turns out that the percentage formula looks like this: C \u003d (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%); C is the desired result.
Formula for calculating a number from a percentage
The following task will demonstrate what a percentage is and how to find a number from a percentage.
The garment factory produced 1,200 dresses, of which 32% are new-style dresses. How many new-style dresses did the clothing factory make?
1. 1200: 100 = 12 (dresses) - 1% of all manufactured items.
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 \u003d (A ∙ 100) / B, where A is the total number of items (in this case, A \u003d 1200); B - the number of percent (in a specific task B = 32%); C is the desired value.
Increase, decrease a number by a given percentage
Students 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 increased by a given percentage will be equal to. 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 \u003d (1 + 40 / 100) ∙ X \u003d 1.4 ∙ X. If we substitute any number instead of X, take, for example, 100, then the whole expression will be equal to : 1.4 ∙ X \u003d 1.4 ∙ 100 \u003d 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% \u003d X ∙ (1-40 / 100) \u003d 0.6 ∙ X. If the value is 100, then 0.6 ∙ X \u003d 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?
Let's say 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 increased by 25%.
Proportion
There are often cases when it is necessary to solve problems for percentages 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 \u003d C / D.
In mathematics textbooks, there is such a rule: the product of the extreme terms is equal to 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 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 percentages. 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.
In this case, the value of 200 rubles corresponds to 125% of the original (sales) 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 population of Vicencia. Of the people, 40% are employed in the maintenance of the mine, the rest are mined for tetanium. How many people mine tetanium?
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 engaged in the extraction of tetanium.
Multiple change of a number by a certain percentage
When it is already clear 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. So, 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 we talk about a half or third increase, a quarter decrease, a 15% increase, 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
This shows that X has increased by 44%.
Examples of tasks for percentages
1. What percentage of the number 36 is the number 9?
According to the formula for finding a percentage of a 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.
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: 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 shared the profits equally. How much in rubles did the first partner lose compared to how much he would have received if they divided the income according to the percentage of invested funds?
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 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 be as low as possible. Then you should apply for a 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 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 is added on the balance of the principal debt. Monthly, the total amount of payments will decrease.
Now you need to consider both methods. 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 can’t make calculations, there are a lot of online calculators that will help you cope with the task.