Percentage is one of the interesting and often used tools in practice. Percentages are partially or fully used in any science, in any job, and even in everyday communication. A person who is well versed in percentages creates the impression of being smart and educated. In this lesson we will learn what a percentage is and what actions you can perform with it.
Lesson contentWhat is percentage?
Fractions are most common in everyday life. 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 the fraction 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 percentage is one hundredth of something
For example, one meter is 1 cm. One meter is divided into one hundred parts, and one part is taken (remember that 1 meter is 100 cm). And one part of these hundred parts is 1 cm. This means that one percent of one meter is 1 cm.
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: one ruble equals one kopeck. The ruble was divided into one hundred parts, and one part was taken from there. And one part of these hundred parts is one kopeck. This means that one percent of one ruble is one kopeck.
Percentages were so common that people replaced the fraction with a special icon that looks like this:
This entry reads "one percent." It replaces a fraction. It also replaces the decimal fraction 0.01 because if we convert a regular fraction to a decimal fraction, we get 0.01. Therefore, between these three expressions we can put an equal sign:
1% = = 0,01
Two percent in fractional form will be written as , in decimal form as 0.02, and using a special icon, two percent is written as 2%.
2% = = 0,02
How to find the percentage?
The principle of finding a percentage is the same as the usual finding of a fraction from a number. To find a 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 the entry 2% mean? The 2% entry replaces the . If we translate this task into a more understandable language, it will look like this:
Find from 10 cm
And we already know how to solve such tasks. This is the usual way of finding a fraction from a number. To find a fraction of a number, you need to divide this number by the denominator of the fraction, and multiply the resulting result by the numerator of the fraction.
So, divide the number 10 by the denominator of the fraction
We got 0.1. Now we multiply 0.1 by the numerator of the fraction
0.1 × 2 = 0.2
We received an answer of 0.2. This means that 2% of 10 cm is 0.2 cm. And if , then we get 2 millimeters:
0.2 cm = 2 mm
This means that 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 resulting result by 50.
So, divide 300 rubles by 100
300: 100 = 3
Now multiply the result by 50
3 × 50 = 150 rub.
This means that 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, but solving such problems is still easier for us
300: 100 = 3
3 × 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 1,200 suits. Of these, 32% are suits of a new style. How many new style suits did the factory produce?
Here you need to find 32% of 1200. The found number will be the answer to the problem. Let's use the rule for finding percentage. Let's divide 1200 by 100 and multiply the resulting result by the desired percentage, i.e. at 32
1200: 100 = 12
12 × 32 = 384
Answer: The factory produced 384 suits of a new style.
Second way to find percentage
The second method of finding the percentage is much simpler and more convenient. It lies in the fact that the number from which the percentage is being sought will immediately be multiplied by the desired percentage, expressed as a decimal fraction.
For example, let's solve the previous problem using this method. Find 50% of 300 rubles.
The entry 50% replaces the entry , and if we convert these to 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 × 0.5 = 150
By the way, the mechanism for finding percentage on calculators works on the same principle. To find a percentage using a calculator, you need to enter into the calculator the number from which the percentage is being sought, then press the multiplication key and enter the desired percentage. Then press the percentage key %
Finding a number by its percentage
Knowing the percentage of a number, you can find out the entire number. For example, an enterprise paid us 60,000 rubles for work, and this amounts to 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 make up 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 formulate 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 resulting result by 100.
Example 2. The number 35 is 7% of some unknown number. Find this unknown number.
Let's 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 × 100 = 500
500 is an unknown number that needed to be found. You can do a check. To do this, we find 7% of 500. If we did everything correctly, we should get 35
500: 100 = 5
5 × 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 a whole number 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 using a fraction. To find a number using a fraction, we divide this number by the numerator of the fraction and multiply the resulting result by the denominator of the fraction. In our example, the number 35 must be divided by 7 and the resulting result multiplied by 100
35: 7 = 5
5 × 100 = 500
In the future we will solve problems involving percentages, 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 a number, and the number by percentage.
Tasks for independent solution
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1% = 0,01.
Finding percentages of a number.
To find the 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 the given number by the resulting decimal fraction.
To find what percentage one number is of another, you can divide one number by another and multiply the resulting product by 100.
How to solve problems involving percentages. Examples.
Finding the percentage of a number is related to finding the fraction of a number. Percentage is a special way of writing a common fraction, so you should begin to reveal the meaning of the concept of percentage by understanding the concept of a common fraction.
Let's take a few ordinary fractions, for example. What is the meaning of each such entry?
- These are examples of proper ordinary 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 proper fraction as an example. For example. The meaning of this expression can be revealed as follows. A certain 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 200 liter barrel was 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 a barrel equal to 200 l, 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 part of a number, i.e. 1% = 0.01.
Then the meaning of the sentence a% of number b can be explained this way. A certain object (a value whose value is equal to b units) divided into 100 equal parts and took from them a parts.
Example 2. Masha had 400 rubles. She spent 24% of this amount. What is the meaning of this statement?
Since 24% = 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 a sum 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 the number b
.
Let x be the number we need to find.
p%
= 0,01p,
x = b
0,01p
To find the percentage of a number, you need to represent the percentage as a decimal fraction and multiply this number by this decimal fraction.
Another approach to this problem. You can use the concept and properties of proportion. If we remember that a 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 - р%,
We have the proportion:
b: 100 = x: p, (b is to 100 as x is to p) whence,
Example 4. Let there be numbers a And b , and 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: “By what percentage 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 the smaller number. 8: 2 = 4 is the ratio of two numbers!
3 Express the ratio as a percentage 4100 = 400%.
The number 10 is 400% greater than the number 2.
If we divide 8 by 10, we will find a ratio showing what part of 10 2 is less than 10 (here the comparison is with the number 10.
The number 2 is 80% less than the number 10.
Example 5. The tractor driver plowed 6 hectares, which is the entire field. What is the area of the entire field?
This is a typical problem of finding a number from its fraction. Let the area of the entire field be equal x,
then we have the equation x= 6. Where does x = 6:; x = 26. The area of the field is 26 hectares.
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 amounts to p% from the number a. Find the number A.
p%
= 0,01p
b
= 0,01pa
a = b: (0.01p)
Given a number b , which is p% from the number a .
Find the number A .
a - 100%
b - p%
a: 100 = b: p
Compound interest formula.
If the amount deposited is a monetary units, and the bank charges R%
per annum, then through n
years, the amount on deposit will be monetary units, or
a(1+0.01p)n
monetary units.
Example 7. Building the house cost 9,800 rubles, of which 35% was paid for labor, and the rest for materials. How many rubles did the materials cost?
Paid for the work:
0,359800 = 3430.
Therefore, the materials cost: 9800 - 3430 = 6370.
Answer: 6370 rub.
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 do you need to add to the tank to fill it?
If the unfilled part of the tank is 6.5% of 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 the number knowing that 25% of it is equal to 45% of 640.
Let x be the desired number. We have
0.25x = 0.45640.
Answer: 1152.
Example 10. Number a is 92% of number b. If number b is increased by 700, then the new number will be 9% larger than number a. Find numbers a and b.
From the problem conditions 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 the number x is of the number 0.5x; Let's make a proportion:
from which we find
Answer: 200%.
Example 12. The lyceum has 260 students, of which 10% are unsuccessful. After the expulsion of a certain number of unsuccessful students, their percentage dropped to 6.4%. How many students were expelled?
Before expulsion, the number of unsuccessful students before expulsion was
Let x people be expelled. Then there were only 260 students left 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 the number 250 greater than the number 200?
Let's do two things.
1) Find out what percentage the number 250 t is from 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. By what percentage is the number 200 smaller than the number 250?
1) Find out what percentage the number 200 is from the number 250 (unlike the previous example, here you need to take the number 250 as 100%!):
2) The number 200 is 100% less than the number 250 - 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 this increase or decrease the volume of the brick and by what percentage?
Let the initial length of the brick be x, the width be y, and the height be z. Then the initial volume of the brick: V 1 = xyz. New brick sizes: 1.3x; 1.2у; 0.6z and new volume: V 2 = 1.3x1.2y0.6z = 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 the product dropped by 40%, then by another 25%. By what percentage did the price of the product decrease compared to the original price?
Let us denote the original price of the product by x. After the first decrease the price will be equal to
x - 0.4x = 0.6x.
The second price reduction is 25% of the new price of 0.6x, so after the second reduction we will have a price
0.6x - 0.250.6x = 0.45x;.
After two reductions, the total price change is:
x - 0.45x = 0.55x.
Since the value is 0.55x; is 55% of the value x, then the price of the product decreased by 55%.
Answer: 55%.
Example 17. The initial cost per 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 cost) by the same number of percent, as a result of which it became equal to 72 rubles. Determine the percentage increase and decrease in unit cost.
Let x% be the percentage increase (and decrease) in unit cost. By definition, x% of 75 is 750.01x. Then after the first increase the price will be 75 + 0.75x.
During the second year the price will decrease by
0.01x(75+0.75x) = 0.75x + 0.0075x 2.
Now we can write the equation for the final price
(75 + 0.75x) - (0.75x + 0.0075x 2) = 72;
x 2 = 400; hence x 1 = - 20, x 2 = 20.
Only one root of this equation is suitable: x 2 = 20.
Answer: 20%.
Example 18. 10 thousand rubles were deposited into the bank account. After the money had been lying there for one year, 1 thousand rubles were withdrawn from the account. A year later, there were 11 thousand rubles in the account. 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 will increase in a year to the amount
10000 + 0.01p10000 = 10000 + 100 rubles.
When 1000 rubles are withdrawn from the account, 9000 + 100 rubles will remain there.
2) In another year, the last value, due to the accrual of interest, will increase to the value of 9000 + 100 rubles + 0.01p (9000 + 100 rubles) = p 2 + 190 rubles + 9000 rubles.
According to the condition, this value is equal to 11,000 rubles, so we have a quadratic equation.
р 2 + 190р + 9000 = 11000;
р 2 + 190р - 2000 = 0
, let's solve this quadratic equation using Viette's theorem, p 1 = 10, p 2 = -200.
A 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 currently is expressed in terms of x using the compound interest formula:
x(1+0.1) 2 = 1.21x.
From the problem statement:
Answer: 40,000 people.
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Anonymous Number A is 56% less than number B, which is 2.2 times less than number C. What percentage of number C is relative to 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 is 5 times more A C is 400% more A Anonymous Help. In 2001, revenue increased by 2 percent compared to 2000, although it was planned to double. By what percentage was 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% (plan fulfilled) 100 - 51 = 49% (plan not fulfilled) Anonymous Help answer the question. Watermelon contains 99% moisture, but after drying (put it in the sun for several days), its moisture content is 98%. By what % will the WEIGHT of the watermelon change after drying? If you calculate it mathematically, it turns out that my watermelon has completely dried out. For example: with a weight of 20 kg, water makes up 99% of the mass, that is, dry weight is 1% = 0.2 kg. Here the watermelon loses liquid and is already 98%, therefore, the dry weight is 2%. But the dry weight cannot change due to water loss, so it remains equal to 0.2 kg. 2%=0.2 => 100%=10 kg. Anonymous Please tell me how to calculate the percentage itself in the range of 2 values? Let's say, what percentage does the number 37 have 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 shrunk). Help out. NMitra It works out like this for me: percentage = (number - z0) ⋅ 100: (z1-z0) z0 - initial value of the range z1 - final value of the range For example, x = (37-22) ⋅ 100: (63-22) = 1500 : 41 = 37% For the example below it 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 |
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 quantity. Interests are found in every area of modern structures. Take banks, for example: the amount of loan overpayment depends on the amount specified in the agreement; the size of the profit is also affected. Therefore, it is vitally important to know what percentage is.
Interest concept
According to one legend, the percentage appeared due to a stupid typo. The typesetter was supposed to set the number 100, but he got it wrong and set it like this: 010. This caused the first zero to rise slightly and the second to fall. The one turned into a backslash. Such manipulations resulted in the appearance of the percent sign. Of course, there are other legends about the origin of this quantity.
Hindus knew about interest back in the 5th century. In Europe, with which our concept is closely interconnected, they appeared a millennium later. For the first time in the Old World, the idea of what interest is was introduced by a scientist from Belgium, Simon Stevin. In 1584, a table of quantities 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, percentage means one hundredth of any value or number. This value is indicated by the % sign.
Thanks to percentages, it became possible to compare parts of one whole without much difficulty. The appearance of shares greatly simplified calculations, which is why they became so common.
Converting fractions to percentages
To convert a decimal fraction into a percentage, you may need the so-called percentage formula: the fraction is multiplied by 100, and % is added to the result.
If you need to convert a common fraction to a percentage, you first need to make it a decimal, and then use the above formula.
Converting percentages to fractions
As such, the percentage formula is quite arbitrary. But you need to know how to convert this value into a fractional expression. To convert fractions (percents) to decimals, you need to remove the % sign and divide the indicator by 100.
Formula for calculating percentage of a number
1) 40 x 30 = 1200.
2) 1200: 100 = 12 (students).
Answer: 12 students wrote the test for “5”.
You can use a ready-made table that shows some fractions and the percentages that correspond to them.
It turns out that the formula for percentages of a number looks like this: C = (A∙B) / 100, where A is the original number (in this particular example, equal to 40); B - number of percents (in this problem B = 30%); C is the 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 using a percentage.
The garment factory produced 1,200 dresses, of which 32% were dresses of a new style. How many dresses of the new style did the garment factory produce?
1. 1200: 100 = 12 (dresses) - 1% of all products released.
2. 12 x 32 = 384 (dresses).
Answer: the factory produced 384 dresses of the new style.
If you need to find a 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 - number of percents (in a specific task B = 32%); C is the desired value.
Increase or decrease a number by a specified percentage
Students must learn what percentages are, how to count them, and solve a variety of problems. To do this, you need to understand how a number increases or decreases by N%.
Often tasks are given, and in life you need to find out what a number will be equal to when increased by a given percentage. For example, given the number X. You need to find out what the value of X will be equal to if it is increased, say, by 40%. First you need to convert 40% into a fraction (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 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 a 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 train speed increased to 100 km/h. By what percentage did the speed of the train increase?
Let's say 80 km/h - 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 involving percentages using proportions. In fact, this method of finding the result greatly simplifies the task for students, teachers and others.
So what is proportion? This term refers to the equality of two ratios, which can be expressed as follows: A / B = C / D.
In mathematics textbooks there is such a rule: the product of the extreme terms is equal to the product of the middle terms. 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 using the proportion method, you need to understand from which number to take percentages. There are cases 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 required value is 200 rubles, which corresponds to 125% of the original (sale) price of the T-shirt. Then, to find out its cost during the sale, you need (200 x 100): 125. The result is 160 rubles.
2. On the planet Vicencia there are 200,000 inhabitants: people and representatives of the humanoid race Naavi. The Na'avi make up 80% of the entire population of Vicencia. Of the people, 40% are engaged in servicing the mine, the rest are extracting tettanium. 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 would equal 160,000. This is how many representatives of the humanoid race live on Vicencia. The number of people, accordingly, is 40,000. Of these, 40%, that is, 16,000, service the mine. This means that 24,000 people are engaged in tettanium mining.
Repeated change of a number by a certain percentage
When it is already clear what percentage is, you need to study the concept of absolute and relative change. An absolute conversion means increasing a number by a specific number. So, X increased by 100. No matter what we substitute 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 increased by 20%. This means that X will be equal to: X+X∙20%. Relative change is implied whenever we talk about an increase by half or a third, a decrease by a quarter, an increase by 15%, etc.
There is another important point: if the value of X is increased by 20%, and then by another 20%, then the resulting 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 increased by 44%.
Examples of problems involving percentages
1. What percentage of the number 36 is the number 9?
According to the formula for finding the 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 divided the profits equally. How much in rubles did the first partner lose, compared to how much he would have received if they had divided the income according to the percentage of the funds invested?
So, together they invested 10,000 rubles. The income for each was 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 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 obtained.
Answer: the first partner lost 280 rubles when dividing the profits.
A bit of economics
Today, a fairly popular question is applying for 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 figure be as low as possible. It should then be applied against the loan.
As a rule, the amount of 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 principal 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. The total payment amount will decrease monthly.
Now you need to consider both methods. So, with the annuity option, the amount of overpayment will be higher, and with the differential option, the amount of the first payments will be higher. Naturally, the loan terms are the same for both cases.
Conclusion
So, percentages. How to count them? Simple enough. However, sometimes they can cause difficulties. This topic begins to be studied in 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 the calculations, there are a lot of online calculators that will help you cope with the task.