One percent is a hundredth of a number. This concept is used when it is necessary to indicate the ratio of a share to a whole. In addition, several values can be compared as percentages, while necessarily indicating which integer the percentages are calculated relative to. For example, expenses are 10% higher than income or the price of train tickets has increased by 15% compared to the fares of the previous year. A percentage above 100 means that the proportion is greater than the whole, as is often the case in statistical calculations.
Interest as a financial concept - payment, the borrower to the lender for the provision of money for temporary use. In business, there is an expression "to work for interest." AT this case it is understood that the amount of remuneration depends on profit or turnover (commission). It is impossible to do without calculating interest in accounting, business, banking. To simplify the calculations, an online percentage calculator has been developed.
The calculator allows you to calculate:
- Percentage of the set value.
- Percentage of the amount (tax on actual salary).
- Percentage of the difference (VAT from ).
- And much more...
When solving problems on a percentage calculator, you need to operate with three values, one of which is unknown (according to given parameters variable is calculated). The calculation scenario should be selected based on the specified conditions.
Calculation examples
1. Calculate the percentage of a number
To find a number that is 25% of 1,000 rubles, you need:
- 1,000 × 25 / 100 = 250 rubles
- Or 1,000 × 0.25 = 250 rubles.
To calculate on a regular calculator, you need to multiply 1,000 by 25 and press the% button.
2. Definition of an integer (100%)
We know that 250 rubles. is 25% of some number. How to calculate it?
Let's make a simple proportion:
- 250 rub. - 25%
- Y rub. - 100 %
- Y \u003d 250 × 100 / 25 \u003d 1,000 rubles.
3. Percentage between two numbers
Suppose a profit of 800 rubles was supposed, but they received 1,040 rubles. What is the overage percentage?
The proportion will be:
- 800 rub. - 100 %
- RUB 1,040 – Y%
- Y = 1040 × 100 / 800 = 130%
Overfulfillment of the plan for profit - 30%, that is, implementation - 130%.
4. Calculation not from 100%
For example, a store with three departments is visited by 100% of customers. In the grocery department - 800 people (67%), in the department of household chemicals - 55. What percentage of buyers come to the department of household chemicals?
Proportion:
- 800 visitors - 67%
- 55 visitors - Y %
- Y = 55 × 67 / 800 = 4.6%
5. What percentage is one number less than another
The price of the goods fell from 2,000 to 1,200 rubles. By what percent did the commodity become cheaper, or by what percentage is 1,200 less than 2,000?
- 2 000 - 100 %
- 1 200 – Y%
- Y = 1200 × 100 / 2000 = 60% (60% to 1200 of 2000)
- 100% − 60% = 40% (number 1200 is 40% less than 2000)
6. By what percentage is one number greater than another
Salary increased from 5,000 to 7,500 rubles. By what percent did the salary increase? How many percent is 7,500 more than 5,000?
- 5 000 rub. - 100 %
- 7 500 rub. - Y%
- Y = 7,500 × 100 / 5,000 = 150% (in the figure 7,500 is 150% of 5,000)
- 150% - 100% = 50% (the number 7,500 is 50% greater than 5,000)
7. Increase the number by a certain percentage
The price of goods S is higher than 1,000 rubles. by 27%. What is the price of the item?
- 1 000 rub. - 100 %
- S - 100% + 27%
- S \u003d 1,000 × (100 + 27) / 100 \u003d 1,270 rubles.
The online calculator makes calculations much easier: you need to select the type of calculation, enter the number and percentage (in the case of calculating percentage- the second number), indicate the accuracy of the calculation and give a command to start actions.
Today we continue a series of video tutorials on percentage problems from the Unified State Examination in mathematics. In particular, we will analyze two completely real tasks from the Unified State Examination and once again we will see how important it is to carefully read the condition of the problem and interpret it correctly.
So the first task is:
A task. Only 95% and 37,500 graduates of the city solved problem B1 correctly. How many people correctly solved problem B1?
At first glance, it seems that this is some kind of task for the caps. Like:
A task. There were 7 birds on the tree. 3 of them flew away. How many birds have flown?
However, let's do the math. We will solve by the method of proportions. So, we have 37,500 students - this is 100%. And also there is a certain number x of students, which is 95% of the very lucky ones who correctly solved problem B1. We write it down:
37 500 — 100%
X - 95%
You need to make a proportion and find x. We get:
Before us classical proportion, but before using the main property and multiplying it crosswise, I propose to divide both parts of the equation by 100. In other words, we cross out two zeros in the numerator of each fraction. Let's rewrite the resulting equation:
According to the basic property of proportion, the product of the extreme terms is equal to the product of the middle terms. In other words:
x = 375 95
It's pretty big numbers, so you have to multiply them by a column. I remind you that it is strictly forbidden to use a calculator on the exam in mathematics. We get:
x = 35625
Total answer: 35,625. That is how many people out of the original 37,500 solved problem B1 correctly. As you can see, these numbers are pretty close, which makes sense because 95% is also very close to 100%. In general, the first task is solved. Let's move on to the second.
Interest problem #2
A task. Only 80% of the city's 45,000 graduates solved problem B9 correctly. How many people solved problem B9 incorrectly?
We solve in the same way. Initially, there were 45,000 graduates - this is 100%. Then, x graduates must be selected from this number, which should be 80% of the original number. We make a proportion and solve:
45 000 — 100%
x - 80%
Let's reduce one zero in the numerator and denominator of the 2nd fraction. Let's rewrite the resulting construction once more:
The main property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:
45,000 8 = x 10
It's the simplest linear equation. Let's express the variable x from it:
x = 45,000 8:10
We reduce one zero at 45,000 and at 10, the denominator remains one, so all we need is to find the value of the expression:
x = 4500 8
You can, of course, do the same as last time, and multiply these numbers by a column. But let's not make life difficult for ourselves, and instead of multiplying by a column, we decompose the eight into factors:
x = 4500 2 2 2 = 9000 2 2 = 36,000
And now - the most important thing that I talked about at the very beginning of the lesson. You need to carefully read the condition of the problem!
What do we need to know? How many people solved problem B9 not properly. And we just found those people who decided correctly. These turned out to be 80% of original number, i.e. 36,000. This means that in order to get the final answer, our 80% must be subtracted from the original number of students. We get:
45 000 − 36 000 = 9000
The resulting number 9000 is the answer to the problem. In total, in this city, out of 45,000 graduates, 9,000 people solved problem B9 incorrectly. Everything, the task is solved.
From the point of view of mathematics, a proportion is the equality of two ratios. Interdependence is characteristic of all parts of the proportion, as well as their unchanging result. You can understand how to make a proportion by familiarizing yourself with the properties and formula of proportion. To understand the principle of solving proportions, it will be sufficient to consider one example. Only directly solving proportions, you can easily and quickly learn these skills. And this article will help the reader in this.
Proportion properties and formula
- Reversal of proportion. In the case when the given equality looks like 1a: 2b = 3c: 4d, write 2b: 1a = 4d: 3c. (Moreover, 1a, 2b, 3c and 4d are prime numbers, other than 0).
- multiplication given members cross proportions. AT literal expression it looks like this: 1a: 2b = 3c: 4d, and writing 1a4d = 2b3c will be equivalent to it. Thus, the product of the extreme parts of any proportion (numbers at the edges of equality) is always equal to the product middle parts (numbers located in the middle of equality).
- When compiling a proportion, such a property of it as a permutation of the extreme and middle terms can also be useful. The equality formula 1a: 2b = 3c: 4d can be displayed in the following ways:
- 1a: 3c = 2b: 4d (when the middle members of the proportion are rearranged).
- 4d: 2b = 3c: 1a (when the extreme members of the proportion are rearranged).
- Perfectly helps in solving the proportion of its property of increase and decrease. With 1a: 2b = 3c: 4d, write:
- (1a + 2b) : 2b = (3c + 4d) : 4d (equality by increasing proportion).
- (1a - 2b) : 2b = (3c - 4d) : 4d (equality by decreasing proportion).
- You can create proportions by adding and subtracting. When the proportion is written as 1a:2b = 3c:4d then:
- (1a + 3c) : (2b + 4d) = 1a: 2b = 3c: 4d (the proportion is added).
- (1a - 3c) : (2b - 4d) = 1a: 2b = 3c: 4d (the proportion is subtracted).
- Also, when solving a proportion containing fractional or large numbers, you can divide or multiply both of its members by the same number. For example, the components of the proportion 70:40=320:60 can be written like this: 10*(7:4=32:6).
- The variant of solving the proportion with percentages looks like this. For example, write down, 30=100%, 12=x. Now you should multiply the middle terms (12 * 100) and divide by the known extreme (30). Thus, the answer is: x=40%. In a similar way it is possible, if necessary, to multiply the known extreme terms and divide them by a given average number, obtaining the desired result.
If you are interested in a specific proportion formula, then in the simplest and most common version, the proportion is such an equality (formula): a / b \u003d c / d, in which a, b, c and d are four non-zero numbers.
§ 125. The concept of proportion.
Proportion is the equality of two ratios. Here are examples of equalities called proportions:
Note. The names of the quantities in the proportions are not indicated.
Proportions are usually read as follows: 2 is related to 1 (one), as 10 is related to 5 (the first proportion). You can read it differently, for example: 2 is so many times greater than 1, how many times 10 is greater than 5. The third proportion can be read as follows: - 0.5 is so many times less than 2, how many times 0.75 is less than 3.
The numbers in a proportion are called members of the proportion. Hence, the proportion consists of four terms. The first and last members, i.e., the members standing at the edges, are called extreme, and the terms of the proportion that are in the middle are called average members. This means that in the first proportion, the numbers 2 and 5 will be the extreme members, and the numbers 1 and 10 will be the middle members of the proportion.
§ 126. The main property of proportion.
Consider the proportion:
We multiply its extreme and middle terms separately. The product of the extreme 6 4 \u003d 24, the product of the average 3 8 \u003d 24.
Consider another proportion: 10: 5 \u003d 12: 6. We also multiply here separately the extreme and middle terms.
The product of the extreme 10 6 \u003d 60, the product of the average 5 12 \u003d 60.
The main property of proportion: the product of the extreme terms of the proportion is equal to the product of its middle terms.
AT general view the main property of the proportion is written as follows: ad = bc .
Let's check it on several proportions:
1) 12: 4 = 30: 10.
This proportion is true, since the ratios of which it is composed are equal. At the same time, taking the product of the extreme terms of the proportion (12 10) and the product of its middle terms (4 30), we will see that they are equal to each other, i.e.
12 10 = 4 30.
2) 1 / 2: 1 / 48 = 20: 5 / 6
The proportion is correct, which is easy to verify by simplifying the first and second relations. The main property of the proportion will take the form:
1 / 2 5 / 6 = 1 / 48 20
It is easy to see that if we write such an equality, in which the product of any two numbers is on the left side, and the product of two other numbers on the right side, then from these four numbers you can make a proportion.
Let us have an equality, which includes four numbers, multiplied in pairs:
these four numbers may be members of a proportion, which is not difficult to write, if we take the first product as the product of the extreme terms, and the second as the product of the middle ones. The published equality can be made, for example, the following proportion:
In general, from equality ad = bc you can get the following proportions:
Do the following exercise on your own. Given the product of two pairs of numbers, write the proportion corresponding to each equality:
a) 1 6 = 2 3;
b) 2 15 = b 5.
§ 127. Calculation of unknown members of the proportion.
The main property of the proportion allows you to calculate any of the terms of the proportion if it is unknown. Let's take the proportion:
X : 4 = 15: 3.
In this proportion, one extreme term is unknown. We know that in every proportion the product of the extreme terms is equal to the product of the middle terms. On this basis, we can write:
x 3 = 4 15.
After multiplying 4 by 15, we can rewrite this equation as follows:
X 3 = 60.
Let's look at this equality. In it, the first factor is unknown, the second factor is known, and the product is known. We know that to find an unknown factor, it is enough to divide the product by another (known) factor. Then it will turn out:
X = 60:3, or X = 20.
Let's check the found result by substituting the number 20 instead of X in this proportion:
The proportion is correct.
Let's think about what actions we had to perform to calculate the unknown extreme term of the proportion. Of the four members of the proportion, only one extreme was unknown to us; two middle and second extreme were known. To find the extreme term of the proportion, we first multiplied the middle terms (4 and 15), and then divided the product found by the known extreme term. Now we will show that the actions would not change if the desired extreme term of the proportion were not in the first place, but in the last. Let's take the proportion:
70: 10 = 21: X .
Let's write down the main property of the proportion: 70 X = 10 21.
Multiplying the numbers 10 and 21, we rewrite the equality in this form:
70 X = 210.
One factor is unknown here, to calculate it, it is enough to divide the product (210) by another factor (70),
X = 210: 70; X = 3.
Thus, we can say that each extreme member of the proportion is equal to the product of the averages divided by the other extreme.
Let us now proceed to the calculation of the unknown mean term. Let's take the proportion:
30: X = 27: 9.
Let's write the main property of the proportion:
30 9 = X 27.
We calculate the product of 30 by 9 and rearrange the parts of the last equality:
X 27 = 270.
Let's find the unknown factor:
X = 270: 27, or X = 10.
Let's check with a substitution:
30:10 = 27:9. The proportion is correct.
Let's take another proportion:
12:b= X : 8. Let's write the main property of the proportion:
12 . 8 = 6 X . Multiplying 12 and 8 and rearranging the parts of the equation, we get:
6 X = 96. Find the unknown factor:
X = 96:6, or X = 16.
In this way, each middle member proportion is equal to the product of the extremes, divided by another average.
Find the unknown terms of the following proportions:
1) a : 3= 10:5; 3) 2: 1 / 2 = x : 5;
2) 8: b = 16: 4; 4) 4: 1 / 3 = 24: X .
Two latest rules in general it can be written like this:
1) If the proportion looks like:
x: a = b: c , then
2) If the proportion looks like:
a: x = b: c , then
§ 128. Simplification of proportion and rearrangement of its members.
In this section, we will derive rules that allow us to simplify the proportion in the case when it includes large numbers or fractional terms. Transformations that do not violate the proportion include the following:
1. Simultaneous increase or decrease of both members of any ratio by the same number of times.
EXAMPLE 40:10 = 60:15.
By multiplying both terms of the first relation by 3 times, we get:
120:30 = 60: 15.
The proportion has not changed.
Decreasing both terms of the second relation by 5 times, we get:
We got the correct proportion again.
2. Simultaneous increase or decrease of both previous or both subsequent terms in the same number of times.
Example. 16:8 = 40:20.
Let's double the previous members of both relations:
Got the right proportion.
Let us reduce the next terms of both relations by 4 times:
The proportion has not changed.
The two conclusions obtained can be summarized as follows: The proportion will not be violated if we simultaneously increase or decrease any extreme member of the proportion and any middle one by the same number of times.
For example, by reducing the 1st extreme and 2nd middle members of the proportion 16:8 = 40:20 by 4 times, we get:
3. Simultaneous increase or decrease of all members of the proportion by the same number of times. Example. 36:12 = 60:20. Let's increase all four numbers by 2 times:
The proportion has not changed. Let's reduce all four numbers by 4 times:
The proportion is correct.
The listed transformations make it possible, firstly, to simplify the proportions, and secondly, to free them from fractional members. Let's give examples.
1) Let there be a proportion:
200: 25 = 56: x .
In it, the terms of the first relation are relatively large numbers, and if we wished to find the value X , then we would have to perform calculations on these numbers; but we know that the proportion is not violated if both terms of the ratio are divided by the same number. Divide each of them by 25. The proportion will take the form:
8:1 = 56: x .
We have thus obtained a more convenient proportion, from which X can be found in the mind:
2) Take the proportion:
2: 1 / 2 = 20: 5.
In this proportion there is a fractional term (1 / 2), from which you can get rid of. To do this, we will have to multiply this term, for example, by 2. But we do not have the right to increase the middle term of the proportion; it is necessary, together with it, to increase one of the extreme terms; then the proportion will not be violated (based on the first two points). Let's increase the first of the extreme terms
(2 2) : (2 1 / 2) = 20: 5, or 4: 1 = 20:5.
Let's increase the second extreme term:
2: (2 1 / 2) = 20: (2 5), or 2: 1 = 20: 10.
Let's consider three more examples of freeing the proportion from fractional terms.
Example 1. 1/4: 3/8 = 20:30.
Bring the fractions to common denominator:
2 / 8: 3 / 8 = 20: 30.
Multiplying both terms of the first relation by 8, we get:
Example 2. 12: 15 / 14 \u003d 16: 10 / 7. Let's bring the fractions to a common denominator:
12: 15 / 14 = 16: 20 / 14
We multiply both subsequent terms by 14, we get: 12:15 \u003d 16:20.
Example 3. 1/2: 1/48 = 20: 5/6.
Let's multiply all the terms of the proportion by 48:
24: 1 = 960: 40.
When solving problems in which some proportions occur, it is often necessary to rearrange the terms of the proportion for different purposes. Consider which permutations are legal, i.e., do not violate proportions. Let's take the proportion:
3: 5 = 12: 20. (1)
Rearranging the extreme terms in it, we get:
20: 5 = 12:3. (2)
We now rearrange the middle terms:
3:12 = 5: 20. (3)
We rearrange both the extreme and middle terms at the same time:
20: 12 = 5: 3. (4)
All of these proportions are correct. Now let's put the first relation in place of the second, and the second in place of the first. Get the proportion:
12: 20 = 3: 5. (5)
In this proportion, we will make the same permutations as we did before, i.e., we will first rearrange the extreme terms, then the middle ones, and, finally, both the extreme and the middle ones at the same time. Three more proportions will turn out, which will also be fair:
5: 20 = 3: 12. (6)
12: 3 = 20: 5. (7)
5: 3 = 20: 12. (8)
So, from one given proportion, by rearranging, you can get 7 more proportions, which together with this one makes 8 proportions.
The validity of all these proportions is especially easily revealed when letter entry. The 8 proportions obtained above take the form:
a: b = c: d; c:d = a:b;
d:b = c:a; b:d = a:c;
a:c = b:d; c:a = d:b;
d:c=b:a; b:a = d:c.
It is easy to see that in each of these proportions the main property takes the form:
ad = b.c.
Thus, these permutations do not violate the fairness of the proportion and they can be used if necessary.
In the last video tutorial, we considered solving percentage problems using proportions. Then, according to the condition of the problem, we needed to find the value of one or another quantity.
This time, the initial and final values are already given to us. Therefore, in tasks it will be required to find percentages. More precisely, by what percentage has this or that value changed. Let's try.
A task. Sneakers cost 3200 rubles. After the price increase, they began to cost 4000 rubles. By what percentage was the price of the sneakers increased?
So, we solve through proportion. The first step - the original price was equal to 3200 rubles. Therefore, 3200 rubles is 100%.
In addition, we were given the final price - 4000 rubles. This is an unknown percentage, so let's denote it as x . We get the following construction:
3200 — 100%
4000 - x%
Well, the condition of the problem is written down. We make a proportion:
The fraction on the left is perfectly reduced by 100: 3200: 100 = 32; 4000: 100 = 40. In addition, you can reduce by 4: 32: 4 = 8; 40: 4 = 10. We get the following proportion:
Let's use the basic property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:
8 x = 100 10;
8x = 1000.
This is the usual linear equation. From here we find x :
x=1000:8=125
So, we got the final percentage x = 125. But is the number 125 the solution to the problem? No way! Because the task requires you to find out by what percentage the price of sneakers was increased.
By how many percent - this means that we need to find a change:
∆ = 125 − 100 = 25
We got 25% - that's how much the original price was raised. This is the answer: 25.
Problem B2 for interest #2
Let's move on to the second task.
A task. The shirt cost 1800 rubles. After the price reduction, it began to cost 1530 rubles. By what percentage was the price of the shirt reduced?
We translate the condition into mathematical language. The initial price of 1800 rubles is 100%. And the final price is 1530 rubles - we know it, but it is not known how many percent it is of the original value. Therefore, we denote it by x. We get the following construction:
1800 — 100%
1530 - x%
Based on the resulting record, we make up the proportion:
Let's separate both parts to simplify further calculations. given equation by 100. In other words, the numerator of the left and right fraction we'll cross out two zeros. We get:
Now let's use the basic property of proportion again: the product of the extreme terms is equal to the product of the average ones.
18 x = 1530 1;
18x = 1530.
It remains to find x :
x = 1530: 18 = (765 2) : (9 2) = 765: 9 = (720 + 45) : 9 = 720: 9 + 45: 9 = 80 + 5 = 85
We got that x = 85. But, as in the previous problem, this number in itself is not the answer. Let's go back to our condition. We now know that the new price after the cut is 85% of the old price. And in order to find the changes, you need from the old price, i.e. 100%, subtract new price, i.e. 85%. We get:
∆ = 100 − 85 = 15
This number will be the answer: Please note: exactly 15, and in no case 85. That's all! Problem solved.
Attentive students will surely ask: why in the first task, when finding the difference, we subtracted the initial number from the final number, and in the second task we did exactly the opposite: from the initial 100% we subtracted the final 85%?
Let's clear this up. Formally, in mathematics, the change in quantity is always the difference between final value and initial. In other words, in the second problem, we should have got not 15, but -15.
However, in no case should this minus be included in the answer, because it has already been taken into account in the condition of the original problem. It says right there about the price reduction. A 15% price decrease is the same as a -15% price increase. That is why in the solution and answer of the problem it is enough to write just 15 - without any minuses.
All, I hope, with this moment we have understood. This concludes our lesson for today. See you soon!
