Interest— a convenient relative measure that allows you to operate with numbers in a format familiar to humans, regardless of the size of the numbers themselves. This is a kind of scale to which any number can be reduced. One percent is one hundredth. The word itself percent comes from the Latin "pro centum", meaning "hundredth part".
Interest is irreplaceable in insurance, financial sector, in economic calculations. Tax rates, return on investment, fees for borrowed funds are expressed as percentages. cash(for example, bank loans), economic growth rates and much more.
1. Formula for calculating the percentage share.
Let two numbers be given: A 1 and A 2. It is necessary to determine what share in percentage is the number A 1 from A 2.
P = A 1 / A 2 * 100.
In financial calculations it is often written
P = A 1 / A 2 * 100%.
Example. What percentage is 10 of 200?P = 10 / 200 * 100 = 5 (percent).
2. Formula for calculating percentage of a number.
Let the number A 2 be given. We need to calculate the number A 1, which is equal to given percentage P from A 2 .
A 1 = A 2 * P / 100.
Example. Bank loan 10,000 rubles at 5 percent interest. The interest amount will be.
P = 10000 * 5 / 100 = 500.
3. Formula for increasing a number by a given percentage. Value with VAT.
Let the number A 1 be given. We need to calculate the number A 2, which more number A 1 on specified percentage P. Using the formula for calculating the percentage of a number, we get:
A 2 = A 1 + A 1 * P / 100.
A 2 = A 1 * (1 + P / 100).
Example 1. Bank loan 10,000 rubles at 5 percent interest. The total amount of debt will be.
A 2 = 10000 * (1 + 5 / 100) = 10000 * 1.05 = 10500.
Example 2. The amount excluding VAT is 1000 rubles, VAT 18 percent. The amount including VAT is:
A 2 = 1000 * (1 + 18 / 100) = 1000 * 1.18 = 1180.
style="center">4. Formula for reducing a number by a given percentage.
Let the number A 1 be given. We need to calculate the number A 2, which less number A 1 by a given percentage P. Using the formula for calculating the percentage of a number, we get:
A 2 = A 1 - A 1 * P / 100.
A 2 = A 1 * (1 - P / 100).
Example. The amount of money to be issued minus income tax (13 percent). Let the salary be 10,000 rubles. Then the amount to be issued is:
A 2 = 10000 * (1 - 13 / 100) = 10000 * 0.87 = 8700.
5. Formula for calculating the initial amount. Price without VAT.
Let a number A 1 be given, equal to some initial number A 2 with an added percentage P. We need to calculate the number A 2 . In other words: we know the monetary amount including VAT, we need to calculate the amount excluding VAT.
Let us denote p = P / 100, then:
A 1 = A 2 + p * A 2 .
A 1 = A 2 * (1 + p).
Then
A 2 = A 1 / (1 + p).
Example. The amount including VAT is 1180 rubles, VAT 18 percent. Cost without VAT is:
A 2 = 1180 / (1 + 0.18) = 1000.
style="center">6. Calculation of interest on a bank deposit. Formula for calculating simple interest.
If interest on a deposit is accrued once at the end of the deposit term, then the amount of interest is calculated using the simple interest formula.
S = K + (K*P*d/D)/100
Sp = (K*P*d/D)/100
Where:
S is the amount of the bank deposit with interest,
Sp - amount of interest (income),
K - initial amount (capital),
d — number of days of accrual of interest on the attracted deposit,
D — number of days in calendar year(365 or 366).
Example 1. The bank accepted a deposit in the amount of 100 thousand rubles for a period of 1 year at a rate of 20 percent.
S = 100000 + 100000*20*365/365/100 = 120000
Sp = 100000 * 20*365/365/100 = 20000
Example 2. The bank accepted a deposit in the amount of 100 thousand rubles for a period of 30 days at a rate of 20 percent.
S = 100000 + 100000*20*30/365/100 = 101643.84
Sp = 100000 * 20*30/365/100 = 1643.84
7. Calculation of interest on a bank deposit when calculating interest on interest. Formula for calculating compound interest.
If interest on a deposit is accrued several times at equal intervals and is credited to the deposit, then the amount of the deposit with interest is calculated using the formula compound interest.
S = K * (1 + P*d/D/100) N
Where:
P—annual interest rate,
When calculating compound interest, it is easier to calculate the total amount with interest, and then calculate the amount of interest (income):
Sp = S - K = K * (1 + P*d/D/100) N - K
Sp = K * ((1 + P*d/D/100) N - 1)
Example 1. A deposit of 100 thousand rubles was accepted for a period of 90 days at a rate of 20 percent per annum with interest accrued every 30 days.
S = 100000 * (1 + 20*30/365/100) 3 = 105 013.02
Sp = 100000 * ((1 + 20*30/365/100) N - 1) = 5 013.02
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Example 2. Let's check the formula for calculating compound interest for the case from the previous example.
Let's divide the deposit period into 3 periods and calculate the interest accrual for each period using the simple interest formula.
S 1 = 100000 + 100000*20*30/365/100 = 101643.84
Sp 1 = 100000 * 20*30/365/100 = 1643.84
S 2 = 101643.84 + 101643.84*20*30/365/100 = 103314.70
Sp 2 = 101643.84 * 20*30/365/100 = 1670.86
S 3 = 103314.70 + 103314.70*20*30/365/100 = 105013.02
Sp 3 = 103314.70 * 20*30/365/100 = 1698.32
The total amount of interest, taking into account the calculation of interest on interest (compound interest)
Sp = Sp 1 + Sp 2 + Sp 3 = 5013.02
Thus, the formula for calculating compound interest is correct.
8. Another compound interest formula.
If the interest rate is not given on an annual basis, but directly for the accrual period, then the compound interest formula looks like this.
S = K * (1 + P/100) N
Where:
S—deposit amount with interest,
K - deposit amount (capital),
P - interest rate,
N is the number of interest periods.
Example. A deposit of 100 thousand rubles was accepted for a period of 3 months with monthly interest accrual at a rate of 1.5 percent per month.
S = 100000 * (1 + 1.5/100) 3 = 104,567.84
Sp = 100000 * ((1 + 1.5/100) 3 - 1) = 4,567.84
One hundredth of any quantity or number is called a percentage.
Percentages are indicated by the % sign.
To convert percentages to fractions, remove the % sign and divide the number by 100
1% (one percent) = 1/100 = 0.01
5% = 5/100 = 0,05
20% = 20/100 = 0,2
To translate decimal as a percentage, you need to multiply the fraction by 100 and add the % sign.
0,4 = 0,4 * 100% = 40%
0,07 = 0,07 * 100% = 7%
To convert a fraction to a percentage, you must first convert it to a decimal.
2/5 = 0,4 = 0,4 * 100% = 40%
IN Everyday life you need to know about the numerical relationship between fractions and percentages. So, half - 50%, a quarter - 25%, three quarters - 75%, one fifth - 20%, and three fifths - 60%.
To find any fraction of a number, you need to multiply the value of this fraction by the number.
For example, 1/5 of the number 40 is equal to 1/5⋅40=8.
Let's look at the problem ON SHARES.
After Antoshka ate half of the peaches from the jar, the level of the compote dropped by one third. By what part (of the obtained level) will the compote level decrease if you eat half of the remaining peaches?
Since half of the peaches make up one third of the whole compote, then half of the remaining peaches make up one sixth of the whole compote. It remains to find which part is 1/6 of 2/3.
1/6:2/3 = 1/6⋅3/2=1/4
Answer. One quarter.
Another problem FOR PERCENTAGES:
The rye planting area has rectangular shape. As part of the restructuring of collective farm lands, one side of the plot was increased by 20% and the other was reduced by 20%. How will the area of the plot change?
Let a and b be the sides of the original rectangle. Then the new sides will be a + 20/100a = 6/5a and b− 20/100b = 4/5b, respectively. That's why new square will be equal
6/5a⋅ 4/5b = 24/25ab = 96/100ab = ab − 4/100ab.
Answer. The area decreased by 4%.
The teacher gave the excellent student Petya and the poor student Vasya tasks for the summer, and Vasya - 4 times more tasks than Pete. After the holidays, it turned out that Petya and Vasya solved equally many problems and the percentage of problems solved by Vasya is equal to the percentage of problems not solved by Petya. What is the percentage of problems solved by Petya?
The solution of the problem
Since Vasya and Petya solved the same number of problems, and asked Vasya four times more, this means that the percentage of problems solved by Petya is 4 times greater than the percentage of problems solved by Vasya. And together they make up 100%, since the percentage of problems solved by Vasya is equal to the percentage of problems NOT solved by Petya. This means Petya solved 80% of the problems, and Vasya - 20%.
Environmentalists protested against the large volume of logging. The chairman of the timber industry enterprise calmed them down in the following way: “The forest is 99% pine trees. Only pine trees will be cut down, and after cutting down the percentage of pine trees will remain almost unchanged - there will be 98% pine trees.” What proportion of trees will be cut down? Give your answer as a percentage.
The solution of the problem
Before cutting down, “non-pine trees” made up 1 percent of all trees in the forest, and after cutting down - two percent. Let there be nn trees in the forest before cutting down, and k trees after cutting down. Since the number of non-pine trees remains the same, 1/100⋅n = 2/100⋅k Hence k = n/2.
The original contains such shares fraction, shows the numerator - in the given example there are three of them, which means the percentage expression of one share (25%) should be tripled 25*3=75. The resulting value will be the desired value. Conclusion: to find the percentage equivalent expressed in ordinary fraction Well, divide one hundred by the denominator and multiply by the numerator.
For the wrong one common fraction use the same calculation algorithm. Distinctive feature The only difference in this case is that the resulting value will always be more than one hundred percent. For example, to convert the fraction 7/4, you need to divide 100 by 4 and multiply the result by 7: 100/4*7 = 175%.
If necessary, round the result to the required number of decimal places. The rounding rules are as follows: if the highest digit to be deleted contains a digit from 0 to 4, then the next highest digit (which is not deleted) does not change, and if the digit is from 5 to 9, it increases by one. If the last of these operations is subjected to the digit with the number 9, the unit is transferred to another, even more senior digit, like a column. Please note that rounding to the available number of familiar places does not always carry out this operation. Sometimes there are hidden bits in its memory that are not displayed on the indicator. Logarithmic, having low accuracy (up to two decimal places), often handles rounding in the right direction better.
If you find that a certain sequence of numbers is repeated after a decimal point, place that sequence in parentheses. They say about it that it is located "" because it repeats periodically. For example, number 53.7854785478547854... can be written as 53,(7854).
Proper fraction, whose value more than one, consists of two parts: integer and fractional. First, divide the numerator of the fraction by its denominator. Then add the result of division with whole part. Then, if necessary, round the result to required quantity decimal places or find the periodicity and highlight it in brackets.
All measurements are expressed by numbers, for example, length, area and volume in geometry, distance and speed in physics, etc. The result does not always turn out to be whole; this is how fractions appear. There are various actions with them and ways to transform them, in particular, you can ordinary fraction convert to decimal.
Instructions
A fraction is a notation of the form m/n, where m belongs to the set of integers, and n belongs to the set of natural numbers. Moreover, if m>n, then the fraction is improper, and a whole part can be separated from it. When the numerator m and denominator n are multiplied by the same number, the result remains unchanged. All transformation operations are based on this rule. Thus, you can turn by selecting the appropriate multiplier.
Choose a number so that the result of multiplying it by the denominator is 10. Reason backwards: is it possible to turn the number 4 into 10? Answer: no, because 10 is not divisible by 4. Then 100? Yes, 100 is divided by 4 without a remainder, the result is 25. Multiply the numerator and denominator by 25 and write the answer in decimal:
¼ = 25/100 = 0.25.
It is not always possible to use the selection method; there are two more ways. Their principle is practically the same, only the recording differs. One of them is the gradual allocation of decimal places. Example: convert the fraction 1/8.
Think about it this way:
1/8 does not have an integer part, therefore it is equal to 0. Write down this number and place a comma after it;
Multiply 1/8 by 10 to get 10/8. From this fraction you can select an integer part equal to 1. Write it after the decimal point. Continue working with the resulting 2/8 residue;
2/8*10 = 20/8. The whole part is equal to 2, – 4/8. Subtotal – 0.12;
4/8*10 = 40/8. From the multiplication table it follows that 40 is divisible by 8. This completes your calculations, the final answer is 0.125 or 125/1000.
And finally, the third method is column division. Every time you have to divide smaller number for more, lower the zero “from above” (see figure).
Today at 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. Percentages are found in any field 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 from 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 translate this value V 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: test 12 students wrote “5”.
You can use ready-made table, which indicates some fractions and the 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 specific 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 in this case A=1200); B - number of percent (in specific task B=32%); C is the desired value.
Increase or decrease a number by a specified percentage
Schoolchildren must learn what percentages are, how to count them and solve them. various tasks. 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 transfer 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 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 the number by given number percent. 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 not known 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 sizes. 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 transformation means an increase in a number by 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 one 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
Quite today popular question- obtaining 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. Monthly total amount payments will decrease.
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 understand the essence 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.
Percentage is one of the interesting and often used tools in practice. Interest is partially or fully applied in any science, in any work, and even in everyday communication. A person who is good at percentages gives 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 got the name 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
How to decide similar tasks we already know. 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, we divide 300 rubles 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, previous task 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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