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Interest- one of the concepts applied mathematics, which are often found in Everyday life. Thus, you can often read or hear that, for example, 56.3% of voters took part in the elections, the rating of the winner of the competition is 74%, industrial production increased by 3.2%, the bank charges 8% per annum, milk contains 1.5% fat, fabric contains 100% cotton, etc. It is clear that understanding such information is necessary in modern society.
One percent of any value - a sum of money, the number of school students, etc. - one hundredth of it is called.
The percentage is denoted by the % sign. Thus,
1% is 0.01, or \(\frac(1)(100)\) part of the value
Here are some examples: - 1% of the minimum wages
2300 rub. (September 2007) - this is 2300/100 = 23 rubles;
- 1% of the population of Russia, equal to approximately 145 million people (2007), is 1.45 million people;
- A 3% concentration of a salt solution is 3 g of salt in 100 g of solution (recall that the concentration of a solution is the part that is the mass of the dissolved substance from the mass of the entire solution).
It is clear that the entire value under consideration is 100 hundredths, or 100% of itself. So, for example, a label saying “100% cotton” means the fabric is pure cotton, and 100% achievement means there are no failing students in the class.
The word "percent" comes from the Latin pro centum, meaning "from a hundred" or "per 100." This phrase can also be found in modern speech. For example, they say: “Out of every 100 lottery participants, 7 participants received prizes.” If we take this expression literally, then this statement is, of course, false: it is clear that it is possible to select 100 people who participated in the lottery and did not receive prizes. In fact, the exact meaning of this expression is that 7% of lottery participants received prizes, and this understanding corresponds to the origin of the word "percentage": 7% is 7 out of 100, 7 people out of 100 people. The "%" sign has become widespread in century. In 1685, the book “Manual of Commercial Arithmetic” by Mathieu de la Porte was published in Paris. In one place it was about percentage, which was then designated “cto” (short for cento). However, the typesetter mistook this “s/o” for a fraction and printed “%”. So, due to a typo, this sign came into use.
Any number of percentages can be written as a decimal fraction expressing a fraction of a quantity.
To express percentages as numbers, you need to divide the number of percentages by 100. For example:
\(58\% = \frac(58)(100) = 0.58; \;\;\; 4.5\% = \frac(4.5)(100) = 0.045; \;\;\; 200\% = \frac(200)(100) = 2\)For a reverse transition, the reverse action is performed. Thus, To express a number as a percentage, you need to multiply it by 100:
IN practical life It is useful to understand the relationship between the simplest percentages and the corresponding fractions: half - 50%, quarter - 25%, three-quarters - 75%, fifth - 20%, three-fifths - 60%, etc.
It is also useful to understand different shapes expressions of the same change in quantity, formulated without percentages and using percentages. For example, the messages “The minimum wage has been increased by 50% since February” and “The minimum wage has been increased by 1.5 times since February” say the same thing. In the same way, to increase by 2 times means to increase by 100%, to increase by 3 times means to increase by 200%, to decrease by 2 times means to decrease by 50%.
Likewise
- increase by 300% - this means increase 4 times,
- reduce by 80% - this means reduce by 5 times.
Percentage problems
Since percentages can be expressed as fractions, percent problems are essentially the same as fraction problems. In the simplest problems involving percentages, a certain value a is taken as 100% (“whole”), and its part b is expressed by the number p%.
Depending on what is unknown - a, b or p, there are three types of problems involving percentages. These problems are solved in the same way as the corresponding fraction problems, but before solving them, the number p% is expressed as a fraction.
1. Finding the percentage of a number.
To find \(\frac(p)(100) \) from a, you need to multiply a by \(\frac(p)(100) \):
So, to find p% of a number, you need to multiply this number by the fraction \(\frac(p)(100)\). For example, 20% of 45 kg is equal to 45 0.2 = 9 kg, and 118% of x is equal to 1.18x
2. Finding a number by its percentage.
To find a number from its part b, expressed as the fraction \(\frac(p)(100) , \; (p \neq 0) \), you need to divide b by \(\frac(p)(100) \):
\(a = b: \frac(p)(100)\)
3. Finding percentage two numbers.
To find what percentage the number b is of a \((a \neq 0) \), you must first find out what part b is of a, and then express this part as a percentage:
For example, 9 g of salt in a solution weighing 180 g is \(\frac(9\cdot 100)(180) = 5\%\) of the solution.
The quotient of two numbers expressed as a percentage is called percentage these numbers. Therefore the last rule is called rule for finding the percentage ratio of two numbers.
It is easy to see that the formulas
\(b = a \cdot \frac(p)(100), \;\; a = b: \frac(p)(100), \;\; p = \frac(b)(a) \cdot 100 \% \;\; (a,b,p \neq 0) \) are interrelated, namely, the last two formulas are obtained from the first if we express the values of a and p from it. Therefore, the first formula is considered the main one and is called percentage formula. The percent formula combines all three types of fraction problems and can be used to find any of the unknowns a, b, and p if desired.Compound problems involving percentages are solved similarly to problems involving fractions.
Simple percentage growth
When a person does not pay his rent on time, he is subject to a fine called a “penalty” (from the Latin roena - punishment). So, if the penalty is 0.1% of the rent amount for each day of delay, then, for example, for 19 days of delay the amount will be 1.9% of the rent amount. Therefore, together with, say, 1000 rubles. rent, a person will have to pay a penalty of 1000 0.019 = 19 rubles, and a total of 1019 rubles.
It is clear that in different cities and at different people the rent, the amount of penalties and the period of delay are different. Therefore, it makes sense to create a general rent formula for sloppy payers, applicable under all circumstances.
Let S be the monthly rent, the penalty is p% of the rent for each day of delay, and n is the number of days overdue. The amount that a person must pay after n days of delay will be denoted by S n.
Then for n days of delay the penalty will be pn% of S, or \(\frac(pn)(100)S\), and in total you will have to pay \(S + \frac(pn)(100)S = \left(1+ \frac(pn)(100) \right) S\)
Thus:
\(S_n = \left(1+ \frac(pn)(100) \right) S \)
This formula describes many specific situations and has a special name: simple percentage growth formula.
A similar formula will be obtained if a certain value decreases over this period time by a certain percentage. As above, it is easy to verify that in this case
\(S_n = \left(1- \frac(pn)(100) \right) S \)
This formula is also called simple percentage growth formula Although set value is actually decreasing. Growth in this case is “negative”.
Compound interest growth
In Russian banks, for some types of deposits (so-called time deposits, which cannot be taken earlier than after a period specified in the agreement, for example, in a year) next system income payments: for the first year that the deposited amount is in the account, the income is, for example, 10% of it. At the end of the year, the depositor can withdraw from the bank the money invested and the income earned - "interest", as it is usually called.
If the depositor has not done this, then the interest is added to the initial deposit (capitalized), and therefore at the end of the next year 10% is added by the bank to the new, increased amount. In other words, with such a system, “interest on interest” is calculated, or, as they are usually called, compound interest.
Let's calculate how much money the investor will receive in 3 years if he deposited 1000 rubles in a fixed-term bank account. and will never take money from the account for three years.
10% from 1000 rub. are 0.1 1000 = 100 rubles, therefore, in a year his account will have
1000 + 100 = 1100 (r.)
10% of the new amount 1100 rub. are 0.1 1100 = 110 rubles, therefore, after 2 years there will be
1100 + 110 = 1210 (r.)
10% of the new amount 1210 rub. are 0.1 1210 = 121 rubles, therefore, after 3 years there will be
1210 + 121 = 1331 (r.)
It is not difficult to imagine how much time, with such a direct, “head-on” calculation, it would take to find the amount of the deposit after 20 years. Meanwhile, the calculation can be done much easier.
Namely, in a year the initial amount will increase by 10%, that is, it will be 110% of the initial one, or, in other words, it will increase by 1.1 times. Next year the new, already increased amount will also increase by the same 10%. Therefore, after 2 years the initial amount will increase by 1.1 1.1 = 1.1 2 times.
In another year, this amount will increase by 1.1 times, so the initial amount will increase by 1.1 1.1 2 = 1.1 3 times. With this method of reasoning, we obtain a much simpler solution to our problem: 1.1 3 1000 = 1.331 1000 - 1331 (r.)
Let us now solve this problem in general view. Let the bank accrue income in the amount of p% per annum, the deposited amount is equal to S rub., and the amount that will be in the account in n years is equal to S n rub.
The value p% of S is \(\frac(p)(100)S \) rub., and after a year the amount will be in the account
\(S_1 = S+ \frac(p)(100)S = \left(1+ \frac(p)(100) \right)S \)
that is, the initial amount will increase by \(1+ \frac(p)(100)\) times.
Behind next year the amount S 1 will increase by the same amount, and therefore after two years the account will have the amount
\(S_2 = \left(1+ \frac(p)(100) \right)S_1 = \left(1+ \frac(p)(100) \right) \left(1+ \frac(p)(100) ) \right)S = \left(1+ \frac(p)(100) \right)^2 S \)
Similarly \(S_3 = \left(1+ \frac(p)(100) \right)^3 S \), etc. In other words, the equality is true
\(S_n = \left(1+ \frac(p)(100) \right)^n S \)
This formula is called compound interest formula, or simply compound interest formula.
In this article we will describe how find the percentage of a number, the proportion of one number to another. Somewhere in the fifth grade, on entertaining lessons Mathematics children begin to study such a topic as "interest". Then for those who like to count it opens fascinating world percentages and fractional numbers. Teachers give a significant number of interesting, exciting problems to solve involving determining percentages. But in school years children think that they will not necessarily need this knowledge, but in vain! After all, this topic is always relevant and is closely related to everyday life and may well be useful in various life situations.
Why is it important to be able to find percentages of numbers?
Everyone definitely needs to be able to calculate percentages. You will ask why? It’s just that any person almost every day encounters prices for goods and services in certain enterprises and establishments. Almost every second person has a loan, an installment plan, many have savings deposits in banks, and perhaps even more than one. Taxes, insurance, purchases - almost everything in our world involves interest. This topic concerns both financial, economic and other areas of our lives. But when solving children's problems from textbooks in grades 5-6, there are not as many pitfalls as when calculating an adult loan.
IN school curriculum There is 3 patterns to solve problems in percentages:
finding percent from number;
finding percentage numbers
finding the number itself based on its percentage.
Do not forget that calculating interest is very often used in everyday life. An example of this is using them in your family's budget calculations. Many families take out loans such as: “Car loan”, “Consumer loan”, “Education loan” and of course “Housing loan”, which also has another name that is more familiar to us - “Mortgage”.
How is percentage of a number indicated?
It is known that the percentage is indicated by the icon «%» . Use different definitions term.
- The first one is known to everyone: a percentage is one hundredth of a number.
- The second is the fee charged by the bank or other persons issuing financial assets on credit for their use. This concept is extremely common for people in everyday life.
Percentage of a number - the history of the origin of the concept
Few people have wondered where this term came from. But the word “percentage” comes from the Roman Empire. Word "pro centum" can tell you little about it. But its literal designation means “from a hundred” or “for a hundred.” The very idea of expressing parts of a whole in many equal shares was born a long time ago in ancient Babylon. Back then, people used sexagesimal fractions in their calculations. People who lived in Babylon left us “as a souvenir” registers, from which they calculated interest to calculate the amount of debt that the borrower had “accumulated” in interest.
Interests were extremely famous even in other states of Antiquity. People who know exact science mathematics, in India they calculated percentages using the triple rule and used proportions in their calculations. The Romans, for example, were professionals in this field, because they called interest the money that the defaulter is forced to return to the one who issued it, and for every hundred. Even then, the Parliament of Rome adopted the maximum permissible interest that was taken from the debtor, because there were cases when lenders tried too hard to get their interest money. And it was from the Romans that the concept of interest passed on to all other peoples.
Who needs to know how to calculate interest?
- Accountant. He just needs to know how to calculate percentages. In any company, in any job, there is a person involved in payroll. Calculating, subtracting, multiplying your hard-earned money, earned through honest labor. Who is this? Of course an accountant. For example, he deals with the deduction of a percentage of wages. This percentage is a tax that is this moment is 13% of income.
- A bank employee. He also just needs to know the percentage. For what? Yes, because it is this employee who deals with loans, mortgages, and financial investments. He calculates where people's money goes. Provides information about how much a person will overpay or receive during a transaction with the bank.
- Oculist. A doctor examining the fundus of the eye, studying how well a person sees. It determines vision. He will write out glasses. But with vision, as with glasses, not everything is so simple - we are all individual, and accordingly, our vision is different. Some have +(-) 1, and some have +(-) 0.75. And the ophthalmologist, like no one else, knows a lot about this. And not only education, but also knowledge of the percentage helps him understand this.
Application of finding percentages in different areas
Financial.
Everything is elementary here - this is the same amount that the borrower pays to the lender for the fact that the second provided the first with funds for temporary use. In this case, both persons negotiate the conditions of issuance in advance and individually, documenting the financial relationship. Business vocabulary.
In business there is such a concept - “work for interest.” This means that a person is ready to work and receive remuneration, which is calculated from the profit and turnover of the enterprise.
Significance in economics. A certain amount of profit that the “lender” pays to the “lender” for the capital borrowed. The source of interest is the surplus value that is formed when using its loan capital.
Loan interest. This is a kind of deduction for the temporary use of finances. A category that functions in credit relations. In short, this is a relationship between the lender and the borrower, where each has their own interest in finding and receiving interest. This is not a loan, because the loan interest is only the cost of the profit from the product. It turns out that the interest itself is simply a deduction of profit from the amount that is at the borrower’s disposal. Deposit interest.
Deduction of interest for storing funds in storage facilities, which a bank or other borrower takes. There are two participants in this relationship. The first person (lender) is the bank's client, the second (borrower) is the bank itself.
How to find percentages - formula for finding percentage of a number (2 formulas with examples)
1. The first formula is how you can calculate the percentage of a number - divide the desired number by one hundred and multiply by the number of percentages that is necessary.
X/100*Y=...
Where X is the total number from which the percentage is to be extracted, Y- the desired percentage of it.
Example from life: You need to transfer 300 rubles to a relative in Kamchatka. You took advantage payment system“Zhmotfinance”, in which the interest for the transfer is 16% of the payment amount. Thus, we need to find out how much 16 percent of the number 300 will be. Divide 300 by 100 and multiply by 16. (300/100*16) = 48. This will be the amount that the greedy payment system will take for itself.
2. And the second, more simple formula- multiply the number from which you need to extract (X) by 0,Y - where Y - this is the number of desired percentages, you will get the required amount of interest.
X* 0, Y... =
Where also: X - total number, Y - the desired percentage of it.
Example from life: Let’s say you again contacted the Zhmotfinance company, which is ready to transfer your funds to anywhere in Russia for the same 16%. But now you need to send another amount to another relative living in Vladivostok - 500 rubles. This means that we need to get a percentage of the number 500. To do this, we simply multiply 500 by 0.16 (500*0.16) = 80. The extortionate 80 rubles as interest for the transfer go to the income of this greedy company.
Finally, remember - algebra, geometry, physics, chemistry and many other sciences will always be useful to you. And learning to find the percentage of a number may even benefit you in the future. Numbers and numbers play vital role in the future of man. And the ability to find percentages of any number in your mind can make your life much easier and help you avoid absurd and awkward situations in everyday life.
Video about calculating the share
A percentage is one hundredth of something. From the definition it follows that anything whole is taken as 100 percent. The percentage is indicated by the "%" sign.
How to solve problems in which you need to calculate percentages of a number? The percentage of a number can be calculated either by a formula or on a calculator.
- Example task: The price of a basket of apples is 160 rubles. The price of a basket of plums is 20% more expensive. How many rubles is more expensive than a basket of plums?
- Solution: In this task, we are required to do nothing more than find out how many rubles are 20% percent of the number 160.
Formula for calculating percentage:
1 way
Since 160 rubles is 100%, we first find out what 1% will be equal to. And then multiply this number by the 20% we need.
- 160 / 100 * 20 = 1,6 * 20 = 32
Answer: a basket of plums is 32 rubles more expensive.
2 way
The second method is a modified version of the first method. Let's multiply the number that is 100% by a decimal fraction. This fraction is obtained by dividing the number of percentages that need to be found by 100. In our case:
- 20% / 100 = 0,2
We multiply 160 by 0.2 and get the same answer 32.
3 way
Method 3 - proportion.
Let's make a proportion of the form:
- x = 20%
- 160 = 100%
We multiply the parts of the proportion cross by cross and get the equation:
- x = (160 * 20) / 100
- x = 32
Calculating percentage of a number on a calculator
In order to calculate 20% of the number 160 on a calculator, you need:
- First, dial the number 160 on the screen - that is, our 100%
- Then press the multiply button "*"
- We will multiply by the number of percents that need to be found, that is, by 20. Press 20
- Now press the % key
- The answer should appear on the screen: 32
Read more about interest calculation algorithms in the article
Percentages of numbers need to be calculated not only when solving problems and equations. You may also need this when making any purchases, obtaining a loan, etc. Therefore, absolutely everyone should be able to find the percentage of a number, regardless of how they plan to study. But it’s worth noting right away that finding percentages is extremely easy. There is no serious theory here.
How to find one percent of a number?
A percentage is one hundredth of a number. That is, if we divide any number by 100, then we get one percent of this very number.
For example, we need to find 1% of 200. We take 200, divide by 100 and get 2. Thus, 1% of 200 is equal to two.
This rule applies to any numbers, both integers and decimals. The main thing is to understand this principle. And you can work with percentages.
How to find a few percent of a number?
To find multiple percentages, you also need to divide the number by 100. This will give you 1%. Then you must multiply the resulting value by the percentage you are looking for.
For example, you need to find 5% of 300. You take 300 and divide by 100. You get 3. That's one percent. And you need to understand how much 5% will be.
So, you multiply 3 by 5 and get 15. Your problem is solved.
How to find percentages on a calculator?
It is worth noting that in difficult situations You can use any calculator. There is a special function for calculating percentages.
You take the percent number, multiply it by the primary number, and click on the "%" sign. In this case, you should not press “equals” or other keys.
For example, you need to find 9% of 851. You take a calculator and enter 851 * 9%. All. You should have the answer you need.
Some important facts
To work better with these actions you need to understand that:
- Half of any number is 50%;
- The fourth part is 25%;
- A fifth is 20%.
- A tenth is 10%.
It is important to know that 30% is not the third part of the number. It seems that this is exactly the case, but there is just a discrepancy here.
It is important to note that it is up to you to decide complex examples with percentages it is necessary using proportions and equations, which are described in detail in the mathematics course. But if you know the basic rules for working with these actions, then it will be easier for you.
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