How to find one percent of the amount. Examples of school assignments
Interest Calculator is designed to calculate basic mathematical problems related to percentages. In particular, it allows:
- Calculate the percentage of a number.
- Determine the percentage of one number from another.
- Add or subtract a percentage from a number.
- Find a number, knowing its certain percentage.
- Calculate by what percent one number is greater than another.
The result can be rounded to the required decimal place.
Interest- one of the concepts of applied mathematics, which are often encountered in everyday life. So, 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, the fabric contains 100% cotton, etc. It is clear that the understanding of such information is necessary in modern society.
One percent of any value - the amount of money, the number of students in the school, etc. - called one hundredth of it. 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 wage 2300 rubles. (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 a solution (recall that the concentration of a solution is the part that makes up the mass of the solute from the mass of the entire solution).
It is clear that the entire value under consideration is 100 hundredths, or 100% of itself. Therefore, for example, the inscription on the label "cotton 100%" means that the fabric consists of pure cotton, and 100% academic performance means that there are no underachieving students in the class.
The word "percent" comes from the Latin pro centum, meaning "from a hundred" or "by 100". This phrase can be found in modern speech. For example, they say: "Out of every 100 participants in the lottery, 7 participants received prizes." If this expression is taken literally, then this statement is, of course, incorrect: it is clear that one can choose 100 people participating in the lottery and not receiving prizes. In fact, the exact meaning of this expression is that 7% of the lottery participants received prizes, and this is the understanding that corresponds to the origin of the word "percentage": 7% is 7 out of 100, 7 people out of 100 people.
The sign "%" became widespread at the end of the 17th century. In 1685, the book "Guide to commercial arithmetic" by Mathieu de la Porta was published in Paris. In one place, it was about percentages, which then stood for "cto" (short for cento). However, the compositor mistook this "c/o" for a fraction and typed "%". So because of a typo, this sign came into use.
Any number of percent can be written as a decimal fraction, expressing a part of the value.
To express a percentage as a number, divide the percentage by 100. For example:
\(58\% = \frac(58)(100) = 0.58; \;\;\; 4.5\% = \frac(4.5)(100) = 0.045; \;\;\; 200\% = \frac(200)(100) = 2\)For the reverse transition, the reverse action is performed. In this way, 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 values of percentages and the corresponding fractions: half - 50%, a quarter - 25%, three quarters - 75%, a fifth - 20%, three fifths - 60%, etc.
It is also useful to understand different forms of expressing the same change in quantity, formulated without percentages and with the help of percentages. For example, in 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" they 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%.
Similarly
- to increase by 300% - this means to increase by 4 times,
- reduce by 80% - this means to reduce by 5 times.
Interest tasks
Since percentages can be expressed as fractions, problems with percentages are essentially the same problems with fractions. In the simplest percentage problems, some value a is taken as 100% ("whole"), and its part b is expressed as p%.
Depending on what is unknown - a, b or p, three types of interest problems are distinguished. These problems are solved in the same way as the corresponding fractional problems, but before solving them, the number p% is expressed as a fraction.
1. Finding a percentage of a number.
To find \(\frac(p)(100) \) from a, 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 equals 45 0.2 = 9 kg and 118% of x equals 1.18x
2. Finding a number by its percentage.
To find a number by its part b, expressed as a fraction \(\frac(p)(100) , \; (p \neq 0) \), divide b by \(\frac(p)(100) \):
\(a = b: \frac(p)(100) \)
3. Finding the percentage of two numbers.
To find how many percent the number b is from a \((a \neq 0) \), you must first find out what part of b is from a, and then express this part as a percentage:
For example, 9 g of salt in a solution of 180 g is \(\frac(9\cdot 100)(180) = 5%\) 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 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 a and p from it. Therefore, the first formula is considered the main one and is called percent formula. The percent formula combines all three types of fraction problems, and you can use it if you want to find any of the unknowns a, b, and p.Compound problems for percentages are solved similarly to problems for fractions.
Simple percentage growth
When a person does not pay the rent on time, a fine is imposed on him, which is called "fine" (from the Latin poena - punishment). So, if the penalty is 0.1% of the amount of the rent for each day of delay, then, for example, for 19 days of delay, the amount will be 1.9% of the amount of the rent. Therefore, together, say, with 1000 r. rent, a person will have to pay a penalty of 1000 0.019 \u003d 19 rubles, and in total 1019 rubles.
It is clear that in different cities and for different people the rent, the size of the penalty fee and the delay time are different. Therefore, it makes sense to draw up a general rent formula for sloppy payers, applicable in any 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, we will denote 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 \(S + \frac(pn)(100)S = \left(1+ \frac(pn)(100) \right) S \)
In this way:
\(S_n = \left(1+ \frac(pn)(100) \right) S \)
This formula describes many specific situations and has a special name: formula for simple percentage growth.
A similar formula will be obtained if a certain value decreases over a given period of time by a certain number of percent. 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 the given value actually decreases. Growth in this case is "negative".
Compound interest growth
In Russian banks, for certain types of deposits (the so-called term deposits, which cannot be taken earlier than after a period specified in the agreement, for example, in a year), the following income payment system has been adopted: for the first year the deposited amount is in the account, the income is, for example, 10% from her. At the end of the year, the depositor can withdraw from the bank the invested money and earned income - "interest", as it is usually called.
If the depositor did not do this, then interest is added to the initial deposit (capitalized), and therefore at the end of the next year 10% is charged by the bank for a new, increased amount. In other words, under such a system, "interest on interest" is charged, or, as they are usually called, compound interest.
Let's calculate how much money the depositor will receive in 3 years if he put 1000 rubles into a fixed-term bank account. and never once within three years will not take money from the account.
10% from 1000 rubles are 0.1 1000 \u003d 100 rubles, therefore, in a year his account will have
1000 + 100 = 1100 (r.)
10% of the new amount of 1100 rubles. are 0.1 1100 \u003d 110 rubles, therefore, after 2 years, his account will have
1100 + 110 = 1210 (p.)
10% of the new amount 1210 rub. are 0.1 1210 \u003d 121 rubles, therefore, after 3 years, his account will have
1210 + 121 = 1331 (p.)
It is not difficult to imagine how much time would be needed with such a direct, "frontal" calculation to find the amount of the deposit in 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 amount, or, in other words, 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 one more year this amount will also increase by 1.1 times, so that 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 \u003d 1.331 1000 - 1331 (r.)
Let us now solve this problem in general form. Let the bank accrue income in the amount of p% per annum, the deposited amount is equal to S p., and the amount that will be in the account in n years is equal to S n p.
The value of p% of S is \(\frac(p)(100)S \) r., and in a year the account will have the amount
\(S_1 = S+ \frac(p)(100)S = \left(1+ \frac(p)(100) \right)S \)
that is, the initial sum will increase by \(1+ \frac(p)(100) \) times.
Over the next year, the amount S 1 will increase by the same amount, and therefore in 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
\(S_n = \left(1+ \frac(p)(100) \right)^n S \)
This formula is called compound interest growth formula, or simply compound interest formula.
A percent is one hundredth of something. It follows from the definition that something whole is taken as 100 percent. The percentage is denoted by the "%" sign.
How to solve problems in which it is required to calculate percentages of a number? The percentage of a number can be calculated both with a formula and on a calculator.
- Task example: The price of a basket of apples is 160 rubles. The price of a basket of plums is 20% more expensive. How much more expensive is a basket of plums?
- Solution: In this task, we are required to do nothing more than find out how many rubles make up 20% of the number 160.
Percentage formula:
1 way
Since 160 rubles is 100%, we first find out what 1% will be equal to. And then we 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. Multiply the number that is 100% by the decimal. This fraction is obtained by dividing the percentage 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
3 way - 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 a 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 percentages that need to be found, that is, by 20. Press 20
- Now press the % key
- The screen should display the answer: 32
Read more about interest calculation algorithms in the article.
The rules for writing numbers that have a fractional part provide for several formats, the main ones being “decimal” and “ordinary”. Common fractions, in turn, can be written in formats called "improper" and "mixed". To select the integer part from the fractional number of each of these notation options, it is more convenient to use different methods.
Instruction
Drop the fractional part if you need to extract from a positive fraction written in mixed format. In such a fraction, the integer part before the fractional - for example, 12 ⅔. In this fraction, the integer part will be the number 12. If the mixed fraction has a sign, then reduce the number obtained in this way by one. The need for this action follows from the definition of the integer part of the number, according to which it cannot be greater than the value of the original fraction. For example, the integer part of -12 ⅔ is -13.
Divide without a remainder the numerator of the original fraction by its denominator, if it is written in the wrong ordinary format. If the original number has a positive sign, then the result will be the integer part. For example, the integer part of the fraction 716/51 is 14. If the original number is negative, then one should be subtracted from the result - for example, calculating the integer part of the fraction -716/51 should give the number -15.
Think of zero as the integer part of a positive fraction written in normal format that is neither mixed nor improper. For example, this is for the fraction 48/51. If the initial fraction is less than zero, then, as in the previous cases, the result is needed by one. For example, the integer part of the fraction -48/51 should be considered the number -1.
Drop all characters after the decimal point if you need to extract from a positive number written in decimal format. In this case, it is the separating
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, and so on. Therefore, absolutely everyone should be able to find the percentage of the number, regardless of how he is going 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 a 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. So 1% of 200 equals 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?
In order to find a few 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 you 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 percentage number, multiply it by the primary number, and click on the "%" sign. In this case, do not press "equal" or other keys.
For example, you need to find 9% of 851. You take a calculator and enter 851 * 9%. All. You should get the answer you need.
Some important facts
To better work with such actions, you need to understand that:
- Half of any number is 50%;
- The fourth part - 25%;
- The fifth part is 20%.
- A tenth is 10% respectively.
It is important to know that 30% is not a third of the number. It seems that it is so, but here is just a discrepancy.
It is important to note that it is necessary to solve complex examples with percentages 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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