To find a number by. Calculating percentages, or everyday mathematics
Calculating interest is a simple mathematical operation that is quite common in everyday life. For example, you need to calculate how much a person saves by using a store discount card or buying an item on sale at a discount, and at what percentage the loan is taken out. Percentages can be calculated using a calculator or proportion; a formula for calculating percentages and knowledge of basic known ratios will be useful.
What is a percentage of a number
Calculation of interest in school curriculum is studied in the 5th grade, if not earlier. By definition, a percentage is one hundredth of a number. The term appeared in Ancient Rome and literally translates as “from a hundred.” The idea of calculating percentages originally originated in Babylon. In parallel in Ancient India learned to calculate percentages using proportions.
In order to find the percentage of a number, you need to divide the number by 100. Obviously, 1% of 100 equals one.
Calculating interest using formulas
The formula to find the percentage of a number is simple. You need to divide the number by 100, then multiply by the desired percentage.
If we take the original number as X, and the desired percentage as Y, then the formula is written as X/100*Y=...
Calculations using proportions
Calculating percentages can be done with an understanding of the proportion method. Let A be the main number taken as 100%, B the number whose relationship with A as a percentage must be calculated, and X the number of the desired percentages. Then:
A - 100%,
B - X%.
Multiplying crosswise will give the equality: A*X=B*100. Therefore, X=B*100/A.
For example, you need to find out what percentage of 300 is the number 75. It turns out: 75*100/300=25%.
Alternative calculation method
Let's represent one percent not as a decimal, but as a simple fraction - 1/100. Similarly, you can write any number of percentages. So, 10% is 0.1 or 1/10, 25% is 0.25 or 25/100=1/4 and so on. Therefore, finding 10% of a number is quite simple - you need to divide the original number by 10. In this way it is convenient to calculate 20, 25 and 50 percent:
- 20% is 1/5, which means you need to divide the original number by 5.
- 25% is 1/4, you need to divide by 4.
- 50% is 1/2, just divide by two.
But not every percentage is convenient to calculate using this method. For example, 33% is 33/100, which when written as a decimal gives 0.3333 with an infinite number of threes after the decimal point.
If you have any doubts about the correctness of your calculations, you can always check yourself using a calculator, which is now available on any mobile device and on any computer.
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How to find the percentage of a number? General rule such. To find the percentage part of a number, you need:
1. Divide the number by 100. Why 100? Because a percentage is one hundredth of a number. And in order to find a few percent, you first need to find 1% (percent). We divide the number by 100 and thus we find 1% (percent) of the number.
2. Multiply the resulting result by the number of percent. This way we will see what part of the number we were looking for.
Let's look at this with specific examples:
1. Calculate 5% of the number 60. Let’s find 1%, so we need to divide the number 60 by 100 (60: 100= 0.6). Now 0.6 needs to be multiplied by the number of percentages we are looking for. We are looking for 5%. We simply multiply 6*5 =30, as a result you need to separate one decimal place with a comma, because the factors have one decimal place, so 0.6*5= 3
2. Calculate 15% of the number 30. Using the same scheme, 30:100 = 0.3. Now 0.3 needs to be multiplied by the number we are looking for. We are looking for 15%. We simply multiply 3*15 =45, but we need to separate 1 digit with a comma. Therefore 0.3*15= 4.5
3. Calculate 75% of the number 150. Using the same scheme, 150:100= 1.5. Now 1.5 needs to be multiplied by the number we are looking for. We are looking for 75%. therefore, in order to multiply these 2 numbers, you need to discard all the commas and simply multiply 15 * 75 = 1125. Now, as a result, you need to separate as many digits with a comma as there are in both factors in total. We have one digit in both factors. That is, only 5 in the number 1.5. Therefore, we also move the comma by one digit 1.5 * 75 = 112.5.
This way it is easier to find out the percentages.
Interest is one of the concepts of applied mathematics that are often encountered 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 wage 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 became widespread at the end of the 17th century. In 1685, the book “Manual of Commercial Arithmetic” by Mathieu de la Porte was published in Paris. In one place there was talk of interest, 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 percents 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 percentage values 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 shapes expressions of the same change in quantity, formulated without percentages and using percentages. For example, in messages "Minimum wages 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 increase by 200%, decrease by 2 times - this means 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, percentage 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 the percentage ratio of 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 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 some types of deposits (the so-called time deposits, which cannot be taken earlier than after a period specified in the agreement, for example, a year), the following income payment system has been adopted: for the first year that 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 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.
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 is true
\(S_n = \left(1+ \frac(p)(100) \right)^n S \)
This formula is called compound interest formula, or just compound interest formula.
Perhaps math wasn't your favorite subject in school, and numbers were scary and boring. But in adult life there is no escape from them. Without calculations, you can’t fill out a receipt for paying for electricity, you can’t draw up a business project, you can’t help your child with his homework. Often in these and other cases it is necessary to calculate the percentage of the amount. How to do this if you have vague memories of what percentage is from your school days? Let's strain our memory and figure it out.
Method one: percentage of the amount by determining the value of one percent
A percentage is one hundredth of a number and is denoted by the % sign. If you divide the amount by 100, you get just one percent. And then everything is simple. We multiply the resulting number by the required percentage. In this way it is easy to calculate the profit on a bank deposit.
For example, you deposited an amount of 30,000 at 9% per annum. What will be the profit? We divide the amount of 30,000 by 100. We get the value of one percent - 300. Multiply 300 by 9 and get 2,700 rubles - an increase to the original amount. If the contribution is for two or three years, then this figure doubles or triples. There are deposits for which interest payments are made monthly. Then you need to divide 2700 by 12 months. 225 rubles will be a monthly profit. If interest is capitalized (added to the total account), then the deposit amount will increase every month. This means that the percentage will not be calculated from down payment, but from the new indicator. Therefore, at the end of the year you will receive a profit of not 2,700 rubles, but more. How many? Try to count.
Method two: convert percentages to decimals
As you remember, a percentage is a hundredth of a number. As a decimal it is 0.01 (zero point one hundredth). Therefore, 17% is 0.17 (zero point, seventeen hundredths), 45% is 0.45 (zero point, forty-five hundredths), etc. We multiply the resulting decimal fraction by the amount of which we calculate the percentage. And we find the answer we are looking for.
For example, let's calculate the amount income tax from salary 35,000 rubles. The tax is 13%. As a decimal it would be 0.13 (zero point one, thirteen hundredths). Let's multiply the amount of 35,000 by 0.13. The result will be 4,550. This means that after deducting income tax, you will receive a salary of 35,000 - 4,550 = 30,050. Sometimes this amount, already without tax, is called “salary in hand” or “net.” In contrast, the amount together with tax is “dirty salary”. It is the “dirty salary” that is indicated in company vacancy announcements and in employment contract. Less is given to your hands. How many? Now you can easily count.
Method three: count on a calculator
If you doubt your mathematical abilities, use a calculator. With its help, it is calculated faster and more accurately, especially when it comes to large amounts. It is easier to work with a calculator that has a button with a percentage sign. Multiply the amount by the percentage and press the % button. The required answer will be displayed on the screen.
For example, you want to calculate what your child care benefit will be for up to 1.5 years. It is 40% of average earnings for the last two closed calendar years. Let’s say the average salary is 30,000 rubles. On the calculator, multiply 30,000 by 40 and press the % button. Key = no need to touch. The answer 12,000 will be displayed on the screen. This will be the amount of the benefit.
As you can see, everything is very simple. Moreover, the “Calculator” application is now available in every cell phone. If the device does not have a special % button, then use one of the two methods described above. And perform multiplication and division on a calculator, which will facilitate and speed up your calculations.
Don't forget: there are online calculators to make calculations easier. They operate in the same way as regular ones, but are always at hand when you work on the computer.
Method four: making a proportion
You can calculate the percentage of the amount using a proportion. This is another scary word from school course mathematics. Proportion – equality between two relations of four quantities For clarity, it is better to immediately understand it with a specific example. You want to buy boots for 8,000 rubles. The price tag indicates that they are sold at a 25% discount. How much is this in rubles? Of the 4 values, we know 3. There is a sum of 8,000, which is equal to 100%, and 25% that needs to be calculated. In mathematics, an unknown quantity is usually called X. The proportion we get is:
For ease of calculation, we convert the percentages to decimals. We get:
The proportion is solved as follows: X = 8,000 * 0.25: 1X = 2,000
2,000 rubles – discount on boots. We subtract this amount from the old price. 8,000 – 2,000= 6,000 rubles (new discounted price). This is such a nice proportion.
This method can also be used to determine the value of 100%, if you know the numerical indicator - say, 70%. At a company-wide meeting, the boss announced that 46,900 units of goods had been sold during the year, while the plan was only 70% fulfilled. How much did you need to sell to fully fulfill the plan? Let's make a proportion:
Converting percentages to decimal fractions, it turns out:
Let’s solve the proportion: X = 46,900 * 1: 0.7X = 67,000. These were the results of the work that the bosses expected.
As you may have guessed, the proportion method can be used to calculate what percentage a numerical indicator is of the amount. For example, while taking a test, you answered 132 out of 150 questions correctly. What percent of the task was completed?
There is no need to convert this proportion into decimal fractions; you can solve it right away.
X = 100 * 132: 150. As a result, X = 88%
As you can see, it's not all that scary. A little patience and attention, and now you have mastered the calculation of percentages.
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. Percentages express tax rates, return on investment, fees for borrowed funds (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 percentage of the number A 1 is 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. It is necessary to calculate the number A 1, which is a given percentage P of 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. Amount including VAT.
Let the number A 1 be given. We need to calculate the number A 2, which more 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 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. Amount excluding VAT.
Let a number A 1 be given, equal to some original number A 2 with an added percentage P. We need to calculate the number A 2 . In other words: we know sum of money with VAT, you 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 regular intervals and is credited to the deposit, then the amount of the deposit with interest is calculated using the compound interest formula.
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
style="center">
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
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