In this short video tutorial, we will learn how to solve percentage problems using a special formula, which is called the simple interest formula. Let's put this formula in the form of a theorem.

The simple interest theorem. Suppose there is some initial value x , which then changes by k%, and a new value y is obtained. Then all three numbers are related by the formula:

Plus or minus in front of the coefficient k is placed depending on the condition of the problem. If, by the condition, x is increasing, then k is preceded by a plus. If the value decreases, then the coefficient k is preceded by a minus.

Despite the apparent sophistication of this formula, many tasks can be solved very quickly and beautifully with its help. Let's try.

A task. The price of the goods was increased by 10% and amounted to 2970 rubles. How much was the product worth before the price increase?

To solve this problem using the simple interest formula, we need three numbers: the original value x , the percentage k and the final value y . Of all three numbers, we know the percentages k = 10 and the final value y = 2970. Please note: 2970 is exactly the final price, i.e. y . Because according to the condition of the problem, the initial price for the goods is unknown (it just needs to be found). But then it was raised, and only then amounted to 2970 rubles.

So we need to find x , i.e. original value. Well, we substitute our numbers into the formula and get:

We add the numbers in the numerator and get:

We reduce one zero in the numerator and denominator, and then multiply both sides of the equation by 10. We get:

11x = 29700

To find x from this simple linear equation, you divide both sides by 11:

x = 29700: 11 = 2700

As you can see, these are quite large numbers, so such calculations cannot be done in the mind. If you meet such a task at the exam, you will have to share a corner. In this case, everything was divided without a trace, and we got the value x :

x=2700

That's what it was worth before the price increase. And it was this number that we needed to find according to the condition of the problem. So that's it: problem solved. Moreover, it was not solved "blank", but with the help of a simple percentage formula - quickly, beautifully and clearly.

Of course, this problem could be solved in a different way. For example, through proportions. Or an exotic method of coefficients. But it will be much better and more reliable if you are armed with several tricks for solving any problem with interest. So be sure to practice using this formula.

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The simplest and most obvious method is to draw up a proportion. All further calculations are based on it. It looks like this:

• 45 is a known number equal to 100%.
• ? - a number that is 15% of 45.

Further, the fraction is simplified to an equation with one unknown. According to mathematical laws, the cross data in proportions are equal to each other, that is: 45*15%=?*100%. To find "?", we use a simple rule and get the following.

The calculation of the proportion formula always takes place according to the principle of multiplying the known data standing on the diagonal and dividing them by a third number.

You can write a formula with any unknown in . In order not to be confused, a percentage or a number is obtained as a result, we recall the rule of reduction in fractions - if the percent sign (%) or currency symbol (rub) is present both above and below, it is reduced. Example:

The result of the calculation is the amount of money.

How to find percentage of a number. Options

Let's consider in order the situations of finding percentages.

How to find 100%. It is necessary to calculate the number, 15% of which is equal to 45. We make up the proportion:

We calculate by the formula: (45*100)/15=300

If it is not known how much is 100%. Sometimes the calculation is carried out with respect to the same initial data, but their exact value is not known. For example: yesterday 15% of the total number of cookies in the amount of 450 rubles, and today 25%.

How much did you sell today? Since the sum for 100% is the total value for both 15% and 25%, it is possible to carry out calculations without looking for the total cost.

We calculate by the formula: (25*450)/15=750

You can complicate the task if there is no confidence in the calculations, or there is a need to check the result. To do this, first find 100%, based on full data (15% costs 450 rubles), and then 25% are counted from 100%.

How much less than another number as a percentage

For example: the usual cost of the powder is 500 rubles. According to the action, the price was reduced to 480 rubles. How much is the share price less than the initial percentage? First, the percentage component of the promotional price from the base price is found, and then their difference is found. We make a proportion:

We calculate by the formula: (480*100)/500=96. 100%-96%=4%. The share price is less than the original price by 4%.

How much a number is greater than another as a percentage. Example: the keyboard cost 300 rubles, and after the appreciation of the dollar, the price rose to 390 rubles. How much has the price of the keyboard changed as a percentage? First, the total interest rate of the new price, relative to the original one, is found, then their difference is calculated. We make a proportion:

We calculate by the formula: (390 * 100) / 300 \u003d 130. 130%-100%=30%. The price has increased by 30%.

The unknown number is greater than the known number by a certain percentage. Example: a product in a store is 15% more expensive than a product in a warehouse. The price of sugar in the warehouse is 50 rubles and is equal to 100%. Shop price - 100% + 15% = 115%. We calculate by the formula: (115 * 50) / 100 \u003d 57.5

The unknown number is less than the known number by a given percentage. Example: wholesale 5% cheaper. The retail price is 60 rubles and is equal to 100 percent, for wholesale - 100% -5% \u003d 95%. We make a proportion:

We calculate by the formula: (60*95)/100=57

Percentage between two numbers. A situation where a number is known that is 100% and a number that is a certain fraction of the original. Example: a batch of 60 boxes was expected, but 53 were delivered. What percentage of the plan was fulfilled. We make a proportion:

We calculate by the formula: (53 * 100) / 60 \u003d 88.3

The most difficult “task” is not to get confused in drawing up a proportion.

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In the process of solving problems 149–156, it is necessary to bring students to an understanding of the rule for finding a part of a number:

To find the part of a number expressed as a fraction, you can divide this number by the denominator of the fraction and multiply the result by its numerator.

Of course, students can formulate this rule only for specific situations: in order to find 3 / 4 number 24, you can divide this number by the denominator fractions 4 and multiply the result by the numerator 3.

149 . a) 12 birds were sitting on a branch; 2/3 of their number flew away. How many birds have flown?

b) There are 32 students in the class; 3/4 of all students went skiing. How many students skied?

150 . a) Cyclists traveled 48 in two days km. On the first day they traveled 2/3 of the way. How many kilometers did they drive on the second day?

b) Someone, having 350 rubles, spent 5/7 of his money. How much money does he have left?

c) There are 24 pages in the notebook. The girl filled out all the pages of the notebook on the 5th/8th. How many unwritten pages are left?

151 . Old problem. Bought a chest of drawers for 36 R., I then had to sell it for 7/12 of the price. How many rubles did I lose in this sale?

152 . Autotourists traveled 360 in three days km; on the first day they traveled 2/5, and on the second day they traveled 3/8 of the entire journey. How many kilometers did the autotourists drive on the third day?

153 . 1) There are 24 girls and several boys in the drama club. The number of boys is 3/8 of the number of girls. How many students are in the drama club?

2) There are 45 commemorative ruble coins in the collection. The number of 3 and 5 ruble coins is 2/9 of the number of ruble coins. How many commemorative coins of 1, 3 and 5 rubles are in the collection?

Students must solve tasks 154–156 by first finding the indicated part of the value, and then increasing or decreasing this value by the found part. Another solution will be shown later.

154 . 1) Reduce 90 rubles by 1/10 of this amount.

2) Increase 80 rubles by 2/5 of this amount.

155 . Last month the price of the item was 90 R. Now it has gone down by 3/10 of that amount. What is the price of the item now?

156 . Last month the salary was 400 R. Now it has increased by 2/5 of that amount. What is the salary now?

In the process of solving problems 157–158 and the following problems, students should be led to understand and correctly apply the rule for finding a number by its part:

To find a number by its part, expressed as a fraction, you can divide this part by the numerator of the fraction and multiply the result by its denominator.

The formulation of this rule is complicated because of the need
somehow call the number that we have named « part » . The authors of textbooks also have to circumvent this difficulty. So in the textbook I.V. Baranova and Z.G. Borchug's rule is formulated only for specific cases: to find a number, 3 / 5 which is 90 km, it is necessary to divide 90 km by the numerator of the fraction 3 and multiply the result by the denominator of the fraction 5.

This is how students can use it. True, when speaking of number, it is better not to use names, since number and magnitude are not the same thing. Later in the same textbook on p. 226, a general rule is formulated in which the term we use « part » corresponding turnover « the number corresponding to it » , which is hardly easier.

157 . a) 120 R. make up 3/4 of the amount of money available. What is this amount?

b) Determine the length of the segment, 3/5 of which are equal to 15 cm.

158 . a) My son is 10 years old. His age is 2/7 of his father's age. How old is father?

b) Daughter 12 years old. Her age is 2/5 of the mother's age. How old is the mother?

For the purchase of vegetables, the hostess spent 6 R., which amounted to 1/6 of the money she had. Then she bought 2 kg apples 7 R. per kilogram. How much money does she have left after these purchases?

160 . Father bought his son a suit for 24 R., on which he spent 1/3 of his money. After that, he bought several books and had 39 left. R. How much did the books cost?

161 . The son is 8 years old, his age is 2/9 of his father's age. And the age of the father is 3/5 of the age of the grandfather. How old is grandpa?

162 .* From the papyrus of Ahmes (Egypt, c. 2000 BC).

A shepherd comes with 70 bulls. He is asked:

How many do you bring from your numerous flock?

The shepherd answers:

I bring two-thirds of a third of the cattle. Count!

How many bulls are in the herd?

Using the percentage calculator you can make all kinds of calculations using percentages. Rounds results to the desired number of decimal places.

How many percent is the number X of the number Y. What number corresponds to X percent of the number Y. Add or subtract percentages from the number.

Interest calculator

clear form

How much is % of number

Calculation

0% of the number 0 = 0

Interest calculator

clear form

What percentage is the number from number

Calculation

The number 15 of the number 3000 = 0.5%

Interest calculator

clear form

add % to number

Calculation

Add 0% to the number 0 = 0

Interest calculator

clear form

Subtract % from the number

Calculation clear all

The calculator is designed specifically for calculating percentages. Allows you to perform various calculations when working with percentages. Functionally consists of 4 different calculators. Examples of calculations on the percentage calculator, see below.

A percent in mathematics is called a hundredth of a number. For example, 5% of 100 equals 5.
This calculator will allow you to accurately calculate the percentage of a given number. There are various calculation modes. You can make various calculations using percentages.

• The first calculator is needed when you want to calculate the percentage of the amount. Those. Do you know the meaning of percentage and amount
• The second is if you need to calculate what percentage is X of Y. X and Y are numbers, and you are looking for the percentage of the first in the second
• The third mode is adding a percentage of the specified number to the given number. For example, Vasya has 50 apples. Misha brought Vasya another 20% of the apples. How many apples does Vasya have?
• The fourth calculator is the opposite of the third. Vasya has 50 apples, and Misha took 30% of the apples. How many apples does Vasya have left?

Frequent tasks

Task 1. An individual entrepreneur receives 100 thousand rubles every month. He works on a simplified basis and pays taxes of 6% per month. How much does an individual entrepreneur have to pay taxes per month?

Solution: We use the first calculator. Enter the bet 6 in the first field, 100000 in the second
We get 6000 rubles. - amount of tax.

Problem 2. Misha has 30 apples. 6 he gave to Katya. What percentage of the total number of apples did Misha give to Katya?

Solution: We use the second calculator - enter 6 in the first field, 30 in the second. We get 20%.

Task 3. At Tinkoff Bank, for replenishing a deposit from another bank, the depositor receives 1% on top of the replenishment amount. Kolya replenished the deposit with a transfer from another bank in the amount of 30,000. What is the total amount Kolya's deposit will be replenished with.

Solution: use the 3rd calculator. Enter 1 in the first field, 10000 in the second. We press the calculation, we get the amount of 10100 rubles.