Tasks with a solution for interest. Start in science. Decrease and increase in interest

Percentage is the form of a decimal fraction. The essence of percent can be understood from the name, which comes from the word "cento", which means "one hundred" in translation. It follows that the percentage is one hundredth of a whole number, taken as one. The% sign is used to represent percentages in mathematics and other fields of science.

Does an ordinary person need this?

Of course, most often, people who work with science have to deal with interest. It is not uncommon for this happiness to go to students as part of the school mathematics curriculum. However, the scope of application of interest is so wide that representatives of various professions and occupations are faced with the need to calculate them. The audience of our site is no exception. Indeed, summer residents are often faced with the task of determining the concentration of a fertilizer solution, calculating the tax on land or other property, determining the size of the loan payment, etc.

In all these cases, you cannot do without the ability to properly handle interest. And they are capricious comrades, they do not like mistakes. Therefore, despite the seeming simplicity of tasks with interest, when solving them, it is necessary to follow a number of certain rules.

Basic reception

All problems involving percentages are fairly easy to solve using the principle of proportion. What is its essence? For example, you need to determine what is equal to 76% of the number 840? For this, an appropriate proportion is drawn up. It equates 840 to 100%. The sought value x is 76%. This makes it possible to compose the following ratio:

840 / x = 100% / 76% or 840 * 76% = x * 100%

Hence it turns out that:
x = 840 * 76% / 100% = 638.4

As you can see, everything is extremely simple.

Basic problem types with percentages

From the point of view of mathematics, there are 3 categories of problems, the solution of which is associated with the calculation of percentages.

First type

This is when you need to find the percentage of a specific number specified in the conditions. Adapting the example to the circumstances of the life of summer residents, the following problem can be cited. Suppose that according to the laws of a certain region, the owner of a private land plot must pay land tax annually. Its size is determined as 2% of the cadastral value of the land. The price of the plot is 327 thousand rubles. What is the annual tax rate? To answer the question posed, a proportion is drawn up:

327 thousand rubles. = 100%;
X thousand rubles = 2%.

Bringing this dependence to the equation, we get: x * 100 = 327 * 2. As a result: x = 327 * 2/100 = 6.54 thousand rubles.

Another example of this kind of tasks is related to the issue that worries the overwhelming majority of summer residents - increases in pensions or wages. Suppose now a person's pension is 7,200 rubles, but from next month they promise to increase it by 15%. How much will it be directly in rubles? The proportion is again compiled:

Second type

In this case, you have to solve the inverse problem, that is, calculate the number from the available percentage. For example, it is known that 10 kg of a certain substance are part of the fertilizer, while representing 40% of its total amount. It is necessary to determine the total mass of the finished fertilizer. For this, a proportion is also compiled, but it will look slightly different:

10 kg - 40%
x kg - 100%

It follows that x = 10 * 100/40 = 25 kg.

Third type

This category includes tasks in which you need to determine the percentage of another through one number. For example, the volume of morning watering of carrots should be - 60 liters. In the evening, 150 liters should be poured onto the beds. What percentage of evening watering is morning watering? The basic relationship is as follows:

150 l - 100%;
60 L cup - x%

Then: x = 60 * 100/150 = 40%

For those summer residents who consider their personal plot as a source of income, the technology for calculating profitability should be interesting. This indicator is used in economics as a measure of the success of an enterprise and is also calculated as a percentage. It is by the level of profitability that they judge how rationally the production process is organized.

So, the calculation is based on two quantities:

* full cost, including all cash costs, including transportation, as well as the purchase of inventory, etc.;

* income received from the sale of the harvested crop.

The difference is the net profit. Pr = D - C. In this case, the profitability formula has the form: P = Pr / C * 100%. Thus, if the total cost of production is 8,200 rubles, and it was sold for 9,000 rubles, the profitability will be: P = (9,000 - 8,200) / 8,200 * 100% = 9.75%. Usually, an acceptable level of profitability in the economy of an enterprise is considered to be 5%. At lower rates, management is advised to look for options for a more rational organization of work.

In any case, you need to know how to solve algebra problems with percentages even at school, and then it will not be difficult for you further.

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To solve most problems in high school mathematics, knowledge of proportioning is required. This simple skill will help you not only perform complex exercises from the textbook, but also delve into the very essence of mathematics. How to make the proportion? Let's take a look at it now.

The simplest example is a problem where three parameters are known, and the fourth must be found. The proportions are, of course, different, but often you need to find some number by percentage. For example, the boy had ten apples in total. He gave the fourth part to his mother. How many apples does the boy have left? This is the simplest example that will allow you to compose the proportion. The main thing is to do it. There were originally ten apples. Let it be 100%. We marked all his apples. He gave away one fourth. 1/4 = 25/100. This means that he has: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the number of fruits remaining to the number of the first available. Now we have three numbers, by which it is already possible to solve the proportion. 10 apples - 100%, NS apples - 75%, where x is the required amount of fruit. How to make the proportion? You need to understand what it is. Mathematically, it looks like this. The equal sign is put for your understanding.

10 apples = 100%;

x apples = 75%.

It turns out that 10 / x = 100% / 75. This is the main property of proportions. After all, the larger x, the more percent this number is from the original. We solve this proportion and we get that x = 7.5 apples. Why the boy decided to give a non-whole amount is unknown to us. Now you know how to proportion. The main thing is to find two relations, one of which contains the unknown unknown.

Solving a proportion often comes down to simple multiplication, and then to division. In schools, children are not explained why this is the case. While it is important to understand that proportional relationships are a mathematical classic, it is the very essence of science. To solve proportions, you need to be able to handle fractions. For example, it is often necessary to convert percentages to fractions. That is, a 95% record will not work. And if you write 95/100 right away, then you can make solid reductions without starting the main count. It should be said right away that if your proportion turned out to be with two unknowns, then it cannot be solved. No professor can help you here. And your task, most likely, has a more complex algorithm of correct actions.

Consider another example where there is no interest. The motorist bought 5 liters of gasoline for 150 rubles. He wondered how much he would pay for 30 liters of fuel. To solve this problem, we denote by x the required amount of money. You can solve this problem yourself and then check the answer. If you have not yet figured out how to proportion, then take a look. 5 liters of gasoline is 150 rubles. As in the first example, we will write down 5L - 150r. Now let's find the third number. Of course, this is 30 liters. Agree that a pair of 30 liters - x rubles is appropriate in this situation. Let's move on to mathematical language.

5 liters - 150 rubles;

30 liters - x rubles;

We solve this proportion:

x = 900 rubles.

So we decided. In your task, do not forget to check the adequacy of the answer. It happens that with the wrong decision, cars reach unrealistic speeds of 5000 kilometers per hour and so on. Now you know how to proportion. You can also solve it. As you can see, this is not difficult.

We continue to study elementary problems in mathematics. This lesson is about interest problems. We will consider several tasks, and also touch on those points that were not mentioned earlier in the study of interest, considering that at first they create difficulties for learning.

Most tasks on percentages boil down to finding a percentage of a number, finding a number by percentage, expressing any part as a percentage, or expressing as a percentage the relationship between several objects, numbers, quantities.

Preliminary skills Lesson content

Methods for finding interest

The percentage can be found in various ways. The most popular way is to divide the number by 100 and multiply the result by the desired percentage.

For example, to find 60% of 200 rubles, you must first divide these 200 rubles into one hundred equal parts:

200 rubles: 100 = 2 rubles.

When we divide a number by 100, we thereby find one percent of that number. So, dividing 200 rubles into 100 parts, we automatically found 1% of two hundred rubles, that is, we found out how many rubles are needed for one part. As you can see from the example, one part (one percent) accounts for 2 rubles.

1% of 200 rubles - 2 rubles

Knowing how many rubles are in one part (by 1%), you can find out how many rubles are in two parts, three, four, five, etc. That is, you can find any number of percentages. To do this, it is enough to multiply these 2 rubles by the required number of parts (percent). Let's find sixty pieces (60%)

2 rubles × 60 = 120 rubles.

2 rubles × 5 = 10 rubles.

Find 90%

2 rubles × 90 = 180 rubles.

Find 100%

2 rubles × 100 = 200 rubles.

100% is all one hundred parts and they are all 200 rubles.

The second way is to represent the percentage as an ordinary fraction and find this fraction from the number from which you want to find the percentage.

For example, let's find the same 60% of 200 rubles. First, let's represent 60% as a fraction. 60% is sixty parts out of a hundred, that is, sixty hundredths:

Now the task can be understood as « find from 200rubles " ... This is the one we studied earlier. Recall that to find the fraction of a number, you need to divide this number by the denominator of the fraction and multiply the result by the numerator of the fraction

200: 100 = 2

2 × 60 = 120

Or multiply the number by the fraction ():

The third way is to represent the percentage as a decimal and multiply the number by that decimal.

For example, let's find the same 60% of 200 rubles. First, we represent 60% as a fraction. 60% percent is sixty parts out of a hundred

Let's divide in this fraction. Move the comma in the number 60 two digits to the left:

Now we find 0.60 from 200 rubles. To find the decimal fraction of a number, you need to multiply this number by a decimal fraction:

200 × 0.60 = 120 rubles.

The given method of finding the percentage is the most convenient, especially if a person is used to using a calculator. This method allows you to find the percentage in one step.

As a rule, it is not difficult to express a percentage in a decimal fraction. Suffice it to prefix "zero integers" before the percentage if the percentage is a two-digit number, or add "zero integers" and another zero if the percentage is a single digit. Examples:

60% = 0.60 - assigned zero integers before 60, since 60 is two-digit

6% = 0.06 - assigned zero integers and one more zero before the number 6, since the number 6 is single-digit.

When dividing by 100, we used the method of moving the comma two digits to the left. In the answer 0.60 the zero after the number 6 is preserved. But if you perform this division with a corner, zero disappears - the answer is 0.6

It must be remembered that decimal fractions 0.60 and 0.6 are equal to the same value:

0,60 = 0,6

In the same "corner", you can continue dividing endlessly, each time assigning zero to the remainder, but this will be a meaningless action:

You can express percentages as a decimal not only by dividing by 100, but also by multiplication. The percent sign (%) itself replaces the 0.01 multiplier. And if we take into account that the number of percent and the percent sign are written together, then between them there is an "invisible" multiplication sign (×).

So, the 45% entry actually looks like this:

Replace the percent sign with a factor of 0.01

This multiplication by 0.01 is performed by moving the comma two digits to the left:

Problem 1... The family's budget is 75 thousand rubles a month. 70% of them are money earned by dad. How much did Mom earn?

Solution

Only 100 percent. If dad earned 70% of the money, then mom earned the remaining 30% of the money.

Problem 2... The family's budget is 75 thousand rubles a month. Of these, 70% is money earned by dad, and 30% is money earned by mom. How much money did everyone make?

Solution

Let's find 70 and 30 percent of 75 thousand rubles. This will determine how much money each earned. For convenience, 70% and 30% will be written as decimal fractions:

75 × 0.70 = 52.5 (thousand rubles earned by dad)

75 × 0.30 = 22.5 (thousand rubles.Mother earned)

Examination

52,5 + 22,5 = 75

75 = 75

Answer: 52.5 thousand rubles. dad earned, 22.5 rubles. Mom earned.

Problem 3... When cooled down, the bread loses up to 4% of its weight as a result of water evaporation. How many kilograms will evaporate when 12 tons of bread cool.

Solution

Let's translate 12 tons into kilograms. There is a thousand kilograms in one ton, and 12 times more in 12 tons:

1000 × 12 = 12,000 kg

Now we will find 4% of 12000. The obtained result will be the answer to the problem:

12,000 × 0.04 = 480 kg

Answer: when 12 tons of bread cool down, 480 kilograms will evaporate.

Problem 4... When dried, apples lose 84% of their weight. How many dried apples will you get from 300 kg of fresh apples?

Find 84% of 300 kg

300: 100 × 84 = 252 kg

As a result of drying, 300 kg of fresh apples will lose 252 kg of their weight. To answer the question how many dried apples you get, you need to subtract 252 from 300

300 - 252 = 48 kg

Answer: 300 kg of fresh apples will make 48 kg of dried apples.

Problem 5... Soybean seeds contain 20% oil. How much oil is in 700 kg of soybeans?

Solution

Find 20% of 700 kg

700 × 0.20 = 140 kg

Answer: 700 kg of soy contains 140 kg of oil

Problem 6... Buckwheat contains 10% protein, 2.5% fat and 60% carbohydrates. How many of these products are contained in 14.4 quintals of buckwheat?

Solution

Convert 14.4 centners to kilograms. In one centner, 100 kilograms, in 14.4 centners, 14.4 times more

100 × 14.4 = 1440 kg

Find 10%, 2.5% and 60% of 1440 kg

1440 × 0.10 = 144 (kg of proteins)

1440 × 0.025 = 36 (kg fat)

1440 × 0.60 = 864 (kg of carbohydrates)

Answer: 14.4 cc of buckwheat contains 144 kg of proteins, 36 kg of fat, 864 kg of carbohydrates.

Problem 7... For the tree nursery, the students collected 60 kg of oak, acacia, linden and maple seeds. Acorns accounted for 60%, maple seeds 15%, linden seeds 20% of all seeds, and the rest were acacia seeds. How many kilograms of acacia seeds were collected by the students?

Solution

Let's take the seeds of oak, acacia, linden and maple as 100%. Subtract from these 100% the percentages that express oak, linden and maple seeds. So we find out how many percent are acacia seeds:

100% − (60% + 15% + 20%) = 100% − 95% = 5%

Now we find the seeds of the acacia:

60 × 0.05 = 3 kg

Answer: Schoolchildren collected 3 kg of acacia seeds.

Examination:

60 × 0.60 = 36

60 × 0.15 = 9

60 x 0.20 = 12

60 × 0.05 = 3

36 + 9 + 12 + 3 = 60

60 = 60

Problem 8... A man bought food. Milk costs 60 rubles, which is 48% of the cost of all purchases. Determine the total amount of money spent on groceries.

Solution

This is the task of finding a number by its percentage, that is, by its known part. This problem can be solved in two ways. The first is to express the known number of percentages as a decimal fraction and find the unknown number from this fraction.

Express 48% as a decimal

48% : 100 = 0,48

Knowing that 0.48 is 60 rubles, we can determine the sum of all purchases. To do this, you need to find an unknown number by decimal fraction:

60: 0.48 = 125 rubles

This means that the total amount of money spent on groceries is 125 rubles.

The second way is to first find out how much money is in one percent, then multiply the result by 100

48% is 60 rubles. If we divide 60 rubles by 48, then we find out how many rubles are 1%

60: 48% = 1.25 rubles

1% accounts for 1.25 rubles. Total percent 100. If we multiply 1.25 rubles by 100, we get the total amount of money spent on food

1.25 × 100 = 125 rubles

Problem 9... 35% of dried plums come out of fresh plums. How many fresh plums do you need to take to get 140 kg of dried ones? How many dried plums will you get from 600 kg of fresh plums?

Solution

We express 35% as a decimal fraction and find the unknown number from this fraction:

35% = 0,35

140: 0.35 = 400 kg

To get 140 kg of dried plums, you need to take 400 kg of fresh ones.

Let's answer the second question of the problem - how many dried plums will turn out from 600 kg of fresh ones? If 35% of dried plums come out of fresh plums, then it is enough to find these 35% of 600 kg of fresh plums

600 × 0.35 = 210 kg

Answer: to get 140 kg of dried plums, you need to take 400 kg of fresh ones. From 600 kg of fresh plums, 210 kg of dried plums will turn out.

Problem 10... The assimilation of fats by the human body is 95%. During the month, the student consumed 1.2 kg of fat. How much fat can his body absorb?

Solution

Convert 1.2 kg to grams

1.2 × 1000 = 1200g

Find 95% of 1200 g

1200 x 0.95 = 1140 g

Answer: 1140 g of fat can be absorbed by the student's body.

Expressing numbers as percentages

Percentage, as mentioned earlier, can be represented as a decimal fraction. To do this, it is enough to divide the number of these percentages by 100. For example, we represent 12% as a decimal fraction:

Comment. Now we do not find a percentage of something, but simply write it down as a decimal fraction..

But the reverse process is also possible. The decimal fraction can be represented as a percentage. To do this, you need to multiply this fraction by 100 and put a percent sign (%)

Rewrite decimal 0.12 as a percentage

0.12 x 100 = 12%

This action is called as a percentage or expressing numbers in hundredths.

Multiplication and division are inverse operations. For example, if 2 × 5 = 10, then 10: 5 = 2

Similarly, the division can be written in reverse order. If 10: 5 = 2, then 2 × 5 = 10:

The same thing happens when we express the decimal fraction as a percentage. So, 12% were expressed as a decimal fraction as follows: 12: 100 = 0.12, but then the same 12% were "returned" using multiplication, writing the expression 0.12 × 100 = 12%.

Similarly, you can express as a percentage any other numbers, including integers. For example, let's express the number 3 as a percentage. Multiply this number by 100 and add a percent sign to the result:

3 × 100 = 300%

Large percentages like 300% can be confusing at first, since people are used to counting 100% as the maximum. From additional information about fractions, we know that one whole object can be denoted by one. For example, if there is a whole uncut cake, then it can be denoted by 1

The same cake can be referred to as 100% cake. In this case, both 1 and 100% will mean the same whole cake:

Cut the cake in half. In this case, one will turn into a decimal number 0.5 (since it is half a one), and 100% will turn into 50% (since 50 is half of a hundred)

Let's return the whole cake back, one unit and 100%

Let's draw two more such cakes with the same designations:

If one cake is a unit, then three cakes are three units. Each cake is one hundred percent whole. If you add these three hundred, you get 300%.

Therefore, when converting integers to percentages, we multiply these numbers by 100.

Problem 2... Express the number 5 as a percentage

5 × 100 = 500%

Problem 3... Express the number 7 as a percentage

7 × 100 = 700%

Problem 4... Express the number 7.5 as a percentage

7.5 × 100 = 750%

Problem 5... Express the number 0.5 as a percentage

0.5 × 100 = 50%

Problem 6... Express the number 0.9 as a percentage

0.9 × 100 = 90%

Example 7... Express the number 1.5 as a percentage

1.5 × 100 = 150%

Example 8... Express the number 2.8 as a percentage

2.8 × 100 = 280%

Problem 9... George walks home from school. For the first fifteen minutes, he covered 0.75 paths. The rest of the time, he covered the remaining 0.25 paths. Express as a percentage the portions of the path that George has traveled.

Solution

0.75 × 100 = 75%

0.25 × 100 = 25%

Problem 10... John was treated to half an apple. Express this half as a percentage.

Solution

Half an apple is written as a fraction of 0.5. To express this fraction as a percentage, multiply it by 100 and add a percent sign to the result.

0.5 × 100 = 50%

Fractional analogs

The value, expressed as a percentage, has its counterpart in the form of a regular fraction. So, an analogue for 50% is a fraction. Fifty percent can also be called half.

The analog for 25% is a fraction. Twenty-five percent can also be called a quarter.

The analog for 20% is a fraction. Twenty percent can also be called a fifth.

The analog for 40% is a fraction.

The analog for 60% is the fraction

Example 1... Five centimeters is 50% of a decimeter, or just half. In all cases, we are talking about the same value - five centimeters out of ten

Example 2... Two and a half centimeters is 25% of a decimeter, or or just a quarter

Example 3... Two centimeters is 20% of a decimeter or

Example 4... Four centimeters is 40% of a decimeter or

Example 5... Six centimeters is 60% of a decimeter or

Decrease and increase in interest

When increasing or decreasing the value, expressed as a percentage, the preposition "on" is used.

Examples of:

• Increase by 50% means increase the value by 1.5 times;
• Increase by 100% - means increase the value by 2 times;
• To increase by 200% means to increase by 3 times;
• Decrease by 50% - means to decrease the value by 2 times;
• Reducing by 80% means reducing by 5 times.

Example 1... Ten centimeters have been increased by 50%. How many centimeters did you get?

To solve such problems, you need to take the initial value as 100%. The initial value is 10 cm. 50% of them are 5 cm

The original 10 cm was increased by 50% (by 5 cm), which means it turned out 10 + 5 cm, that is, 15 cm

An analogue of increasing ten centimeters by 50% is a multiplier of 1.5. If you multiply 10 cm by it, you get 15 cm

10 × 1.5 = 15 cm

Therefore, the expressions “increase by 50%” and “increase by 1.5 times” say the same thing.

Example 2... Five centimeters have been increased by 100%. How many centimeters did you get?

Let's take the original five centimeters as 100%. One hundred percent of these five centimeters will be 5 cm themselves.If you increase 5 cm by the same 5 cm, you get 10 cm

The analogue of an increase of five centimeters by 100% is a factor of 2. If you multiply 5 cm by it, you get 10 cm

5 × 2 = 10 cm

Therefore, the expressions “increase by 100%” and “increase by 2 times” mean the same thing.

Example 3... Five centimeters have increased by 200%. How many centimeters did you get?

Let's take the original five centimeters as 100%. Two hundred percent is two times one hundred percent. That is, 200% of 5 cm will be 10 cm (5 cm for every 100%). If you increase 5 cm by these 10 cm, you get 15 cm

The analogue of an increase of five centimeters by 200% is a factor of 3. If you multiply 5 cm by it, you get 15 cm

5 × 3 = 15 cm

Therefore, the expressions “increase by 200%” and “increase by 3 times” mean the same thing.

Example 4... Ten centimeters have been reduced by 50%. How many centimeters are left?

Let's take the original 10 cm as 100%. Fifty percent of 10 cm is 5 cm.If you reduce 10 cm by these 5 cm, there will be 5 cm

The analogue of reducing ten centimeters by 50% is the divider 2. If you divide 10 cm by it, you get 5 cm

10: 2 = 5 cm

Therefore, the expressions "reduce by 50%" and "reduce by 2 times" say the same thing.

Example 5... Ten centimeters have been reduced by 80%. How many centimeters are left?

Let's take the original 10 cm as 100%. Eighty percent of 10 cm is 8 cm.If you reduce 10 cm by this 8 cm, you will have 2 cm

The analogue of reducing ten centimeters by 80% is the divisor 5. If you divide 10 cm by it, you get 2 cm

10: 5 = 2 cm

Therefore, the expressions "reduce by 80%" and "reduce by 5 times" say the same thing.

When solving problems for decreasing and increasing percentages, you can multiply / divide the value by the multiplier specified in the problem.

Problem 1... How much has the value changed as a percentage, if it has increased by 1.5 times?

The value referred to in the task can be designated as 100%. Then multiply this 100% by a factor of 1.5

100% × 1.5 = 150%

Now, subtract the initial 100% from the received 150% and get the answer to the problem:

150% − 100% = 50%

Problem 2... How much has the value changed as a percentage if it has decreased by 4 times?

This time there will be a decrease in the value, so we will perform division. The value referred to in the problem is denoted as 100%. Next, divide this 100% by a divisor of 4

Subtract the received 25% from the initial 100% and get the answer to the problem:

100% − 25% = 75%

This means that with a decrease in the value by 4 times, it decreased by 75%.

Problem 3... How much has the value changed if it has decreased by 5 times?

The value referred to in the problem is denoted as 100%. Next, divide this 100% by divisor 5

Subtract the resulting 20% ​​from the initial 100% and get the answer to the problem:

100% − 20% = 80%

This means that with a decrease in the value by 5 times, it decreased by 80%.

Problem 4... How much has the value changed as a percentage if it has decreased 10 times?

The value referred to in the problem is denoted as 100%. Next, divide this 100% by a divisor of 10

Subtract the received 10% from the initial 100% and get the answer to the problem:

100% − 10% = 90%

This means that with a decrease in the value by 10 times, it decreased by 90%.

The problem of finding the percentage

To express something as a percentage, you first need to write a fraction showing how much the first number is from the second, then divide in this fraction and express the result as a percentage.

For example, let's say there are five apples. In this case, two apples are red, three are green. Let's express the red and green apples as a percentage.

First you need to find out what part are red apples. There are five apples in total, and two red ones. This means that two out of five or two-fifths are red apples:

There are three green apples. This means that three out of five or three-fifths are green apples:

We have two fractions and. Let's divide in these fractions

We got decimals 0.4 and 0.6. Now let's express these decimal fractions as a percentage:

0.4 × 100 = 40%

0.6 × 100 = 60%

This means that 40% are red apples, 60% are green.

And all five apples are 40% + 60%, that is, 100%

Problem 2... Mother gave two sons 200 rubles. Mom gave the younger brother 80 rubles, and the older one 120 rubles. Express as a percentage the money given to each brother.

Solution

The younger brother received 80 rubles out of 200 rubles. We write down the fraction eighty two hundredth:

The elder brother received 120 rubles out of 200 rubles. We write down the fraction one hundred twenty two hundredth:

We have fractions and. Let's divide in these fractions

Let us express the results obtained as a percentage:

0.4 × 100 = 40%

0.6 × 100 = 60%

This means that 40% of the money was received by the younger brother, and 60% - by the older one.

Some fractions, showing how much the first number is from the second, can be abbreviated.

So the fractions could be reduced. From this, the answer to the problem would not change:

Problem 3... The family's budget is 75 thousand rubles a month. Of these, 52.5 thousand rubles. - money earned by dad. 22.5 thousand rubles - money earned by mom. Express as a percentage of the money mom and dad earned.

Solution

This task, like the previous one, is the task of finding the percentage.

Let's express as a percentage the money dad earned. He earned 52.5 thousand rubles out of 75 thousand rubles

Let's divide in this fraction:

0.7 × 100 = 70%

This means that dad earned 70% of the money. Further, it is easy to guess that the mother earned the remaining 30% of the money. After all, 75 thousand rubles is all 100% of the money. To be sure, we will do a check. Mom earned 22.5 thousand rubles. from 75 thousand rubles. We write down the fraction, perform division and express the result as a percentage:

Problem 4... The student is practicing doing pull-ups on the bar. Last month, he could do 8 pull-ups per set. This month, he can do 10 pull-ups per set. By what percentage did he increase the number of pull-ups?

Solution

Find out how many more pull-ups the student does in the current month than in the past

Find out what part two pull-ups are from eight pull-ups. To do this, we find the ratio of 2 to 8

Let's divide in this fraction

Let's express the result as a percentage:

0.25 × 100 = 25%

This means that the student has increased the number of pull-ups by 25%.

This problem can be solved by the second, faster method - find out how many times 10 pull-ups are more than 8 pull-ups and express the result as a percentage.

To find out how many times ten pull-ups are more than eight pull-ups, you need to find a ratio of 10 to 8

Divide the resulting fraction

Let's express the result as a percentage:

1.25 × 100 = 125%

The pull-up rate this month is 125%. This statement must be understood exactly as "Is 125%", not how "The indicator increased by 125%"... These are two different statements expressing different quantities.

The saying "is 125%" should be understood as "eight pull-ups, which are 100% plus two pull-ups, which are 25% of the eight pull-ups." Graphically, it looks like this:

And the statement “increased by 125%” should be understood as “to the current eight pull-ups, which were 100%, another 100% (8 more pull-ups) plus another 25% (2 pull-ups) were added”. A total of 18 pull-ups are obtained.

100% + 100% + 25% = 8 + 8 + 2 = 18 pull-ups

Graphically, this statement looks like this:

All in all, it turns out to be 225%. If we find 225% of eight pull-ups, we get 18 pull-ups.

8 × 2.25 = 18

Problem 5... Last month, the salary was 19.2 thousand rubles. In the current month, it amounted to 20.16 thousand rubles. How much has the salary increased?

This problem, like the previous one, can be solved in two ways. The first is to first find out how many rubles the salary has increased. Next, find out what part of this increase is from the salary of the last month

Find out how many rubles the salary has increased:

20.16 - 19.2 = 0.96 thousand rubles.

Find out what part of 0.96 thousand rubles. ranges from 19.2. To do this, we find the ratio of 0.96 to 19.2

Let's perform division in the resulting fraction. On the way, remember:

Let's express the result as a percentage:

0.05 × 100 = 5%

This means that the salary has increased by 5%.

Let's solve the problem in the second way. Find out how many times 20.16 thousand rubles. more than 19.2 thousand rubles. To do this, we find the ratio of 20.16 to 19.2

Let's perform division in the resulting fraction:

Let's express the result as a percentage:

1.05 × 100 = 105%

The salary is 105%. That is, this includes 100%, which amounted to 19.2 thousand rubles, plus 5% which is 0.96 thousand rubles.

100% + 5% = 19,2 + 0,96

Problem 6... The price of a laptop is up 5% this month. What is its price if last month it cost 18.3 thousand rubles?

Solution

Finding 5% of 18.3:

18.3 × 0.05 = 0.915

Add this 5% to 18.3:

18.3 + 0.915 = 19.215 thousand rubles.

Answer: the price of a laptop is 19.215 thousand rubles.

Problem 7... The price of a laptop is down 10% this month. What is its price if last month it cost 16.3 thousand rubles?

Solution

Find 10% of 16.3:

16.3 x 0.10 = 1.63

Subtract this 10% from 16.3:

16.3 - 1.63 = 14.67 (thousand rubles)

Similar tasks can be written briefly:

16.3 - (16.3 × 0.10) = 14.67 (thousand rubles)

Answer: the price of a laptop is 14.67 thousand rubles.

Problem 8... Last month, the price of a laptop was 21 thousand rubles. This month the price has risen to 22.05 thousand rubles. How much has the price increased?

Solution

Determine how much rubles the price has increased

22.05 - 21 = 1.05 (thousand rubles)

Find out what part of 1.05 thousand rubles. is from 21 thousand rubles.

Express the result as a percentage

0.05 × 100 = 5%

Answer: laptop price increased by 5%

Problem 8... The worker had to make 600 parts according to the plan, and he made 900 parts. By what percentage has he fulfilled the plan?

Solution

We find out how many times 900 parts are more than 600 parts. To do this, we find the ratio of 900 to 600

The value of this fraction is 1.5. Let's express this value as a percentage:

1.5 × 100 = 150%

This means that the worker fulfilled the plan by 150%. That is, he completed it 100%, having produced 600 parts. Then he made another 300 parts, which is 50% of the original plan.

Answer: the worker fulfilled the plan by 150%.

Percentage comparison

We have compared values ​​many times in different ways. Our first tool was the difference. So, for example, to compare 5 rubles and 3 rubles, we wrote down the difference 5−3. Having received the answer 2, one could say that "five rubles is more than three rubles for two rubles."

The answer obtained as a result of subtraction in everyday life is called not "difference", but "difference".

So, the difference between five and three rubles is two rubles.

The next tool we used to compare values ​​was ratio. The ratio allowed us to find out how many times the first number is greater than the second (or how many times the first number contains the second).

So, for example, ten apples are five times more than two apples. Or put another way, ten apples contains two apples five times. This comparison can be written using the relation

But the values ​​can also be compared in percentages. For example, to compare the price of two goods not in rubles, but to estimate how much the price of one good is more or less than the price of the other in percentage.

To compare the values ​​in percent, one of them must be designated as 100%, and the second based on the conditions of the problem.

For example, let's find out by how many percent ten apples are more than eight apples.

For 100%, you need to designate the value with which we compare something. We are comparing 10 apples to 8 apples. So, for 100% we designate 8 apples:

Now our task is to compare by how many percent 10 apples are more than these 8 apples. 10 apples are 8 + 2 apples. This means that by adding two more apples to eight apples, we will increase 100% by a certain number of percent. To find out which one, let's determine how many percent of eight apples are two apples

By adding this 25% to eight apples, we get 10 apples. And 10 apples is 8 + 2, that is, 100% and another 25%. In total, we get 125%

This means that ten apples are more than eight apples by 25%.

Now let's solve the inverse problem. Let's find out how many percent eight apples are less than ten apples. The answer immediately suggests itself that eight apples are 25% less. However, it is not.

We are comparing eight apples to ten apples. We agreed that for 100% we will take what we compare with. Therefore, this time we take 10 apples for 100%:

Eight apples is 10−2, that is, decreasing 10 apples by 2 apples, we will decrease them by a certain number of percent. To find out which one, let's determine how many percent of ten apples are two apples

Subtracting this 20% from ten apples, we get 8 apples. And 8 apples are 10−2, that is, 100% and minus 20%. In total, we get 80%

This means that eight apples are less than ten apples by 20%.

Problem 2... By what percentage is 5000 rubles more than 4000 rubles?

Solution

Let's take 4000 rubles for 100%. 5 thousand more than 4 thousand per 1 thousand. This means that by increasing four thousand by one thousand, we will increase four thousand by a certain amount of percent. Let's find out which one. To do this, let's determine what part one thousand is from four thousand:

Let's express the result as a percentage:

0.25 × 100 = 25%

1000 rubles from 4000 rubles are 25%. If you add this 25% to 4000, you get 5000 rubles. This means that 5000 rubles is 25% more than 4000 rubles

Problem 3... How many percent is 4000 rubles less than 5000 rubles?

This time we compare 4000 with 5000. Let's take 5000 as 100%. Five thousand is more than four thousand for one thousand rubles. Find out what part one thousand is from five thousand

A thousand from five thousand is 20%. If we subtract this 20% from 5,000 rubles, we get 4,000 rubles.

This means that 4000 rubles is less than 5000 rubles by 20%

Concentration problems, alloys and mixtures

Let's say there is a desire to make some kind of juice. We have water and raspberry syrup at our disposal

Pour 200 ml of water into a glass:

Add 50 ml of raspberry syrup and stir the resulting liquid. As a result, we get 250 ml of raspberry juice. (200 ml water + 50 ml syrup = 250 ml juice)

How much of the resulting juice is raspberry syrup?

Raspberry syrup makes up the juice. We calculate this ratio, we get the number 0.20. This number shows the amount of dissolved syrup in the resulting juice. Let's call this number concentration of syrup.

The concentration of a solute is the ratio of the amount of a solute or its mass to the volume of a solution.

Concentration is usually expressed as a percentage. Let's express the concentration of the syrup as a percentage:

0.20 × 100 = 20%

Thus, the concentration of syrup in raspberry juice is 20%.

Substances in solution can be heterogeneous. For example, mix 3 liters of water and 200 g of salt.

The mass of 1 liter of water is 1 kg. Then the mass of 3 liters of water will be 3 kg. We translate 3 kg into grams, we get 3 kg = 3000 g.

Now put 200 g of salt in 3000 g of water and mix the resulting liquid. The result will be a saline solution, the total mass of which will be 3000 + 200, that is, 3200 g. Let's find the salt concentration in the resulting solution. To do this, we find the ratio of the mass of the dissolved salt to the mass of the solution

This means that when mixing 3 liters of water and 200 g of salt, you get a 6.25% salt solution.

Similarly, the amount of substance in the alloy or in the mixture can be determined. For example, the alloy contains tin with a mass of 210 g, and silver with a mass of 90 g. Then the mass of the alloy will be 210 + 90, that is, 300 g. The alloy will contain tin, and silver. The percentage of tin will be 70%, and silver 30%

When two solutions are mixed, a new solution is obtained, consisting of the first and second solutions. A new solution may have a different concentration of the substance. A useful skill is the ability to solve concentration, alloy and mixture problems. In general, the meaning of such tasks is to track the changes that occur when mixing solutions of different concentrations.

Mix two raspberry juices. The first 250 ml juice contains 12.8% raspberry syrup. And the second juice with a volume of 300 ml contains 15% raspberry syrup. Pour these two juices into a large glass and mix. As a result, we get a new 550 ml juice.

Now let's determine the concentration of syrup in the resulting juice. The first drained juice with a volume of 250 ml contained 12.8% syrup. And 12.8% of 250 ml is 32 ml. This means that the first juice contained 32 ml of syrup.

The second drained juice with a volume of 300 ml contained 15% syrup. And 15% of 300 ml is 45 ml. This means that the second juice contained 45 ml of syrup.

Let's add the amounts of syrups:

32 ml + 45 ml = 77 ml

This 77 ml of syrup is contained in the new juice, which has a volume of 550 ml. Let's determine the concentration of syrup in this juice. To do this, we find the ratio of 77 ml of dissolved syrup to the volume of juice of 550 ml:

This means that when mixing 12.8% raspberry juice with a volume of 250 ml and 15% ‍ raspberry juice with a volume of 300 ml, you get 14% raspberry juice with a volume of 550 ml.

Problem 1... There are 3 solutions of sea salt in water: the first solution contains 10% salt, the second contains 15% salt and the third contains 20% salt. Mixed 130 ml of the first solution, 200 ml of the second solution and 170 ml of the third solution. Determine the percentage of sea salt in the resulting solution.

Solution

Determine the volume of the resulting solution:

130 ml + 200 ml + 170 ml = 500 ml

Since the first solution contained 130 × 0.10 = 13 ml of sea salt, in the second solution 200 × 0.15 = 30 ml of sea salt, and in the third - 170 × 0.20 = 34 ml of sea salt, the resulting solution will contain contain 13 + 30 + 34 = 77 ml of sea salt.

Let's determine the concentration of sea salt in the resulting solution. To do this, we find the ratio of 77 ml of sea salt to the volume of a solution of 500 ml

This means that the resulting solution contains 15.4% sea salt.

Problem 2... How many grams of water should be added to a 50 g solution containing 8% salt to obtain a 5% solution?

Solution

Note that if you add water to the existing solution, the amount of salt in it will not change. Only its percentage will change, since the addition of water to the solution will lead to a change in its mass.

We need to add such an amount of water that eight percent of the salt would be five percent.

Determine how many grams of salt are contained in 50 g of solution. For this we find 8% of 50

50g × 0.08 = 4g

8% of 50 g is 4 g. In other words, for eight parts out of a hundred, there are 4 grams of salt. Let's make it so that these 4 grams are not in eight parts, but in five parts, that is, 5%

4 grams - 5%

Now knowing that there are 4 grams per 5% solution, we can find the mass of the entire solution. For this you need:

4g: 5 = 0.8g
0.8g × 100 = 80g

80 grams of solution is the mass at which 4 grams of salt will be in a 5% solution. And to get these 80 grams, you need to add 30 grams of water to the original 50 grams.

This means that to obtain a 5% salt solution, you need to add 30 g of water to the existing solution.

Problem 2... Grapes contain 91% moisture and raisins 7%. How many kilograms of grapes does it take to get 21 kilograms of raisins?

Solution

Grapes are composed of moisture and pure substance. If fresh grapes contain 91% moisture, then the remaining 9% will account for the pure substance of these grapes:

Raisins contain 93% pure substance and 7% moisture:

Note that in the process of turning grapes into raisins, only the moisture of this grape disappears. The pure substance remains unchanged. After the grapes turn into raisins, the resulting raisins will have 7% moisture and 93% pure substance.

Let's determine how much pure substance is contained in 21 kg of raisins. For this we find 93% of 21 kg

21 kg × 0.93 = 19.53 kg

Now let's go back to the first picture. Our task was to determine how many grapes you need to take to get 21 kg of raisins. The pure substance weighing 19.53 kg will account for 9% of the grapes:

Now, knowing that 9% of the pure substance is 19.53 kg, we can determine how many grapes are required to obtain 21 kg of raisins. To do this, you need to find the number by its percentage:

19.53 kg: 9 = 2.17 kg
2.17 kg × 100 = 217 kg

This means that to get 21 kg of raisins, you need to take 217 kg of grapes.

Problem 3... In the alloy of tin and copper, copper is 85%. How much alloy should you take to contain 4.5 kg of tin?

Solution

If the alloy contains 85% copper, then the remaining 15% will be tin:

The question is how much alloy should be taken so that it contains 4.5 tin. Since the alloy contains 15% tin, 4.5 kg of tin will account for these 15%.

And knowing that 4.5 kg of alloy is 15%, we can determine the mass of the entire alloy. To do this, you need to find the number by its percentage:

4.5 kg: 15 = 0.3 kg
0.3 kg × 100 = 30 kg

This means that you need to take 30 kg of the alloy so that it contains 4.5 kg of tin.

Problem 4... A certain amount of a 12% hydrochloric acid solution was mixed with the same amount of a 20% solution of the same acid. Find the concentration of the resulting hydrochloric acid.

Solution

Let's depict the first solution in the form of a straight line in the figure and select 12% on it

Since the number of solutions is the same, you can draw the same figure next to it, illustrating the second solution with a hydrochloric acid content of 20%

We got two hundred parts of the solution (100% + 100%), thirty-two parts of which are hydrochloric acid (12% + 20%)

Determine which part 32 parts are from 200 parts

This means that when mixing a 12% solution of hydrochloric acid with the same amount of a 20% solution of the same acid, a 16% solution of hydrochloric acid will be obtained.

To check, let's imagine that the mass of the first solution was 2 kg. The mass of the second solution will also be 2 kg. Then, when these solutions are mixed, 4 kg of solution will be obtained. In the first solution of hydrochloric acid there was 2 × 0.12 = 0.24 kg, and in the second - 2 × 0.20 = 0.40 kg. Then in a new solution of hydrochloric acid there will be 0.24 + 0.40 = 0.64 kg. The concentration of hydrochloric acid will be 16%

Tasks for independent solution

on, we will find 60% of the number

Now we will increase the number by the found 60%, i.e. by the number

Answer: the new value is

Problem 12. Answer the following questions:

1) Spent 80% of the amount. How much percent of this amount is left?
2) Men make up 75% of all factory workers. What percentage of the plant workers are women?
3) Girls make up 40% of the class. What percentage of the class are boys?

A Solution

Let's use a variable. Let be P this is the original number referred to in the problem. Let's take this initial number P for 100%

Reduce this original number P by 50%

The new number is now 50% of the original number. Find out how many times the original number P more than the new number. To do this, we find the ratio of 100% to 50%

The original number is twice the new one. This can be seen even from the picture. And to make the new number equal to the original, it must be doubled. And doubling the number means increasing it by 100%.

This means that the new number, which is half of the original number, needs to be increased by 100%.

Considering the new number, it is also taken as 100%. So, in the figure shown, the new number is half of the original number and is signed as 50%. In relation to the original number, the new number is half. But if we consider it separately from the original, it must be taken as 100%.

Therefore, in the figure, the new number, which is depicted as a line, was initially designated as 50%. But then we designated this number as 100%.

Answer: to get the original number, the new number must be increased by 100%.

Problem 16. Last month, 15 accidents occurred in the city.
This month, this figure has dropped to 6. By what percentage has the number of accidents decreased?

Solution

There were 15 accidents last month. This month 6. This means that the number of accidents decreased by 9.
Let's take 15 accidents as 100%. By reducing 15 accidents by 9, we will reduce them by a certain number of percent. To find out which one, we find out which part of the 9 accidents is from 15 accidents

Answer: the concentration of the resulting solution is 12%.

Problem 18. A certain amount of an 11% solution of a certain substance was mixed with the same amount of a 19% solution of the same substance. Find the concentration of the resulting solution.

Solution

The mass of both solutions is the same. Each solution can be taken as 100%. After adding the solutions, you get a 200% solution. The first solution contained 11% of the substance, and the second 19% of the substance. Then in the resulting 200% solution there will be 11% + 19% = 30% of the substance.

Determine the concentration of the resulting solution. To do this, we find out which part thirty parts of a substance make up from two hundred parts of a substance:

1,10. This means that the price for the first month will become 1.10.

In the second month, the price also increased by 10%. Add ten percent of this price to the current price of 1.10, we get 1.10 + 0.10 x 1.10. This sum is equal to the expression 1.21 . This means that the price for the second month will become 1.21.

In the third month, the price also increased by 10%. Add to the current price 1.21 ten percent of this price, we get 1.21 + 0.10 x 1.21. This sum is equal to 1.331 . Then the price for the third month will become 1.331.

Let's calculate the difference between the new and old prices. If the original price was 1, then it increased by 1.331 - 1 = 0.331. Express this result as a percentage, we get 0.331 × 100 = 33.1%

Answer: for 3 months food prices increased by 33.1%.

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Interest-based problems first appear in the life of young mathematicians in grade 5 and accompany them until the final exams. Percentage-related tasks are in the USE options (in particular, task number 17 of the profile exam) and the OGE. Interest will inevitably be found in physics, chemistry, and economics courses. After all, in our daily life we ​​constantly come across this concept (think, for example, the rates on loans or the generous promises of 90% discounts in stores).

In this article, we will start with the simplest definitions and examples, we will gradually increase the level of complexity and by the 4th part we will get to quite difficult problems.

Interest. Initial information.

How to find a percentage of a number

Surprisingly, many graduates fail to explain clearly what is percent... But everything is very simple:

Percent is one hundredth of the number.

Why exactly the hundredth? Yes, simply because it is convenient to divide by 100, and a hundred is not too much and not too little (not a very strict definition).

To find 1% of a number, you just need to divide that number by 100.

Example 1... Find 1% of 1200, 1% of 2.1% of 98765.

1% of 1200 is 12, since 1200: 100 = 12;
1% of 2 is 0.02, since 2: 100 = 0.02;
1% of 98765 = 98765: 100 = 987.65.

Exercise 1... Calculate 1% of 450, 1% of 12000, 1% of 9.

Assignment 2... Calculate 1% of 1% of 6700.

How to find a few percent of a number

Now suppose we need to find not 1% of the number, but, say, 12%. How to do it? You can, of course, first find one percent, and then multiply the result by 12. But why do two things if you can get by with one? One percent is one hundredth, and t percent is t hundredths. To find, for example, 12 hundredths of a number, you need to multiply the number by 0.12. We get a universal rule:

To find t% of a number, you need to multiply this number by t 100.
t percent of A = A ⋅ t 100

Example 2... Find 17% of 300, 86% of 20, 140% of 2, 0.1% of 4000.

17% of 300 is 51, since 300 * 0.17 = 51 (multiply the number by seventeen hundredths);
86% of 20 is 17.2, because 20 * 0.86 = 17.2 (multiply by 86/100);
140% of 2 = 2 * 1.4 = 2.8 (1.4 is just 140/100);
0.1% of 4000 = 0.001 * 4000 = 4 (0.001 is 0.1 / 100).

Assignment 3... Calculate 14% of 1200, 57% of 50, 250% of 4, 0.02% of 1,000,000.

Example 3... Calculate 18% of 80% of 1000. Is it true that this is the same as 98% of 1000?

Let's find first 80% of 1000: 1000 * 0.8 = 800.
We are looking for 18% of the resulting number: 800 * 0.18 = 144.
Find now 98% of 1000. Multiply 1000 by 98/100 and get 980.
As you can see, the results are different.

Assignment 4... Calculate 120% of 40% of 350.

How to find "percent of interest"

What if we need to calculate a long sequence of "percent of percent"? Let's say 10% of 10% of 10% of 10% of 200. You can, of course, act sequentially and divide the task into 4 actions, but there is an easier way.

Example 4... Calculate 20% of 30% of 40% of 10,000.

Why do several consecutive multiplications when everything can be reduced to one line:
0,2*0,3*0,4*10000 = 24.

See how simple it is! By the way, no parentheses are needed in this case.

Assignment 5... Calculate 50% of 50% of 40% of 2000.

Assignment 6... In the first week of January, 40% of the monthly snowfall (90 mm) fell, with 90% of this amount falling on Wednesday, and 70% of the precipitation fell in the first half of this day. How many mm of snow fell on Wednesday morning?

So, let's summarize some of the results:

• Percentage is one hundredth of a number.
• To calculate 1%, divide the number by 100 (or multiply by 0.01).
• To find t% of a number, you need to multiply the number by t hundredths.

A small test on the topic "Percentage"

Take a couple of minutes and take a little test on the "Interest" topic. Please enter a whole number or a decimal in your answer. Always use a comma as a decimal separator (for example, 1.2, but not 1.2!) Good luck!

The concept of percentage occurs too often in our life, so it is very important to know how to solve problems with interest. In principle, this is not a difficult matter, the main thing is to understand the principle of working with interest.

What is percentage

We operate with the concept of 100 percent, and accordingly, one percent is a hundredth of a certain number. And all calculations are based on this ratio.

For example, 1% of 50 is 0.5, 15 of 700 is 7.

How to solve

1. Knowing that one percent is one hundredth of the presented number, you can find any number of required percentages. To make it clearer, let's try to find 6 percent of the number 800. This is done simply.
   • First, we find one percent. To do this, divide 800 by 100. It turns out 8.
   • Now this very one percent, that is, 8, is multiplied by the number of percent we need, that is, by 6. It turns out 48.
   • Let's fix the result by repetition.

   15% of 150. Solution: 150/100 * 15 = 22.

   28% of 1582. Solution: 1582/100 * 28 = 442.

2. There are other problems when you are given values ​​and you need to find percentages. For example, you know that there are 5 red roses out of 75 white in the store, and you need to find out what the percentage of red roses is. If we do not know this percentage, then we will designate it as x.

   There is a formula for this: 75 - 100%

   In this formula, the numbers are multiplied cross by cross, that is, x = 5 * 100/75. It turns out that x = 6% This means that the percentage of scarlet roses is 6%.

3. There is another type of percentage problems, when you need to find how many percent one number is greater or less than another. How to solve problems with interest in this case?

   There are 30 students in the class, 16 of them are boys. The question is, by what percentage there are more boys than girls. First you need to calculate what percentage of the students are boys, then you need to find out how many percent are girls. Find the difference at the end.

   So let's get started. We make up the proportion of 30 per. - 100%

   16 account. -NS %

   Now we count. X = 16 * 100/30, x = 53.4% ​​of all students in the class are boys.

   Now let's find the percentage of girls in the same class. 100-53.4 = 46.6%

Now all that remains is to find the difference. 53.4-46.6 = 6.8%. Answer: there are 6.8% more boys than girls.

Highlights in solving interest

So, so that you don't have problems with how to solve problems with interest, remember a few basic rules:

1. In order not to get confused in interest problems, always be vigilant: go from specific values ​​to percentages and vice versa, if necessary. The main thing is never to confuse one with the other.
2. Be careful when calculating interest. It is important to know from which specific value you need to count. With successive changes in values, the percentage is calculated from the last value.
3. Before writing down the answer, read the entire task again, because it may be that you have found only an intermediate answer, and you need to perform one or a couple of actions.

Thus, solving problems with percentages is not so difficult, the main thing in it is attentiveness and accuracy, as, indeed, in all mathematics. And don't forget that it takes practice to perfect any skill. So decide more and everything will be fine or even great for you.