Task 1. The thickness of 300 sheets of printer paper is 3.3 cm. How thick would a stack of 500 sheets of the same paper be?

Decision. Let x cm be the thickness of a 500-sheet paper ream. In two ways we find the thickness of one sheet of paper:

3,3: 300 or x : 500.

Since the sheets of paper are the same, these two ratios are equal to each other. We get the proportion reminder: proportion is the equality of two ratios):

x=(3.3 · 500): 300;

x=5.5. Answer: pack 500 sheets of paper has a thickness 5.5 cm.

This is a classic reasoning and formulation of a solution to a problem. Such problems are often included in graduate tests, which usually write the solution in this form:

or they decide orally, arguing as follows: if 300 sheets have a thickness of 3.3 cm, then 100 sheets have a thickness 3 times smaller. We divide 3.3 by 3, we get 1.1 cm. This is the thickness of a 100 sheet of paper. Therefore, 500 sheets will have a thickness 5 times greater, therefore, we multiply 1.1 cm by 5 and we get the answer: 5.5 cm.

Of course, this is justified, since the time for testing graduates and applicants is limited. However, in this lesson we will reason and write the solution as it should be done in 6 class.

Task 2. How much water is contained in 5 kg of watermelon if it is known that watermelon consists of 98% water?

Decision.

The entire mass of watermelon (5 kg) is 100%. Water will be x kg or 98%. In two ways, you can find how many kg fall on 1% of the mass.

5: 100 or x : 98. We get the proportion:

5: 100 = x : 98.

x=(5 · 98): 100;

x=4.9 Answer: in 5kg watermelon contains 4.9 kg of water.

The mass of 21 liters of oil is 16.8 kg. What is the mass of 35 liters of oil?

Decision.

Let the mass of 35 liters of oil be x kg. Then in two ways you can find the mass of 1 liter of oil:

16,8: 21 or x : 35. We get the proportion:

16,8: 21=x : 35.

Find the middle term of the proportion. To do this, we multiply the extreme terms of the proportion ( 16,8 and 35 ) and divide by the known middle term ( 21 ). Reduce the fraction by 7 .

Multiply the numerator and denominator of the fraction by 10 so that the numerator and denominator contain only natural numbers. We reduce the fraction by 5 (5 and 10) and on 3 (168 and 3).

Answer: 35 liters of oil have a mass 28 kg.

After 82% of the entire field had been plowed, 9 hectares remained to be plowed. What is the area of ​​the entire field?

Decision.

Let the area of ​​the entire field be x ha, which is 100%. It remains to plow 9 hectares, which is 100% - 82% = 18% of the entire field. Let's express 1% of the field area in two ways. This is:

X : 100 or 9 : 18. We make a proportion:

X : 100 = 9: 18.

We find the unknown extreme term of the proportion. To do this, we multiply the average terms of the proportion ( 100 and 9 ) and divide by the known extreme term ( 18 ). We reduce the fraction.

Answer: area of ​​the whole field 50 ha.

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In the last video tutorial, we considered solving percentage problems using proportions. Then, according to the condition of the problem, we needed to find the value of one or another quantity.

This time, the initial and final values ​​are already given to us. Therefore, in tasks it will be required to find percentages. More precisely, by what percentage has this or that value changed. Let's try.

Task. Sneakers cost 3200 rubles. After the price increase, they began to cost 4000 rubles. By what percentage was the price of the sneakers increased?

So, we solve through proportion. The first step - the original price was equal to 3200 rubles. Therefore, 3200 rubles is 100%.

In addition, we were given the final price - 4000 rubles. This is an unknown percentage, so let's denote it as x . We get the following construction:

3200 — 100%
4000 - x%

Well, the condition of the problem is written down. We make a proportion:

The fraction on the left is perfectly reduced by 100: 3200: 100 = 32; 4000: 100 = 40. In addition, you can reduce by 4: 32: 4 = 8; 40: 4 = 10. We get the following proportion:

Let's use the basic property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:

8 x = 100 10;
8x = 1000.

This is the usual linear equation. From here we find x :

x=1000:8=125

So, we got the final percentage x = 125. But is the number 125 the solution to the problem? No way! Because the task requires you to find out by what percentage the price of sneakers was increased.

By how many percent - this means that we need to find a change:

∆ = 125 − 100 = 25

We got 25% - that's how much the original price was increased. This is the answer: 25.

Problem B2 for interest #2

Let's move on to the second task.

Task. The shirt cost 1800 rubles. After the price reduction, it began to cost 1530 rubles. By what percentage was the price of the shirt reduced?

We translate the condition into mathematical language. The initial price of 1800 rubles is 100%. And the final price is 1530 rubles - we know it, but it is not known how many percent it is of the original value. Therefore, we denote it by x. We get the following construction:

1800 — 100%
1530 - x%

Based on the resulting record, we make up the proportion:

To simplify further calculations, let's divide both parts of this equation by 100. In other words, we will cross out two zeros at the numerator of the left and right fractions. We get:

Now let's use the basic property of proportion again: the product of the extreme terms is equal to the product of the average ones.

18 x = 1530 1;
18x = 1530.

It remains to find x :

x = 1530: 18 = (765 2) : (9 2) = 765: 9 = (720 + 45) : 9 = 720: 9 + 45: 9 = 80 + 5 = 85

We got that x = 85. But, as in the previous problem, this number in itself is not the answer. Let's go back to our condition. We now know that the new price after the cut is 85% of the old price. And in order to find the changes, you need from the old price, i.e. 100%, subtract the new price, i.e. 85%. We get:

∆ = 100 − 85 = 15

This number will be the answer: Please note: exactly 15, and in no case 85. That's all! Problem solved.

Attentive students will probably ask: why in the first task, when finding the difference, we subtracted the initial number from the final number, and in the second task we did exactly the opposite: from the initial 100% we subtracted the final 85%?

Let's clear this up. Formally, in mathematics, the change in value is always the difference between the final value and the initial one. In other words, in the second problem, we should have got not 15, but -15.

However, in no case should this minus be included in the answer, because it has already been taken into account in the condition of the original problem. It says right there about the price reduction. A 15% price decrease is the same as a -15% price increase. That is why in the solution and answer of the problem it is enough to write just 15 - without any minuses.

All, I hope, with this moment we have understood. This concludes our lesson for today. See you soon!

A proportion is a mathematical expression in which two or more numbers are compared to each other. In proportions, absolute values ​​​​and quantities can be compared or parts of a larger whole. Proportions can be written and calculated in several different ways, but the basic principle is the same.

Steps

Part 1

What is proportion

Find out what proportions are for. Proportions are used both in scientific research and in everyday life to compare different values ​​and quantities. In the simplest case, two numbers are compared, but a proportion can include any number of values. When comparing two or more quantities, you can always apply a proportion. Knowing how quantities relate to each other makes it possible, for example, to write down chemical formulas or recipes for various dishes. Proportions will come in handy for a variety of purposes.

1. Learn what proportion means. As noted above, proportions allow you to determine the relationship between two or more quantities. For example, if it takes 2 cups of flour and 1 cup of sugar to make cookies, we say that there is a 2 to 1 ratio between the amount of flour and sugar.

   • With proportions, you can show how different quantities relate to each other, even if they are not directly related to each other (unlike a recipe). For example, if there are five girls and ten boys in the class, the ratio of the number of girls to the number of boys is 5 to 10. In this case, one number does not depend on the other and is not directly related to it: the proportion can change if someone leaves the class or vice versa , new students will come to it. Proportion simply allows you to compare two quantities.

2. Pay attention to the different ways of expressing proportions. Proportions can be written in words or mathematical symbols can be used.

   • In everyday life, proportions are more often expressed in words (as above). Proportions are used in a wide variety of areas, and if your profession is not related to mathematics or another science, most often you will come across this way of writing proportions.
   • Proportions are often written with a colon. When comparing two numbers using a proportion, they can be written with a colon, such as 7:13. If more than two numbers are being compared, a colon is inserted consecutively between each two numbers, for example 10:2:23. In the class example above, we are comparing the number of girls and boys, with 5 girls: 10 boys. Thus, in this case, the proportion can be written as 5:10.
   • Sometimes when writing proportions, a fraction sign is used. In our class example, the ratio of 5 girls to 10 boys would be written as 5/10. In this case, the “divide” sign should not be read and it must be remembered that this is not a fraction, but the ratio of two different numbers.

   Part 2

   Operations with proportions

   1. Bring the proportion to its simplest form. Proportions can be simplified, like fractions, by reducing their members by a common divisor. To simplify a proportion, divide all the numbers in it by common divisors. However, one should not forget about the initial values ​​\u200b\u200bthat led to this proportion.

      • In the example above with a class of 5 girls and 10 boys (5:10), both sides of the proportion have a common divisor of 5. Dividing both by 5 (greatest common divisor), we get a ratio of 1 girl to 2 boys (i.e. 1:2) . However, when using a simplified proportion, one should remember the initial numbers: there are not 3 students in the class, but 15. The reduced proportion only shows the ratio between the number of girls and boys. There are two boys for every girl, but this does not mean that there are 1 girl and 2 boys in the class.
      • Some proportions are not amenable to simplification. For example, the ratio 3:56 cannot be reduced, since the quantities included in the proportion do not have a common divisor: 3 is a prime number, and 56 is not divisible by 3.

   2. For "scaling" proportions can be multiplied or divided. Proportions are often used to increase or decrease numbers in proportion to each other. Multiplying or dividing all the quantities in a proportion by the same number keeps the ratio between them unchanged. Thus, the proportions can be multiplied or divided by the “scale” factor.

      • Suppose a baker needs to triple the amount of cookies they bake. If flour and sugar are taken in a ratio of 2 to 1 (2:1), to increase the number of cookies by three times this proportion should be multiplied by 3. The result will be 6 cups of flour for 3 cups of sugar (6:3).
      • You can also do the opposite. If the baker needs to halve the amount of cookies, both parts of the proportion should be divided by 2 (or multiplied by 1/2). The result is 1 cup of flour for half a cup (1/2, or 0.5 cup) of sugar.

   3. Learn how to find an unknown quantity using two equivalent proportions. Another common problem for which proportions are widely used is finding an unknown quantity in one of the proportions, if a second proportion similar to it is given. The multiplication rule for fractions greatly simplifies this task. Write each proportion as a fraction, then equate these fractions to each other and find the desired value.

      • Suppose we have a small group of students of 2 boys and 5 girls. If we want to keep the ratio between boys and girls, how many boys should there be in a class with 20 girls? First, let's make up both proportions, one of which contains an unknown value: 2 boys: 5 girls \u003d x boys: 20 girls. If we write proportions as fractions, we get 2/5 and x/20. After multiplying both sides of the equation by the denominators, we get the equation 5x=40; we divide 40 by 5 and as a result we find x=8.

   Part 3

   Error detection

   1. When dealing with proportions, avoid addition and subtraction. Many proportion problems sound like this: “It takes 4 potatoes and 5 carrots to make a dish. If you want to use 8 potatoes, how many carrots do you need?” Many make the mistake of simply trying to add up the corresponding values. However, to maintain the same proportion, you should multiply, not add. Here is the wrong and right solution for this problem:

      • Wrong method: “8 - 4 = 4, that is, 4 potatoes were added to the recipe. So, you need to take the previous 5 carrots and add 4 to them, so that ... something is not right! Proportions work differently. Let's try again".
      • The correct method is: “8/4 = 2, that is, the number of potatoes has doubled. This means that the number of carrots should also be multiplied by 2. 5 x 2 = 10, that is, 10 carrots must be used in the new recipe.

   2. Convert all values ​​to the same units. Sometimes the problem arises because the values ​​have different units. Before writing down the proportion, convert all quantities to the same units of measurement. For example:

      • The dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in dragon reserves?
      • Grams and kilograms are different units of measurement, so they should be unified. 1 kilogram = 1,000 grams, so 10 kilograms = 10 kilograms x 1,000 grams/1 kilogram = 10 x 1,000 grams = 10,000 grams.
      • So the dragon has 500 grams of gold and 10,000 grams of silver.
      • The ratio of the mass of gold to the mass of silver is 500 grams of gold / 10,000 grams of silver = 5/100 = 1/20.

   3. Write down units of measurement in the solution of the problem. In problems with proportions, it is much easier to find an error if you write down after each value its unit of measurement. Remember that if the numerator and denominator have the same units of measure, they are reduced. After all possible abbreviations, the correct units of measurement should be obtained in the answer.

      • For example: given 6 boxes, and in every three boxes there are 9 balls; how many balls are there?
      • Wrong method: 6 boxes x 3 boxes / 9 marbles = ... Hmm, nothing is reduced, and the answer is “boxes x boxes / marbles“. This makes no sense.
      • Correct method: 6 boxes x 9 balls / 3 boxes = 6 boxes x 3 balls / 1 box = 6 x 3 balls / 1 = 18 balls.

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 mathematical science. How to make a proportion? Now let's figure it out.

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 make a proportion. The main thing is to do it. There were originally ten apples. Let it be 100%. This we marked all his apples. He gave one-fourth. 1/4=25/100. So, he has left: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the amount of fruit left over the amount of fruit that was available first. Now we have three numbers by which we can already solve the proportion. 10 apples - 100%, X apples - 75%, where x is the desired amount of fruit. How to make a proportion? It is necessary to understand what it is. Mathematically it looks like this. The equal sign is 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 more x, the more percent is this number from the original. We solve this proportion and get that x=7.5 apples. Why the boy decided to give a non-integer amount, we do not know. Now you know how to make a proportion. The main thing is to find two ratios, one of which contains the desired unknown.

Solving a proportion often comes down to simple multiplication and then division. Children are not taught in schools why this is so. While it is important to understand that proportional relationships are mathematical classics, 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 ordinary fractions. That is, a record of 95% will not work. And if you immediately write 95/100, then you can make solid reductions without starting the main count. It’s worth saying right away that if your proportion turned out with two unknowns, then it cannot be solved. No professor can help you here. And your task, most likely, has a more complex algorithm for correct actions.

Consider another example where there are no percentages. The motorist bought 5 liters of gasoline for 150 rubles. He thought about 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 make a proportion, then look. 5 liters of gasoline is 150 rubles. As in the first example, let's write 5l - 150r. Now let's find the third number. Of course, it's 30 liters. Agree that a pair of 30 l - 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.

That's what 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 make a proportion. Also you can solve it. As you can see, there is nothing complicated in this.

Today we continue a series of video tutorials on percentage problems from the Unified State Examination in mathematics. In particular, we will analyze two very real problems from the Unified State Examination and once again see how important it is to carefully read the condition of the problem and interpret it correctly.

So the first task is:

Task. Only 95% and 37,500 graduates of the city solved problem B1 correctly. How many people correctly solved problem B1?

At first glance, it seems that this is some kind of task for the caps. Like:

Task. There were 7 birds on the tree. 3 of them flew away. How many birds have flown?

However, let's do the math. We will solve by the method of proportions. So, we have 37,500 students - this is 100%. And also there is a certain number x of students, which is 95% of the very lucky ones who correctly solved problem B1. We write it down:

37 500 — 100%
X - 95%

You need to make a proportion and find x. We get:

Before us is a classic proportion, but before using the main property and multiplying it crosswise, I propose to divide both parts of the equation by 100. In other words, we cross out two zeros in the numerator of each fraction. Let's rewrite the resulting equation:

According to the basic property of proportion, the product of the extreme terms is equal to the product of the middle terms. In other words:

x = 375 95

These are quite large numbers, so you have to multiply them by a column. I remind you that it is strictly forbidden to use a calculator on the exam in mathematics. We get:

x = 35625

Total answer: 35,625. That is how many people out of the original 37,500 solved problem B1 correctly. As you can see, these numbers are pretty close, which makes sense because 95% is also very close to 100%. In general, the first task is solved. Let's move on to the second.

Interest problem #2

Task. Only 80% of the city's 45,000 graduates solved problem B9 correctly. How many people solved problem B9 incorrectly?

We solve in the same way. Initially, there were 45,000 graduates - this is 100%. Then, x graduates must be selected from this number, which should be 80% of the original number. We make a proportion and solve:

45 000 — 100%
x - 80%

Let's reduce one zero in the numerator and denominator of the 2nd fraction. Let's rewrite the resulting construction once more:

The main property of proportion: the product of the extreme terms is equal to the product of the middle ones. We get:

45,000 8 = x 10

This is the simplest linear equation. Let's express the variable x from it:

x = 45,000 8:10

We reduce one zero at 45,000 and at 10, the denominator remains one, so all we need is to find the value of the expression:

x = 4500 8

You can, of course, do the same as last time, and multiply these numbers in a column. But let's not make life difficult for ourselves, and instead of multiplying by a column, we decompose the eight into factors:

x = 4500 2 2 2 = 9000 2 2 = 36,000

And now - the most important thing that I talked about at the very beginning of the lesson. You need to carefully read the condition of the problem!

What do we need to know? How many people solved problem B9 not right. And we just found those people who decided correctly. These turned out to be 80% of the original number, i.e. 36,000. This means that in order to get the final answer, our 80% must be subtracted from the original number of students. We get:

45 000 − 36 000 = 9000

The resulting number 9000 is the answer to the problem. In total, in this city, out of 45,000 graduates, 9,000 people solved problem B9 incorrectly. Everything, the task is solved.