How proportion is calculated. How to make the proportion? Any student and adult will understand Proportions 1 1 in the form

A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute values or parts of a whole. Ratios are calculated and written in different ways, but the basic principles are the same for all ratios.

Steps

Part 1

Determination of ratios

Using ratios. Ratios are used both in science and in everyday life to compare values. The simplest ratios relate only two numbers, but there are ratios that compare three or more values. In any situation in which more than one quantity is present, a ratio can be written. By linking some values, ratios can, for example, suggest how to increase the amount of ingredients in a recipe or substances in a chemical reaction.

Determination of ratios. A ratio is a relationship between two (or more) values of the same kind. For example, if you need 2 cups of flour and 1 cup of sugar to make a cake, then the ratio of flour to sugar is 2 to 1.
- The ratios can also be used in cases where the two quantities are not related to each other (as in the example with the cake). For example, if there are 5 girls and 10 boys in a class, then the ratio of girls to boys is 5 to 10. These values (the number of boys and the number of girls) do not depend on each other, that is, their values will change if someone leaves the class or a new student will come to the class. Ratios simply compare the values of quantities.
Pay attention to the different ways of representing ratios. Relationships can be expressed in words or using mathematical symbols.
- Very often the ratios are expressed in words (as shown above). Especially this form of representation of ratios is used in everyday life, far from science.
- Also, ratios can be expressed through a colon. When comparing two numbers in a ratio, you will use one colon (for example, 7:13); when comparing three or more values, put a colon between each pair of numbers (for example, 10: 2: 23). In our class example, you can express the ratio of girls to boys like this: 5 girls: 10 boys. Or like this: 5:10.
- Less commonly, ratios are expressed using a slash. In the class example, it can be written like this: 5/10. Nevertheless, this is not a fraction and such a ratio is not read as a fraction; Moreover, remember that in the ratio, the numbers do not represent part of a whole.
Part 2
Using ratios
1. Simplify the ratio. The ratio can be simplified (similar to fractions) by dividing each term (number) of the ratio by. However, do not lose sight of the original ratio values when doing this.
  - In our example, there are 5 girls and 10 boys in the class; the ratio is 5:10. The greatest common divisor of the terms of the ratio is 5 (since both 5 and 10 are divisible by 5). Divide each ratio number by 5 to get the ratio of 1 girl to 2 boys (or 1: 2). However, keep the original values in mind when simplifying the ratio. In our example, there are not 3 students in the class, but 15. The simplified ratio compares the number of boys and the number of girls. That is, for every girl there are 2 boys, but there are not 2 boys and 1 girl in the class.
  - Some relationships are not simplified. For example, the ratio 3:56 is not simplified because these numbers have no common divisors (3 is a prime number, and 56 is not divisible by 3).
2. Use multiplication or division to increase or decrease the ratio. Common tasks in which it is necessary to increase or decrease two values proportional to each other. If you are given a ratio and need to find a larger or smaller ratio corresponding to it, multiply or divide the original ratio by some given number.
  - For example, a baker needs to triple the amount of ingredients given in a recipe. If the recipe has a flour to sugar ratio of 2 to 1 (2: 1), then the baker will multiply each term in the ratio by 3 to get a 6: 3 ratio (6 cups flour to 3 cups sugar).
  - On the other hand, if a baker needs to halve the amount of ingredients given in a recipe, then the baker will divide each term in the ratio by 2 and get a 1: ½ ratio (1 cup flour to 1/2 cup sugar).
3. Finding an unknown value when given two equivalents. This is a problem in which you need to find an unknown variable in one relation using the second relation, which is equivalent to the first. To solve such problems, use. Write down each ratio as an ordinary fraction, put an equal sign between them and multiply their terms crosswise.
  - For example, given a group of students, in which there are 2 boys and 5 girls. What will be the number of boys if the number of girls is increased to 20 (the proportion remains the same)? First, write down two ratios - 2 boys: 5 girls and X boys: 20 girls. Now write these ratios as fractions: 2/5 and x / 20. Multiply the terms of the fractions crosswise to get 5x = 40; therefore, x = 40/5 = 8.
Part 3
Common mistakes
1. Avoid addition and subtraction in ratio word problems. Many word problems look something like this: “In the recipe, you need to use 4 potato tubers and 5 carrot root crops. If you want to add 8 potato tubers, how many carrots do you need to keep the ratio unchanged? " When solving such problems, students often make the mistake of adding the same amount of ingredients to the original number. However, to keep the ratio, you need to use multiplication. Here are examples of right and wrong decisions:
  - False: “8 - 4 = 4 - so we added 4 potato tubers. So, you need to take 5 carrot root crops and add 4 more to them ... Stop! Relationships are not calculated that way. It is worth trying again. "
  - It is true: "8 ÷ 4 = 2 - so we multiplied the amount of potatoes by 2. Accordingly, 5 carrots must be multiplied by 2. 5 x 2 = 10 - 10 carrots must be added to the recipe."

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 left: 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%, X 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 proportions often comes down to simple multiplication, and then to division. In schools, children are not explained why this is exactly 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, let x denote 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 the 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.

A relationship is called a certain relationship between the entities of our world. These can be numbers, physical quantities, objects, products, phenomena, actions, and even people.

In everyday life, when it comes to ratios, we say "The ratio of this and that"... For example, if there are 4 apples and 2 pears in a vase, then we say "The ratio of apples and pears" "The ratio of pears and apples".

In mathematics, the ratio is often used as "The attitude of so-and-so to that-and-so"... For example, the ratio of four apples and two pears, which we considered above, in mathematics will read as "The ratio of four apples to two pears" or if you swap apples and pears, then "The ratio of two pears to four apples".

The ratio is expressed as a To b(where instead of a and b any numbers), but more often you can find an entry that is composed using a colon as a: b... You can read this entry in different ways:

a To b
a refers to b
attitude a To b

Let's write the ratio of four apples to two pears using the ratio symbol:

4: 2

If we swap the places of apples and pears, then we will have a ratio of 2: 4. This ratio can be read as "Two to four" or either "Two pears refer to four apples" .

In what follows, we will call the ratio a ratio.

Lesson content

What is an attitude?

The relation, as mentioned earlier, is written in the form a: b... It can also be written as a fraction. And we know that such a notation in mathematics means division. Then the result of the relationship will be the quotient a and b.

A ratio in mathematics is called the quotient of two numbers.

The ratio allows you to find out how much of one entity falls on the unit of another. Let's go back to the ratio of four apples to two pears (4: 2). This ratio will allow us to find out how many apples are there per unit of pear. A unit means one pear. First, let's write the ratio 4: 2 as a fraction:

This ratio is the division of the number 4 by the number 2. If we perform this division, we will get an answer to the question how many apples are there per unit of pear

Received 2. So four apples and two pears (4: 2) correlate (are interconnected with each other) so that there are two apples per pear

The figure shows how four apples and two pears relate to each other. It can be seen that there are two apples for each pear.

The relationship can be reversed by writing as. Then we get the ratio of two pears to four apples, or "the ratio of two pears to four apples." This ratio will show how many pears are there per unit of apple. The unit of apple means one apple.

To find the value of a fraction, you need to remember how to divide a smaller number by a larger one.

Received 0.5. Let's convert this decimal fraction to an ordinary one:

Reduce the resulting fraction by 5

Received an answer (half a pear). This means that two pears and four apples (2: 4) correlate (are interconnected with each other) so that one apple accounts for half of the pear

The figure shows how two pears and four apples relate to each other. It can be seen that for each apple there is a half of a pear.

The numbers that make up the ratio are called members of the relationship... For example, in a 4: 2 ratio, the members are the numbers 4 and 2.

Let's consider other examples of relationships. To prepare something, a recipe is drawn up. The recipe is built from the relationship between the products. For example, making oatmeal usually requires a glass of cereal for two glasses of milk or water. The ratio is 1: 2 ("one to two" or "one glass of cereal for two glasses of milk").

We convert the ratio 1: 2 to a fraction, we get. Calculating this fraction, we get 0.5. This means that one glass of cereal and two glasses of milk are correlated (interconnected with each other) so that one glass of milk accounts for half a glass of cereal.

If you flip the ratio 1: 2, you get a 2: 1 ratio ("two to one" or "two glasses of milk for one glass of cereal"). Convert the ratio 2: 1 to a fraction, we get. Calculating this fraction, we get 2. So two glasses of milk and one glass of cereals are correlated (interconnected with each other) so that there are two glasses of milk for one glass of cereals.

Example 2. There are 15 students in the class. 5 of them are boys, 10 are girls. You can write down the ratio of girls to boys 10: 5 and convert that ratio to a fraction. Calculating this fraction, we get 2. That is, girls and boys are related to each other in such a way that for every boy there are two girls

The figure shows how ten girls and five boys relate to each other. It can be seen that there are two girls for every boy.

The ratio cannot always be converted into a fraction and the quotient can be found. In some cases, this will not be logical.

So, if you turn over the attitude, it turns out, and this is the attitude of boys to girls. If you calculate this fraction, you get 0.5. It turns out that five boys relate to ten girls in such a way that for every girl there is half a boy. Mathematically, this is of course true, but from the point of view of reality it is not entirely reasonable, because a boy is a living person and you cannot just take and divide him, like a pear or an apple.

Building the right attitude is an important problem solving skill. So in physics, the ratio of the distance traveled to time is the speed of movement.

The distance is denoted by the variable S, time - through the variable t, speed - through the variable v... Then the phrase "The ratio of the distance traveled to time is the speed of movement" will be described by the following expression:

Suppose the car has traveled 100 kilometers in 2 hours. Then the ratio of the traversed one hundred kilometers to two hours will be the speed of the car:

It is customary to call speed the distance traveled by the body per unit of time. The unit of time means 1 hour, 1 minute or 1 second. And the relationship, as mentioned earlier, allows you to find out how much of one entity falls on the unit of another. In our example, the ratio of one hundred kilometers to two hours shows how many kilometers are there for one hour of movement. We see that there are 50 kilometers for every hour of movement.

Therefore, the speed is measured in km / h, m / min, m / s... The fraction symbol (/) indicates the ratio of distance to time: kilometers per hour , meters per minute and meters per second respectively.

Example 2... The ratio of the value of a product to its quantity is the price of one unit of a product

If we took 5 chocolate bars from the store and their total cost was 100 rubles, then we can determine the price of one bar. To do this, you need to find the ratio of one hundred rubles to the number of bars. Then we get that there are 20 rubles for one bar.

Comparison of quantities

Earlier we learned that the relationship between quantities of different natures form a new quantity. So, the ratio of the distance traveled to time is the speed of movement. The ratio of the value of a commodity to its quantity is the price of one unit of a commodity.

But the ratio can also be used to compare values. The result of such a relationship is a number showing how many times the first value is greater than the second, or how much of the first value is from the second.

To find out how many times the first value is greater than the second, a larger value must be written in the numerator of the ratio, and a smaller value in the denominator.

To find out what part of the first value is from the second, you need to write a smaller value in the numerator of the ratio, and a larger value in the denominator.

Consider the numbers 20 and 2. Let's find out how many times the number 20 is greater than the number 2. To do this, we find the ratio of the number 20 to the number 2. In the numerator of the ratio we write the number 20, and in the denominator - the number 2

The value of this ratio is ten

The ratio of the number 20 to the number 2 is the number 10. This number shows how many times the number 20 is greater than the number 2. So the number 20 is ten times greater than the number 2.

Example 2. There are 15 students in the class. 5 of them are boys, 10 are girls. Determine how many times there are more girls than boys.

We write down the attitude of girls towards boys. We write down the number of girls in the numerator of the relationship, and the number of boys in the denominator of the relationship:

The value of this ratio is 2. This means that there are twice as many girls in a class of 15 as boys.

There is no longer the question of how many girls there are for one boy. In this case, the ratio is used to compare the number of girls with the number of boys.

Example 3... What part of the number 2 is from the number 20.

We find the ratio of the number 2 to the number 20. In the numerator of the ratio we write the number 2, and in the denominator - the number 20

To find the meaning of this relationship, you need to remember

The value of the ratio of the number 2 to the number 20 is the number 0.1

In this case, the decimal fraction 0.1 can be converted to an ordinary one. This answer will be easier to understand:

So the number 2 of the number 20 is one tenth.

You can check. To do this, we find from the number 20. If we did everything correctly, then we should get the number 2

20: 10 = 2

2 × 1 = 2

We got the number 2. So one tenth of the number 20 is the number 2. Hence we conclude that the problem is solved correctly.

Example 4. There are 15 people in the class. 5 of them are boys, 10 are girls. Determine what proportion of the total number of schoolchildren are boys.

We write down the ratio of boys to the total number of schoolchildren. We write down five boys in the numerator of the relationship, and the total number of students in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write 15 in the denominator of the relationship

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 5 must be divided by the number 15

When you divide 5 by 15, you get a periodic fraction. Let's convert this fraction to an ordinary one

We got the final answer. So boys make up one third of the class.

The figure shows that in a class of 15 students, 5 boys make up a third of the class.

If we find from 15 schoolchildren for verification, then we get 5 boys

15: 3 = 5

5 × 1 = 5

Example 5. How many times is 35 greater than 5?

We write down the ratio of the number 35 to the number 5. In the numerator of the ratio, you need to write the number 35, in the denominator - the number 5, but not vice versa

The value of this ratio is 7. So the number 35 is seven times the number 5.

Example 6. There are 15 people in the class. 5 of them are boys, 10 are girls. Determine what proportion of the total number are girls.

We write down the ratio of girls to the total number of schoolchildren. We write ten girls in the numerator of the relationship, and the total number of schoolchildren in the denominator. The total number of schoolchildren is 5 boys plus 10 girls, so we write 15 in the denominator of the relationship

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 10 must be divided by the number 15

When you divide 10 by 15, you get a periodic fraction. Let's convert this fraction to an ordinary one

Reduce the resulting fraction by 3

We got the final answer. So girls make up two-thirds of the class.

The figure shows that in a class of 15 students, two thirds of the class are 10 girls.

If we find from 15 schoolchildren for verification, then we get 10 girls

15: 3 = 5

5 × 2 = 10

Example 7. What part of 10 cm is 25 cm

We write down the ratio of ten centimeters to twenty-five centimeters. We write 10 cm in the numerator of the ratio, 25 cm in the denominator

To find the meaning of this ratio, you need to remember how to divide a smaller number by a larger one. In this case, the number 10 must be divided by the number 25

Let's convert the resulting decimal fraction to an ordinary

Reduce the resulting fraction by 2

We got the final answer. This means that 10 cm are from 25 cm.

Example 8. How many times is 25 cm more than 10 cm

We write down the ratio of twenty-five centimeters to ten centimeters. In the numerator of the ratio we write 25 cm, in the denominator - 10 cm

The answer was 2.5. Means 25 cm more than 10 cm 2.5 times (two and a half times)

Important note. When finding the ratio of physical quantities of the same name, these quantities must necessarily be expressed in one unit of measurement, otherwise the answer will be incorrect.

For example, if we are dealing with two lengths and we want to know how many times the first length is greater than the second, or what part of the first length is from the second, then both lengths must first be expressed in one unit of measurement.

Example 9. How many times is 150 cm more than 1 meter?

First, let's make it so that both lengths are expressed in the same unit of measurement. To do this, let's convert 1 meter to centimeters. One meter is one hundred centimeters

1 m = 100 cm

Now we find the ratio of one hundred and fifty centimeters to one hundred centimeters. In the numerator of the ratio we write 150 centimeters, in the denominator - 100 centimeters

Let's find the value of this ratio

The answer was 1.5. This means that 150 cm is 1.5 times more than 100 cm (one and a half times).

And if they did not convert meters to centimeters and immediately tried to find the ratio of 150 cm to one meter, then we would get the following:

It would turn out that 150 cm is more than one meter one hundred and fifty times, but this is not true. Therefore, it is imperative to pay attention to the units of measurement of physical quantities that are involved in the relationship. If these quantities are expressed in different units of measurement, then to find the ratio of these quantities, you need to go to one unit of measurement.

Example 10. Last month, a person's salary was 25,000 rubles, and this month, the salary has increased to 27,000 rubles. Determine how many times the salary has grown

We write down the ratio of twenty-seven thousand to twenty-five thousand. We write 27000 in the numerator of the ratio, 25000 in the denominator.

Let's find the value of this ratio

The answer was 1.08. This means that the salary has increased by 1.08 times. In the future, when we get to know percentages, we will express such indicators as salaries as percentages.

Example 11... The width of the apartment building is 80 meters and the height is 16 meters. How many times is the house wider than its height?

We write down the ratio of the width of the house to its height:

The value of this ratio is 5. This means that the width of the house is five times its height.

Relationship property

The ratio will not change if its members are multiplied or divided by the same number.

This is one of the most important properties of the relationship follows from the property of the particular. We know that if the dividend and divisor are multiplied or divided by the same number, then the quotient will not change. And since the relation is nothing more than division, the property of the particular works for it too.

Let's go back to girls' attitudes towards boys (10: 5). This attitude showed that there are two girls for every boy. Let's check how the relationship property works, namely, let's try to multiply or divide its members by the same number.

In our example, it is more convenient to divide the members of the relationship by their greatest common divisor (GCD).

The gcd of members 10 and 5 is the number 5. Therefore, the members of the relationship can be divided by the number 5

Got a new attitude. This is a two-to-one ratio (2: 1). This ratio, like the past ratio of 10: 5, shows that there are two girls for one boy.

The figure shows a 2: 1 (two to one) ratio. As in the past, the ratio of 10: 5 per boy has two girls. In other words, the attitude has not changed.

Example 2... There are 10 girls and 5 boys in one class. In another class there are 20 girls and 10 boys. How many times are there more girls in the first grade than boys? How many times are there more girls in the second grade than boys?

In both classes, there are twice as many girls as boys, because the relationships and are equal to the same number.

The relationship property allows you to build various models that have similar parameters to the real object. Suppose an apartment building is 30 meters wide and 10 meters high.

To draw a similar house on paper, you need to draw it in the same ratio of 30: 10.

Divide both terms of this ratio by the number 10. Then we get the ratio 3: 1. This ratio is 3, just like the previous ratio is 3

Let's convert meters to centimeters. 3 meters is 300 centimeters, and 1 meter is 100 centimeters

3 m = 300 cm

1 m = 100 cm

We have a ratio of 300 cm: 100 cm. Divide the terms of this ratio by 100. We obtain a ratio of 3 cm: 1 cm. Now we can draw a house with a width of 3 cm and a height of 1 cm.

Of course, the drawn house is much smaller than the real house, but the ratio of width and height remains unchanged. This allowed us to draw a house as close as possible to the real one.

Attitude can be understood in other ways as well. It was originally said that a real house has a width of 30 meters and a height of 10 meters. The total is 30 + 10, that is, 40 meters.

These 40 meters can be understood as 40 parts. A ratio of 30: 10 means there are 30 pieces for the width and 10 pieces for the height.

Further, the members of the ratio 30: 10 were divided by 10. The result was a ratio of 3: 1. This ratio can be understood as 4 parts, three of which are for the width, one for the height. In this case, you usually need to find out how many meters are in the width and height.

In other words, you need to find out how many meters are in 3 parts and how many meters are in 1 part. First you need to find out how many meters are in one part. To do this, the total 40 meters must be divided by 4, since in a 3: 1 ratio there are only four parts

Let's determine how many meters are in the width:

10 m × 3 = 30 m

Let's determine how many meters are at the height:

10 m × 1 = 10 m

Multiple relationship members

If several members are given in a relation, then they can be understood as parts of something.

Example 1... Purchased 18 apples. These apples were divided between mom, dad and daughter in a ratio of 2: 1: 3. How many apples did each get?

The ratio 2: 1: 3 means that mom got 2 parts, dad - 1 part, daughter - 3 parts. In other words, each member of the 2: 1: 3 ratio is a specific fraction of 18 apples:

If you add up the members of the ratio 2: 1: 3, then you can find out how many parts there are in total:

2 + 1 + 3 = 6 (parts)

Find out how many apples are in one part. To do this, divide 18 apples by 6

18: 6 = 3 (apples per slice)

Now let's determine how many apples each got. By multiplying three apples by each member of the 2: 1: 3 ratio, you can determine how many apples mom got, how much dad got, and how much daughter got.

Let's find out how many apples mom got:

3 × 2 = 6 (apples)

Find out how many apples dad got:

3 × 1 = 3 (apples)

Let's find out how many apples my daughter received:

3 × 3 = 9 (apples)

Example 2... New silver (alpaca) is an alloy of nickel, zinc and copper in a ratio of 3: 4: 13. How many kilograms of each metal do you need to take to get 4 kg of new silver?

4 kilograms of new silver will contain 3 parts nickel, 4 parts zinc and 13 parts copper. First, we find out how many parts there will be in four kilograms of silver:

3 + 4 + 13 = 20 (parts)

Let's determine how many kilograms will be in one part:

4 kg: 20 = 0.2 kg

Let's determine how many kilograms of nickel will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, three parts of the alloy are indicated to contain nickel. Therefore, we multiply 0.2 by 3:

0.2 kg × 3 = 0.6 kg nickel

Now let's determine how many kilograms of zinc will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, the four parts of the alloy are said to contain zinc. Therefore, we multiply 0.2 by 4:

0.2kg × 4 = 0.8kg zinc

Now let's determine how many kilograms of copper will be contained in 4 kg of new silver. In a ratio of 3: 4: 13, thirteen parts of the alloy are said to contain copper. Therefore, we multiply 0.2 by 13:

0.2 kg × 13 = 2.6 kg copper

This means that to get 4 kg of new silver, you need to take 0.6 kg of nickel, 0.8 kg of zinc and 2.6 kg of copper.

Example 3... Brass is an alloy of copper and zinc, the weight of which is 3: 2. To make a piece of brass, 120 g of copper is required. How much zinc does it take to make this piece of brass?

Let's determine how many grams of alloy are in one part. The condition says that 120 g of copper is required to make a piece of brass. It is also said that the three parts of the alloy contain copper. If we divide 120 by 3, we find out how many grams of alloy are in one part:

120: 3 = 40 grams per portion

Now let's determine how much zinc is required to make a piece of brass. To do this, multiply 40 grams by 2, since in the ratio 3: 2 it is indicated that two parts contain zinc:

40 g × 2 = 80 grams of zinc

Example 4... We took two alloys of gold and silver. In one, the amount of these metals is in a ratio of 1: 9, and in the other 2: 3. How much of each alloy should be taken to get 15 kg of a new alloy, in which gold and silver would be in a ratio of 1: 4?

Solution

15 kg of the new alloy should be in a ratio of 1: 4. This ratio suggests that one part of the alloy will be gold, and four parts will be silver. There are five parts in total. This can be schematically represented as follows

Let's determine the mass of one part. To do this, first add all parts (1 and 4), then divide the mass of the alloy by the number of these parts

1 + 4 = 5
15 kg: 5 = 3 kg

One part of the alloy will have a mass of 3 kg. Then 15 kg of the new alloy will contain 3 × 1 = 3 kg of gold and silver 3 × 4 = 12 kg of silver.

Therefore, to obtain an alloy weighing 15 kg, we need 3 kg of gold and 12 kg of silver.

Now let's answer the question of the problem - " How much of each alloy should you take? »

We will take 10 kg of the first alloy, since gold and silver in it are in a ratio of 1: 9. That is, this first alloy will give us 1 kg of gold and 9 kg of silver.

We will take 5 kg of the second alloy, since gold and silver are in it in a ratio of 2: 3. That is, this second alloy will give us 2 kg of gold and 3 kg of silver.

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Formula of proportions

Proportion is the equality of two ratios when a: b = c: d

ratio 1 : 10 is equal to the ratio 7 : 70, which can also be written as a fraction: 1 10 = 7 70 reads as: "one refers to ten as well as seven refers to seventy"

Basic proportion properties

The product of the extreme terms is equal to the product of the middle terms (crosswise): if a: b = c: d, then a⋅d = b⋅c

1 10 ✕ 7 70 1 ⋅ 70 = 10 ⋅ 7

Inversion of proportion: if a: b = c: d then b: a = d: c

1 10 7 70 10 1 = 70 7

Permutation of middle terms: if a: b = c: d, then a: c = b: d

1 10 7 70 1 7 = 10 70

Permutation of extreme terms: if a: b = c: d, then d: b = c: a

1 10 7 70 70 10 = 7 1

Solving proportions with one unknown | The equation

1 : 10 = x : 70 or 1 10 = x 70

To find x, you need to multiply two known numbers crosswise and divide by the opposite value

x = 1 ⋅ 70 10 = 7

How to calculate the proportion

Task: you need to drink 1 tablet of activated carbon per 10 kilograms of weight. How many tablets should you take if a person weighs 70 kg?

Let's make a proportion: 1 tablet - 10 kg x tablets - 70 kg To find x, you need to multiply two known numbers crosswise and divide by the opposite value: 1 tablet x pills✕ 10 kg 70 Kg x = 1 ⋅ 70 : 10 = 7 Answer: 7 tablets

Task: Vasya writes two articles in five hours. How many articles will he write in 20 hours?

Let's make a proportion: 2 articles - 5 hours x articles - 20 hours x = 2 ⋅ 20 : 5 = 8 Answer: 8 articles

I can say to future school graduates that the ability to make proportions came in handy for me both, and in order to proportionally reduce pictures, and in HTML layout of a web page, and in everyday situations.

Proportions are such a familiar combination, which is probably known from the elementary grades of a comprehensive school. In the most general sense, proportion is the equality of two or more relationships.

That is, if there are some numbers A, B and C

then the proportion

if there are four numbers A, B, C and D

then or is also a proportion

The simplest example where proportion is used is to calculate percentages.

In general, the use of proportions is so wide that it is easier to say where they are not used.

Proportions can be used to determine distances, masses, volumes, and the amount of anything, with one important condition: in proportion, there must be linear dependencies between different objects... Below, using the example of building a model of the Bronze Horseman, you will see how proportions should be calculated where there are nonlinear dependencies.

Determine how many kilograms of rice will be if you take 17 percent of the total rice in 150 kilograms?

Let's put together the proportion in words: 150 kilograms is the total amount of rice. So let's take it as 100%. Then 17% of 100% will be calculated as a proportion of two ratios: 100 percent refers to 150 kilograms as well as 17 percent to an unknown number.

Now the unknown number can be calculated elementary

That is, our answer is 25.5 kilograms of rice.

Interesting riddles are also associated with proportions, which show that you should not recklessly apply proportions for all occasions.

Here is one of them, slightly modified:

For demonstration in the office of the company, the director ordered to create a model of the sculpture "The Bronze Horseman" without a granite pedestal. One of the conditions is that the layout must be made of the same materials as the original, the proportions are observed and the height of the layout was exactly 1 meter. Question: What will be the mass of the layout?

First, let's turn to the reference books.

The rider's height is 5.35 meters, and his weight is 8,000 kg.

If we use the very first thought - to make a proportion: 5.35 meters refers to 8,000 kilograms as 1 meter to an unknown value, then we may not even start calculating, since the answer will be wrong.

It's all about a small nuance that must be taken into account. It's all about the connection between mass and height sculptors nonlinear, that is, we cannot say that by increasing, for example, a cube by 1 meter (observing the proportions so that it remains a cube), we will increase its weight by the same amount.

It is easy to check with examples:

1.Glue a cube with an edge length of 10 centimeters. How much water will go in there? It is logical that 10 * 10 * 10 = 1000 cubic centimeters, that is, 1 liter. Well, since water was poured there (the density is equal to one), and not another liquid, then the mass will be equal to 1 kg.

2. we glue a similar cube but with a rib length of 20 cm. The volume of water poured there will be equal to 20 * 20 * 20 = 8000 cubic centimeters, that is, 8 liters. Well, the weight is naturally 8 kg.

It is easy to see that the relationship between the mass and the change in the length of the edge of the cube is nonlinear, or rather cubic.

Recall that volume is the product of height, width, and depth.

That is, when the figure changes (subject to the proportions / shape) of the linear size (height, width, depth), the mass / volume of the volumetric figure changes cubically.

We argue:

Our linear size has changed from 5.35 meters to 1 meter, then the mass (volume) will change as the cube root of 8000 / x

And we get that the layout Bronze Horseman in the office of the company with a height of 1 meter it will weigh 52 kilograms 243 grams.

But on the other hand, if the task were posed like this " the layout must be made of the same materials as the original, the proportions and volume 1 cubic meter “knowing that there is a linear relationship between volume and mass, we would just use the standard ratio, the old volume to the new, and the old mass to the unknown number.

But our bot helps to calculate proportions in other more common and practical cases.

Surely, it will come in handy for all housewives who prepare food.

Situations arise when a recipe for an amazing 10 kg cake is found, but its volume is too large to prepare .. I would like to be smaller, for example, only two kilograms, but how to calculate all the new weights and volumes of ingredients?

This is where a bot will help you, which will be able to calculate new parameters of a 2 kg cake.

Also, the bot will help in calculations for hardworking men who are building a house and they need to calculate how much to take ingredients for concrete if he only has 50 kilograms of sand.

Syntax

For XMPP client users: pro<строка>

where string has required elements

number1 / number2 - finding the proportion.

In order not to be intimidated by such a short description, we will give an example here.

200 300 100 3 400/100

Which says, for example, about the following:

200 grams of flour, 300 milliliters of milk, 100 grams of butter, 3 eggs - the output of pancakes is 400 grams.

How many ingredients do you need to take to bake only 100 grams of pancakes?

How easy it is to notice

400/100 is the ratio of a typical recipe and the output we want to get.

We will consider examples in more detail in the corresponding section.

Examples of

A friend shared a wonderful recipe

Dough: 200 grams of poppy seeds, 8 eggs, 200 powdered sugar, 50 grams of grated rolls, 200 grams of ground nuts, 3 cups of honey.
Boil poppy seeds for 30 minutes on low heat, grind with a pestle, add melted honey, ground crackers, nuts.
Beat eggs with powdered sugar, add to the mass.
Gently mix the dough, pour into a mold, bake.
Cut the cooled cake into 2 layers, coat with sour jam, then cream.
Decorate with berries from jam.
Cream: 1 cup sour cream, 1/2 cup sugar, beat.