Direct and inverse proportionality. Direct and inverse proportionality

Example

1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8, etc.

Proportionality factor

A constant relationship of proportional quantities is called proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionally, in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction.

Mathematically, direct proportionality is written as a formula:

f(x) = ax,a = const

Inverse proportionality

Inverse proportionality- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

Wikimedia Foundation. 2010.

Example

1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8, etc.

Proportionality factor

A constant relationship of proportional quantities is called proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionally, in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction.

Mathematically, direct proportionality is written as a formula:

f(x) = ax,a = const

Inverse proportionality

Inverse proportionality- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

Wikimedia Foundation. 2010.

• Newton's second law
• Coulomb barrier

See what “Direct proportionality” is in other dictionaries:

direct proportionality- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN direct ratio ... Technical Translator's Guide

direct proportionality- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: engl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f … Fizikos terminų žodynas

PROPORTIONALITY- (from Latin proportionalis proportionate, proportional). Proportionality. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. PROPORTIONALITY lat. proportionalis, proportional. Proportionality. Explanation 25000... ... Dictionary of foreign words of the Russian language

PROPORTIONALITY- PROPORTIONALITY, proportionality, plural. no, female (book). 1. abstract noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between quantities when they are proportional (see proportional ... Ushakov's Explanatory Dictionary

Proportionality- Two mutually dependent quantities are called proportional if the ratio of their values ​​remains unchanged. Contents 1 Example 2 Proportionality coefficient ... Wikipedia

PROPORTIONALITY- PROPORTIONALITY, and, female. 1. see proportional. 2. In mathematics: such a relationship between quantities in which an increase in one of them entails a change in the other by the same amount. Straight line (with a cut with an increase in one value... ... Ozhegov's Explanatory Dictionary

proportionality- And; and. 1. to Proportional (1 digit); proportionality. P. parts. P. physique. P. representation in parliament. 2. Math. Dependence between proportionally changing quantities. Proportionality factor. Direct line (in which with... ... encyclopedic Dictionary

Along with directly proportional quantities in arithmetic, inversely proportional quantities were also considered.

Let's give examples.

1) The length of the base and the height of a rectangle with a constant area.

Suppose you need to allocate a rectangular plot of land with an area of

We “can arbitrarily set, for example, the length of the section. But then the width of the area will depend on what length we have chosen. The different (possible) lengths and widths are shown in the table.

In general, if we denote the length of the section by x and the width by y, then the relationship between them can be expressed by the formula:

Expressing y through x, we get:

Giving x arbitrary values, we will obtain the corresponding y values.

2) Time and speed of uniform motion at a certain distance.

Let the distance between two cities be 200 km. The higher the speed, the less time it will take to cover a given distance. This can be seen from the following table:

In general, if we denote the speed by x, and the time of movement by y, then the relationship between them will be expressed by the formula:

Definition. The relationship between two quantities expressed by the equality , where k is a certain number (not equal to zero), is called an inversely proportional relationship.

The number here is also called the proportionality coefficient.

Just as in the case of direct proportionality, in equality the quantities x and y in the general case can take on positive and negative values.

But in all cases of inverse proportionality, none of the quantities can be equal to zero. In fact, if at least one of the quantities x or y is equal to zero, then the left side of the equality will be equal to

And the right one - to some number that is not equal to zero (by definition), that is, the result will be an incorrect equality.

2. Graph of inverse proportionality.

Let's build a dependence graph

Expressing y through x, we get:

We will give x arbitrary (valid) values ​​and calculate the corresponding y values. We get the table:

Let's construct the corresponding points (Fig. 28).

If we take the values ​​of x at smaller intervals, then the points will be located closer together.

For all possible values ​​of x, the corresponding points will be located on two branches of the graph, symmetrical with respect to the origin of coordinates and passing in the first and third quarters of the coordinate plane (Fig. 29).

So, we see that the graph of inverse proportionality is a curved line. This line consists of two branches.

One branch will be obtained for positive, the other - for negative values ​​of x.

The graph of an inverse relationship is called a hyperbola.

To get a more accurate graph, you need to build as many points as possible.

A hyperbole can be drawn with fairly high accuracy using, for example, patterns.

In drawing 30, a graph of an inversely proportional relationship with a negative coefficient is plotted. For example, by creating a table like this:

we obtain a hyperbola, the branches of which are located in the II and IV quarters.

Example

1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8, etc.

Proportionality factor

A constant relationship of proportional quantities is called proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionally, in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction.

Mathematically, direct proportionality is written as a formula:

f(x) = ax,a = const

Inverse proportionality

Inverse proportionality- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

Wikimedia Foundation. 2010.

§ 129. Preliminary clarifications.

A person constantly deals with a wide variety of quantities. An employee and a worker are trying to get to work by a certain time, a pedestrian is in a hurry to get to a certain place by the shortest route, a steam heating stoker is worried that the temperature in the boiler is slowly rising, a business executive is making plans to reduce the cost of production, etc.

One could give any number of such examples. Time, distance, temperature, cost - all these are various quantities. In the first and second parts of this book, we became acquainted with some particularly common quantities: area, volume, weight. We encounter many quantities when studying physics and other sciences.

Imagine that you are traveling on a train. Every now and then you look at your watch and notice how long you've been on the road. You say, for example, that 2, 3, 5, 10, 15 hours have passed since your train departed, etc. These numbers represent different periods of time; they are called the values ​​of this quantity (time). Or you look out the window and follow the road posts to see the distance your train travels. The numbers 110, 111, 112, 113, 114 km flash in front of you. These numbers represent the different distances the train has traveled from its departure point. They are also called values, this time of a different magnitude (path or distance between two points). Thus, one quantity, for example time, distance, temperature, can take on as many different meanings.

Please note that a person almost never considers only one quantity, but always connects it with some other quantities. He has to simultaneously deal with two, three or more quantities. Imagine that you need to get to school by 9 o'clock. You look at your watch and see that you have 20 minutes. Then you quickly figure out whether you should take the tram or whether you can walk to school. After thinking, you decide to walk. Notice that while you were thinking, you were solving some problem. This task has become simple and familiar, since you solve such problems every day. In it you quickly compared several quantities. It was you who looked at the clock, which means you took into account the time, then you mentally imagined the distance from your home to the school; Finally, you compared two values: the speed of your step and the speed of the tram, and concluded that in a given time (20 minutes) you will have time to walk. From this simple example you can see that in our practice some quantities are interconnected, that is, they depend on each other

Chapter twelve talked about the relationship of homogeneous quantities. For example, if one segment is 12 m and the other is 4 m, then the ratio of these segments will be 12: 4.

We said that this is the ratio of two homogeneous quantities. Another way to say this is that it is the ratio of two numbers one name.

Now that we are more familiar with quantities and have introduced the concept of the value of a quantity, we can express the definition of a ratio in a new way. In fact, when we considered two segments of 12 m and 4 m, we were talking about one value - length, and 12 m and 4 m were only two different values ​​of this value.

Therefore, in the future, when we start talking about ratios, we will consider two values ​​of one quantity, and the ratio of one value of a quantity to another value of the same quantity will be called the quotient of dividing the first value by the second.

§ 130. Values ​​are directly proportional.

Let's consider a problem whose condition includes two quantities: distance and time.

Task 1. A body moving rectilinearly and uniformly travels 12 cm every second. Determine the distance traveled by the body in 2, 3, 4, ..., 10 seconds.

Let's create a table that can be used to track changes in time and distance.

The table gives us the opportunity to compare these two series of values. We see from it that when the values ​​of the first quantity (time) gradually increase by 2, 3,..., 10 times, then the values ​​of the second quantity (distance) also increase by 2, 3,..., 10 times. Thus, when the values ​​of one quantity increase several times, the values ​​of another quantity increase by the same amount, and when the values ​​of one quantity decrease several times, the values ​​of another quantity decrease by the same number.

Let us now consider a problem that involves two such quantities: the amount of matter and its cost.

Task 2. 15 m of fabric costs 120 rubles. Calculate the cost of this fabric for several other quantities of meters indicated in the table.

Using this table, we can trace how the cost of a product gradually increases depending on the increase in its quantity. Despite the fact that this problem involves completely different quantities (in the first problem - time and distance, and here - the quantity of goods and its value), nevertheless, great similarities can be found in the behavior of these quantities.

In fact, in the top line of the table there are numbers indicating the number of meters of fabric; under each of them there is a number expressing the cost of the corresponding quantity of goods. Even a quick glance at this table shows that the numbers in both the top and bottom rows are increasing; upon closer examination of the table and when comparing individual columns, it is discovered that in all cases the values ​​of the second quantity increase by the same number of times as the values ​​of the first increase, i.e. if the value of the first quantity increases, say, 10 times, then the value of the second quantity also increased 10 times.

If we look through the table from right to left, we will find that the indicated values ​​​​of quantities will decrease by the same number of times. In this sense, there is an unconditional similarity between the first task and the second.

The pairs of quantities that we encountered in the first and second problems are called directly proportional.

Thus, if two quantities are related to each other in such a way that as the value of one of them increases (decreases) several times, the value of the other increases (decreases) by the same amount, then such quantities are called directly proportional.

Such quantities are also said to be related to each other by a directly proportional relationship.

There are many similar quantities found in nature and in the life around us. Here are some examples:

1. Time work (day, two days, three days, etc.) and earnings, received during this time with daily wages.

2. Volume any object made of a homogeneous material, and weight this item.

§ 131. Property of directly proportional quantities.

Let's take a problem that includes the following two quantities: working time and earnings. If daily earnings are 20 rubles, then earnings for 2 days will be 40 rubles, etc. It is most convenient to create a table in which a certain number of days will correspond to a certain earnings.

Looking at this table, we see that both quantities took 10 different values. Each value of the first value corresponds to a certain value of the second value, for example, 2 days correspond to 40 rubles; 5 days correspond to 100 rubles. In the table these numbers are written one below the other.

We already know that if two quantities are directly proportional, then each of them, in the process of its change, increases as many times as the other increases. It immediately follows from this: if we take the ratio of any two values ​​of the first quantity, then it will be equal to the ratio of the two corresponding values ​​of the second quantity. Indeed:

Why is this happening? But because these values ​​are directly proportional, i.e. when one of them (time) increased by 3 times, then the other (earnings) increased by 3 times.

We have therefore come to the following conclusion: if we take two values ​​of the first quantity and divide them one by the other, and then divide by one the corresponding values ​​of the second quantity, then in both cases we will get the same number, i.e. i.e. the same relationship. This means that the two relations that we wrote above can be connected with an equal sign, i.e.

There is no doubt that if we took not these relations, but others, and not in that order, but in the opposite order, we would also obtain equality of relations. In fact, we will consider the values ​​of our quantities from left to right and take the third and ninth values:

60:180 = 1 / 3 .

So we can write:

This leads to the following conclusion: if two quantities are directly proportional, then the ratio of two arbitrarily taken values ​​of the first quantity is equal to the ratio of the two corresponding values ​​of the second quantity.

§ 132. Formula of direct proportionality.

Let's make a table of the cost of different quantities of sweets, if 1 kg of them costs 10.4 rubles.

Now let's do it this way. Take any number in the second line and divide it by the corresponding number in the first line. For example:

You see that in the quotient the same number is obtained all the time. Consequently, for a given pair of directly proportional quantities, the quotient of dividing any value of one quantity by the corresponding value of another quantity is a constant number (i.e., not changing). In our example, this quotient is 10.4. This constant number is called the proportionality factor. In this case, it expresses the price of a unit of measurement, i.e. one kilogram of goods.

How to find or calculate the proportionality coefficient? To do this, you need to take any value of one quantity and divide it by the corresponding value of the other.

Let us denote this arbitrary value of one quantity by the letter at , and the corresponding value of another quantity - the letter X , then the proportionality coefficient (we denote it TO) we find by division:

In this equality at - divisible, X - divisor and TO- quotient, and since, by the property of division, the dividend is equal to the divisor multiplied by the quotient, we can write:

y= K x

The resulting equality is called formula of direct proportionality. Using this formula, we can calculate any number of values ​​of one of the directly proportional quantities if we know the corresponding values ​​of the other quantity and the coefficient of proportionality.

Example. From physics we know that weight R of any body is equal to its specific gravity d , multiplied by the volume of this body V, i.e. R = d V.

Let's take five iron bars of different volumes; Knowing the specific gravity of iron (7.8), we can calculate the weights of these ingots using the formula:

R = 7,8 V.

Comparing this formula with the formula at = TO X , we see that y = R, x = V, and the proportionality coefficient TO= 7.8. The formula is the same, only the letters are different.

Using this formula, let's make a table: let the volume of the 1st blank be equal to 8 cubic meters. cm, then its weight is 7.8 8 = 62.4 (g). The volume of the 2nd blank is 27 cubic meters. cm. Its weight is 7.8 27 = 210.6 (g). The table will look like this:

Calculate the numbers missing in this table using the formula R= d V.

§ 133. Other methods of solving problems with directly proportional quantities.

In the previous paragraph, we solved a problem whose condition included directly proportional quantities. For this purpose, we first derived the direct proportionality formula and then applied this formula. Now we will show two other ways to solve similar problems.

Let's create a problem using the numerical data given in the table in the previous paragraph.

Task. Blank with a volume of 8 cubic meters. cm weighs 62.4 g. How much will a blank with a volume of 64 cubic meters weigh? cm?

Solution. The weight of iron, as is known, is proportional to its volume. If 8 cu. cm weigh 62.4 g, then 1 cu. cm will weigh 8 times less, i.e.

62.4:8 = 7.8 (g).

Blank with a volume of 64 cubic meters. cm will weigh 64 times more than a 1 cubic meter blank. cm, i.e.

7.8 64 = 499.2(g).

We solved our problem by reducing to unity. The meaning of this name is justified by the fact that to solve it we had to find the weight of a unit of volume in the first question.

2. Method of proportion. Let's solve the same problem using the proportion method.

Since the weight of iron and its volume are directly proportional quantities, the ratio of two values ​​of one quantity (volume) is equal to the ratio of two corresponding values ​​of another quantity (weight), i.e.

(letter R we designated the unknown weight of the blank). From here:

(G).

The problem was solved using the method of proportions. This means that to solve it, a proportion was compiled from the numbers included in the condition.

§ 134. Values ​​are inversely proportional.

Consider the following problem: “Five masons can lay the brick walls of a house in 168 days. Determine in how many days 10, 8, 6, etc. masons could complete the same work.”

If 5 masons laid the walls of a house in 168 days, then (with the same labor productivity) 10 masons could do it in half the time, since on average 10 people do twice as much work as 5 people.

Let's draw up a table by which we could monitor changes in the number of workers and working hours.

For example, to find out how many days it takes 6 workers, you must first calculate how many days it takes one worker (168 5 = 840), and then how many days it takes six workers (840: 6 = 140). Looking at this table, we see that both quantities took on six different values. Each value of the first quantity corresponds to a specific one; the value of the second quantity, for example, 10 corresponds to 84, the number 8 corresponds to the number 105, etc.

If we consider the values ​​of both quantities from left to right, we will see that the values ​​of the upper quantity increase, and the values ​​of the lower quantity decrease. The increase and decrease are subject to the following law: the values ​​of the number of workers increase by the same times as the values ​​of the spent working time decrease. This idea can be expressed even more simply as follows: the more workers are engaged in any task, the less time they need to complete a certain job. The two quantities we encountered in this problem are called inversely proportional.

Thus, if two quantities are related to each other in such a way that as the value of one of them increases (decreases) several times, the value of the other decreases (increases) by the same amount, then such quantities are called inversely proportional.

There are many similar quantities in life. Let's give examples.

1. If for 150 rubles. If you need to buy several kilograms of sweets, the number of sweets will depend on the price of one kilogram. The higher the price, the less goods you can buy with this money; this can be seen from the table:

As the price of candy increases several times, the number of kilograms of candy that can be bought for 150 rubles decreases by the same amount. In this case, two quantities (the weight of the product and its price) are inversely proportional.

2. If the distance between two cities is 1,200 km, then it can be covered in different times depending on the speed of movement. There are different ways to travel: on foot, on horseback, by bicycle, by boat, in a car, by train, by plane. The lower the speed, the more time it takes to move. This can be seen from the table:

With an increase in speed several times, the travel time decreases by the same amount. This means that under these conditions, speed and time are inversely proportional quantities.

§ 135. Property of inversely proportional quantities.

Let's take the second example, which we looked at in the previous paragraph. There we dealt with two quantities - speed and time. If we look at the table of values ​​of these quantities from left to right, we will see that the values ​​of the first quantity (speed) increase, and the values ​​of the second (time) decrease, and the speed increases by the same amount as the time decreases. It is not difficult to understand that if you write the ratio of some values ​​of one quantity, then it will not be equal to the ratio of the corresponding values ​​of another quantity. In fact, if we take the ratio of the fourth value of the upper value to the seventh value (40: 80), then it will not be equal to the ratio of the fourth and seventh values ​​of the lower value (30: 15). It can be written like this:

40:80 is not equal to 30:15, or 40:80 =/=30:15.

But if instead of one of these relations we take the opposite, then we get equality, i.e., from these relations it will be possible to create a proportion. For example:

80: 40 = 30: 15,

40: 80 = 15: 30."

Based on the foregoing, we can draw the following conclusion: if two quantities are inversely proportional, then the ratio of two arbitrarily taken values ​​of one quantity is equal to the inverse ratio of the corresponding values ​​of another quantity.

§ 136. Inverse proportionality formula.

Consider the problem: “There are 6 pieces of silk fabric of different sizes and different grades. All pieces cost the same. One piece contains 100 m of fabric, priced at 20 rubles. per meter How many meters are in each of the other five pieces, if a meter of fabric in these pieces costs 25, 40, 50, 80, 100 rubles, respectively?” To solve this problem, let's create a table:

We need to fill in the empty cells in the top row of this table. Let's first try to determine how many meters there are in the second piece. This can be done as follows. From the conditions of the problem it is known that the cost of all pieces is the same. The cost of the first piece is easy to determine: it contains 100 meters and each meter costs 20 rubles, which means that the first piece of silk is worth 2,000 rubles. Since the second piece of silk contains the same amount of rubles, then, dividing 2,000 rubles. for the price of one meter, i.e. 25, we find the size of the second piece: 2,000: 25 = 80 (m). In the same way we will find the size of all other pieces. The table will look like:

It is easy to see that there is an inversely proportional relationship between the number of meters and the price.

If you do the necessary calculations yourself, you will notice that each time you have to divide the number 2,000 by the price of 1 m. On the contrary, if you now start multiplying the size of the piece in meters by the price of 1 m, you will always get the number 2,000. This and it was necessary to wait, since each piece costs 2,000 rubles.

From here we can draw the following conclusion: for a given pair of inversely proportional quantities, the product of any value of one quantity by the corresponding value of another quantity is a constant number (i.e., not changing).

In our problem, this product is equal to 2,000. Check that in the previous problem, which talked about the speed of movement and the time required to move from one city to another, there was also a constant number for that problem (1,200).

Taking everything into account, it is easy to derive the inverse proportionality formula. Let us denote a certain value of one quantity by the letter X , and the corresponding value of another quantity is represented by the letter at . Then, based on the above, the work X on at must be equal to some constant value, which we denote by the letter TO, i.e.

x y = TO.

In this equality X - multiplicand at - multiplier and K- work. According to the property of multiplication, a multiplier is equal to the product divided by the multiplicand. Means,

This is the inverse proportionality formula. Using it, we can calculate any number of values ​​of one of the inversely proportional quantities, knowing the values ​​of the other and the constant number TO.

Let's consider another problem: “The author of one essay calculated that if his book is in a regular format, then it will have 96 pages, but if it is a pocket format, then it will have 300 pages. He tried different options, started with 96 pages, and then he ended up with 2,500 letters per page. Then he took the page numbers shown in the table below and again calculated how many letters there would be on the page.”

Let's try to calculate how many letters there will be on a page if the book has 100 pages.

There are 240,000 letters in the entire book, since 2,500 96 = 240,000.

Taking this into account, we use the inverse proportionality formula ( at - number of letters on the page, X - number of pages):

In our example TO= 240,000 therefore

So there are 2,400 letters on the page.

Similarly, we learn that if a book has 120 pages, then the number of letters on the page will be:

Our table will look like:

Fill in the remaining cells yourself.

§ 137. Other methods of solving problems with inversely proportional quantities.

In the previous paragraph, we solved problems whose conditions included inversely proportional quantities. We first derived the inverse proportionality formula and then applied this formula. We will now show two other solutions for such problems.

1. Method of reduction to unity.

Task. 5 turners can do some work in 16 days. In how many days can 8 turners complete this work?

Solution. There is an inverse relationship between the number of turners and working hours. If 5 turners do the job in 16 days, then one person will need 5 times more time for this, i.e.

5 turners complete the work in 16 days,

1 turner will complete it in 16 5 = 80 days.

The problem asks how many days it will take 8 turners to complete the job. Obviously, they will cope with the work 8 times faster than 1 turner, i.e. in

80: 8 = 10 (days).

This is the solution to the problem by reducing it to unity. Here it was necessary first of all to determine the time required to complete the work by one worker.

2. Method of proportion. Let's solve the same problem in the second way.

Since there is an inversely proportional relationship between the number of workers and working time, we can write: duration of work of 5 turners new number of turners (8) duration of work of 8 turners previous number of turners (5) Let us denote the required duration of work by the letter X and substitute the necessary numbers into the proportion expressed in words:

The same problem is solved by the method of proportions. To solve it, we had to create a proportion from the numbers included in the problem statement.

Note. In the previous paragraphs we examined the issue of direct and inverse proportionality. Nature and life give us many examples of direct and inverse proportional dependence of quantities. However, it should be noted that these two types of dependence are only the simplest. Along with them, there are other, more complex dependencies between quantities. In addition, one should not think that if any two quantities increase simultaneously, then there is necessarily a direct proportionality between them. This is far from true. For example, railway fares increase depending on the distance: the further we travel, the more we pay, but this does not mean that the fare is proportional to the distance.