How to distinguish direct and inverse proportionality. Linear function

Dependency Types

Let's look at charging the battery. As the first quantity, let's take the time it takes to charge. The second value is the time it will work after charging. The longer you charge the battery, the longer it will last. The process will continue until the battery is fully charged.

Dependence of battery operating time on the time it is charged

Note 1

This dependence is called straight:

As one value increases, so does the second. As one value decreases, the second value also decreases.

Let's look at another example.

The more books a student reads, the fewer mistakes he will make in the dictation. Or the higher you rise in the mountains, the lower the atmospheric pressure will be.

Note 2

This dependence is called reverse:

As one value increases, the second decreases. As one value decreases, the second value increases.

Thus, in case direct dependence both quantities change equally (both either increase or decrease), and in the case inverse relationship– opposite (one increases and the other decreases, or vice versa).

Determining dependencies between quantities

Example 1

The time it takes to visit a friend is $20$ minutes. If the speed (first value) increases by $2$ times, we will find how the time (second value) that will be spent on the path to a friend changes.

Obviously, the time will decrease by $2$ times.

Note 3

This dependence is called proportional:

The number of times one quantity changes, the number of times the second quantity changes.

Example 2

For $2$ loaves of bread in the store you need to pay 80 rubles. If you need to buy $4$ loaves of bread (the quantity of bread increases by $2$ times), how many times more will you have to pay?

Obviously, the cost will also increase $2$ times. We have an example of proportional dependence.

In both examples, proportional dependencies were considered. But in the example with loaves of bread, the quantities change in one direction, therefore, the dependence is straight. And in the example of going to a friend’s house, the relationship between speed and time is reverse. Thus there is directly proportional relationship And inversely proportional relationship.

Direct proportionality

Let's consider $2$ proportional quantities: the number of loaves of bread and their cost. Let $2$ loaves of bread cost $80$ rubles. If the number of buns increases by $4$ times ($8$ buns), their total cost will be $320$ rubles.

The ratio of the number of buns: $\frac(8)(2)=4$.

Bun cost ratio: $\frac(320)(80)=$4.

As you can see, these relations are equal to each other:

$\frac(8)(2)=\frac(320)(80)$.

Definition 1

The equality of two ratios is called proportion.

With a directly proportional dependence, a relationship is obtained when the change in the first and second quantities coincides:

$\frac(A_2)(A_1)=\frac(B_2)(B_1)$.

Definition 2

The two quantities are called directly proportional, if when one of them changes (increases or decreases), the other value also changes (increases or decreases, respectively) by the same amount.

Example 3

The car traveled $180$ km in $2$ hours. Find the time during which he will cover $2$ times the distance at the same speed.

Solution.

Time is directly proportional to distance:

$t=\frac(S)(v)$.

How many times will the distance increase, at a constant speed, by the same amount will the time increase:

$\frac(2S)(v)=2t$;

$\frac(3S)(v)=3t$.

The car traveled $180$ km in $2$ hours

The car will travel $180 \cdot 2=360$ km - in $x$ hours

The further the car travels, the longer it will take. Consequently, the relationship between the quantities is directly proportional.

Let's make a proportion:

$\frac(180)(360)=\frac(2)(x)$;

$x=\frac(360 \cdot 2)(180)$;

Answer: The car will need $4$ hours.

Inverse proportionality

Definition 3

Solution.

Time is inversely proportional to speed:

$t=\frac(S)(v)$.

By how many times does the speed increase, with the same path, the time decreases by the same amount:

$\frac(S)(2v)=\frac(t)(2)$;

$\frac(S)(3v)=\frac(t)(3)$.

Let's write the problem condition in the form of a table:

The car traveled $60$ km - in $6$ hours

The car will travel $120$ km – in $x$ hours

The faster the car speeds, the less time it will take. Consequently, the relationship between the quantities is inversely proportional.

Let's make a proportion.

Because the proportionality is inverse, the second relation in the proportion is reversed:

$\frac(60)(120)=\frac(x)(6)$;

$x=\frac(60 \cdot 6)(120)$;

Answer: The car will need $3$ hours.

§ 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 various 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 value, 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, the 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 job 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.

Direct and inverse proportionality

If t is the pedestrian’s time of movement (in hours), s is the distance traveled (in kilometers), and he moves uniformly at a speed of 4 km/h, then the relationship between these quantities can be expressed by the formula s = 4t. Since each value t corresponds to a single value s, we can say that a function is defined using the formula s = 4t. It is called direct proportionality and is defined as follows.

Definition. Direct proportionality is a function that can be specified using the formula y=kx, where k is a non-zero real number.

The name of the function y = k x is due to the fact that in the formula y = k x there are variables x and y, which can be values ​​of quantities. And if the ratio of two quantities is equal to some number different from zero, they are called directly proportional . In our case = k (k≠0). This number is called proportionality coefficient.

The function y = k x is a mathematical model of many real situations considered already in the initial mathematics course. One of them is described above. Another example: if one bag of flour contains 2 kg, and x such bags were purchased, then the entire mass of purchased flour (denoted by y) can be represented as the formula y = 2x, i.e. the relationship between the number of bags and the total mass of purchased flour is directly proportional with coefficient k=2.

Let us recall some properties of direct proportionality that are studied in a school mathematics course.

1. The domain of definition of the function y = k x and the range of its values ​​is the set of real numbers.

2. The graph of direct proportionality is a straight line passing through the origin. Therefore, to construct a graph of direct proportionality, it is enough to find only one point that belongs to it and does not coincide with the origin of coordinates, and then draw a straight line through this point and the origin of coordinates.

For example, to construct a graph of the function y = 2x, it is enough to have a point with coordinates (1, 2), and then draw a straight line through it and the origin of coordinates (Fig. 7).

3. For k > 0, the function y = khx increases over the entire domain of definition; at k< 0 - убывает на всей области определения.

4. If the function f is direct proportionality and (x 1, y 1), (x 2, y 2) are pairs of corresponding values ​​of the variables x and y, and x 2 ≠0 then.

Indeed, if the function f is direct proportionality, then it can be given by the formula y = khx, and then y 1 = kh 1, y 2 = kh 2. Since at x 2 ≠0 and k≠0, then y 2 ≠0. That's why and that means .

If the values ​​of the variables x and y are positive real numbers, then the proven property of direct proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y increases (decreases) by the same amount.

This property is inherent only in direct proportionality, and it can be used when solving word problems in which directly proportional quantities are considered.

Problem 1. In 8 hours, a turner produced 16 parts. How many hours will it take a lathe operator to produce 48 parts if he works at the same productivity?

Solution. The problem considers the following quantities: the turner’s work time, the number of parts he makes, and productivity (i.e., the number of parts produced by the turner in 1 hour), with the last value being constant, and the other two taking on different values. In addition, the number of parts made and the work time are directly proportional quantities, since their ratio is equal to a certain number that is not equal to zero, namely, the number of parts made by a turner in 1 hour. If the number of parts made is denoted by the letter y, the work time is x, and productivity is k, then we get that = k or y = khx, i.e. The mathematical model of the situation presented in the problem is direct proportionality.

The problem can be solved in two arithmetic ways:

1st way: 2nd way:

1) 16:8 = 2 (children) 1) 48:16 = 3 (times)

2) 48:2 = 24 (h) 2) 8-3 = 24 (h)

Solving the problem in the first way, we first found the coefficient of proportionality k, it is equal to 2, and then, knowing that y = 2x, we found the value of x provided that y = 48.

When solving the problem in the second way, we used the property of direct proportionality: as many times as the number of parts made by a turner increases, the amount of time for their production increases by the same amount.

Let us now move on to consider a function called inverse proportionality.

If t is the pedestrian’s time of movement (in hours), v is his speed (in km/h) and he walked 12 km, then the relationship between these quantities can be expressed by the formula v∙t = 20 or v = .

Since each value t (t ≠ 0) corresponds to a single speed value v, we can say that a function is specified using the formula v =. It is called inverse proportionality and is defined as follows.

Definition. Inverse proportionality is a function that can be specified using the formula y =, where k is a real number that is not equal to zero.

The name of this function is due to the fact that y = there are variables x and y, which can be values ​​of quantities. And if the product of two quantities is equal to some number different from zero, then they are called inversely proportional. In our case xy = k(k ≠0). This number k is called the proportionality coefficient.

Function y = is a mathematical model of many real situations considered already in the initial mathematics course. One of them is described before the definition of inverse proportionality. Another example: if you bought 12 kg of flour and put it in l: y kg cans each, then the relationship between these quantities can be represented as x-y = 12, i.e. it is inversely proportional with coefficient k=12.

Let us recall some properties of inverse proportionality, known from the school mathematics course.

1.Domain of function definition y = and the range of its values ​​x is the set of real numbers other than zero.

2. The graph of inverse proportionality is a hyperbola.

3. For k > 0, the branches of the hyperbola are located in the 1st and 3rd quarters and the function y = is decreasing over the entire domain of definition of x (Fig. 8).

Rice. 8 Fig.9

At k< 0 ветви гиперболы расположены во 2-й и 4-й четвертях и функция y = is increasing over the entire domain of definition of x (Fig. 9).

4. If the function f is inverse proportionality and (x 1, y 1), (x 2, y 2) are pairs of corresponding values ​​of the variables x and y, then.

Indeed, if the function f is inverse proportionality, then it can be given by the formula y = ,and then . Since x 1 ≠0, x 2 ≠0, x 3 ≠0, then

If the values ​​of the variables x and y are positive real numbers, then this property of inverse proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y decreases (increases) by the same amount.

This property is inherent only in inverse proportionality, and it can be used when solving word problems in which inversely proportional quantities are considered.

Problem 2. A cyclist, moving at a speed of 10 km/h, covered the distance from A to B in 6 hours. How much time will the cyclist spend on the way back if he travels at a speed of 20 km/h?

Solution. The problem considers the following quantities: the speed of the cyclist, the time of movement and the distance from A to B, the last quantity being constant, while the other two take different values. In addition, the speed and time of movement are inversely proportional quantities, since their product is equal to a certain number, namely the distance traveled. If the time of movement of the cyclist is denoted by the letter y, the speed by x, and the distance AB by k, then we obtain that xy = k or y =, i.e. The mathematical model of the situation presented in the problem is inverse proportionality.

There are two ways to solve the problem:

1st way: 2nd way:

1) 10-6 = 60 (km) 1) 20:10 = 2 (times)

2) 60:20 = 3(4) 2) 6:2 = 3(h)

Solving the problem in the first way, we first found the coefficient of proportionality k, it is equal to 60, and then, knowing that y =, we found the value of y provided that x = 20.

When solving the problem in the second way, we used the property of inverse proportionality: the number of times the speed of movement increases, the time to cover the same distance decreases by the same number.

Note that when solving specific problems with inversely proportional or directly proportional quantities, some restrictions are imposed on x and y; in particular, they can be considered not on the entire set of real numbers, but on its subsets.

Problem 3. Lena bought x pencils, and Katya bought 2 times more. Denote the number of pencils purchased by Katya by y, express y by x, and construct a graph of the established correspondence provided that x≤5. Is this correspondence a function? What is its domain of definition and range of values?

Solution. Katya bought = 2 pencils. When plotting the function y=2x, it is necessary to take into account that the variable x denotes the number of pencils and x≤5, which means that it can only take the values ​​0, 1, 2, 3, 4, 5. This will be the domain of definition of this function. To obtain the range of values ​​of this function, you need to multiply each x value from the range of definition by 2, i.e. this will be the set (0, 2, 4, 6, 8, 10). Therefore, the graph of the function y = 2x with the domain of definition (0, 1, 2, 3, 4, 5) will be the set of points shown in Figure 10. All these points belong to the straight line y = 2x.

Proportionality is a relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.

Proportionality can be direct or inverse. In this lesson we will look at each of them.

Direct proportionality

Let's assume that the car is moving at a speed of 50 km/h. We remember that speed is the distance traveled per unit of time (1 hour, 1 minute or 1 second). In our example, the car is moving at a speed of 50 km/h, that is, in one hour it will cover a distance of fifty kilometers.

Let us depict in the figure the distance traveled by the car in 1 hour.

Let the car drive for another hour at the same speed of fifty kilometers per hour. Then it turns out that the car will travel 100 km

As can be seen from the example, doubling the time led to an increase in the distance traveled by the same amount, that is, twice.

Quantities such as time and distance are called directly proportional. And the relationship between such quantities is called direct proportionality.

Direct proportionality is the relationship between two quantities in which an increase in one of them entails an increase in the other by the same amount.

and vice versa, if one quantity decreases by a certain number of times, then the other decreases by the same number of times.

Let's assume that the original plan was to drive a car 100 km in 2 hours, but after driving 50 km, the driver decided to rest. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, reducing the distance traveled will lead to a decrease in time by the same amount.

An interesting feature of directly proportional quantities is that their ratio is always constant. That is, when the values ​​of directly proportional quantities change, their ratio remains unchanged.

In the example considered, the distance was initially 50 km and the time was one hour. The ratio of distance to time is the number 50.

But we increased the travel time by 2 times, making it equal to two hours. As a result, the distance traveled increased by the same amount, that is, it became equal to 100 km. The ratio of one hundred kilometers to two hours is again the number 50

The number 50 is called coefficient of direct proportionality. It shows how much distance there is per hour of movement. In this case, the coefficient plays the role of movement speed, since speed is the ratio of the distance traveled to the time.

Proportions can be made from directly proportional quantities. For example, the ratios make up the proportion:

Fifty kilometers is to one hour as one hundred kilometers is to two hours.

Example 2. The cost and quantity of goods purchased are directly proportional. If 1 kg of sweets costs 30 rubles, then 2 kg of the same sweets will cost 60 rubles, 3 kg 90 rubles. As the cost of a purchased product increases, its quantity increases by the same amount.

Since the cost of a product and its quantity are directly proportional quantities, their ratio is always constant.

Let's write down what is the ratio of thirty rubles to one kilogram

Now let’s write down what the ratio of sixty rubles to two kilograms is. This ratio will again be equal to thirty:

Here the coefficient of direct proportionality is the number 30. This coefficient shows how many rubles are per kilogram of sweets. In this example, the coefficient plays the role of the price of one kilogram of goods, since price is the ratio of the cost of the goods to its quantity.

Inverse proportionality

Consider the following example. The distance between the two cities is 80 km. The motorcyclist left the first city and, at a speed of 20 km/h, reached the second city in 4 hours.

If a motorcyclist's speed was 20 km/h, this means that every hour he covered a distance of twenty kilometers. Let us depict in the figure the distance traveled by the motorcyclist and the time of his movement:

On the way back, the motorcyclist's speed was 40 km/h, and he spent 2 hours on the same journey.

It is easy to notice that when the speed changes, the time of movement changes by the same amount. Moreover, it changed in the opposite direction - that is, the speed increased, but the time, on the contrary, decreased.

Quantities such as speed and time are called inversely proportional. And the relationship between such quantities is called inverse proportionality.

Inverse proportionality is the relationship between two quantities in which an increase in one of them entails a decrease in the other by the same amount.

and vice versa, if one quantity decreases by a certain number of times, then the other increases by the same number of times.

For example, if on the way back the motorcyclist’s speed was 10 km/h, then he would cover the same 80 km in 8 hours:

As can be seen from the example, a decrease in speed led to an increase in movement time by the same amount.

The peculiarity of inversely proportional quantities is that their product is always constant. That is, when the values ​​of inversely proportional quantities change, their product remains unchanged.

In the example considered, the distance between cities was 80 km. When the speed and time of movement of the motorcyclist changed, this distance always remained unchanged

A motorcyclist could travel this distance at a speed of 20 km/h in 4 hours, and at a speed of 40 km/h in 2 hours, and at a speed of 10 km/h in 8 hours. In all cases, the product of speed and time was equal to 80 km

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The concept of direct proportionality

Imagine that you are planning to buy your favorite candies (or anything that you really like). Sweets in the store have their own price. Let's say 300 rubles per kilogram. The more candies you buy, the more money you pay. That is, if you want 2 kilograms, pay 600 rubles, and if you want 3 kilograms, pay 900 rubles. This seems to be all clear, right?

If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the relationship of two quantities dependent on each other. And the ratio of these quantities remains unchanged and constant: by how many parts one of them increases or decreases, by the same number of parts the second increases or decreases proportionally.

Direct proportionality can be described with the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our example about candy, the price is a constant value, a constant. It does not increase or decrease, no matter how many candies you decide to buy. The independent variable (argument)x is how many kilograms of candy you are going to buy. And the dependent variable f(x) (function) is how much money you end up paying for your purchase. So we can substitute the numbers into the formula and get: 600 rubles. = 300 rub. * 2 kg.

The intermediate conclusion is this: if the argument increases, the function also increases, if the argument decreases, the function also decreases

Function and its properties

Direct proportional function is a special case of a linear function. If the linear function is y = k*x + b, then for direct proportionality it looks like this: y = k*x, where k is called the proportionality coefficient, and it is always a non-zero number. It is easy to calculate k - it is found as a quotient of a function and an argument: k = y/x.

To make it clearer, let's take another example. Imagine that a car is moving from point A to point B. Its speed is 60 km/h. If we assume that the speed of movement remains constant, then it can be taken as a constant. And then we write the conditions in the form: S = 60*t, and this formula is similar to the function of direct proportionality y = k *x. Let's draw a parallel further: if k = y/x, then the speed of the car can be calculated knowing the distance between A and B and the time spent on the road: V = S /t.

And now, from the applied application of knowledge about direct proportionality, let’s return back to its function. The properties of which include:

its domain of definition is the set of all real numbers (as well as its subsets);

function is odd;

the change in variables is directly proportional along the entire length of the number line.

Direct proportionality and its graph

The graph of a direct proportionality function is a straight line that intersects the origin. To build it, it is enough to mark only one more point. And connect it and the origin of coordinates with a straight line.

In the case of a graph, k is the slope. If the slope is less than zero (k< 0), то угол между графиком функции прямой пропорциональности и осью абсцисс тупой, а функция убывающая. Если угловой коэффициент больше нуля (k >0), the graph and the x-axis form an acute angle, and the function is increasing.

And one more property of the graph of the direct proportionality function is directly related to the slope k. Suppose we have two non-identical functions and, accordingly, two graphs. So, if the coefficients k of these functions are equal, their graphs are located parallel to the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.

Sample problems

Now let's solve a couple direct proportionality problems

Let's start with something simple.

Problem 1: Imagine that 5 hens laid 5 eggs in 5 days. And if there are 20 hens, how many eggs will they lay in 20 days?

Solution: Let's denote the unknown by kx. And we will reason as follows: how many times more chickens have there become? Divide 20 by 5 and find out that it is 4 times. How many times more eggs will 20 hens lay in the same 5 days? Also 4 times more. So, we find ours like this: 5*4*4 = 80 eggs will be laid by 20 hens in 20 days.

Now the example is a little more complicated, let’s paraphrase the problem from Newton’s “General Arithmetic”. Problem 2: A writer can compose 14 pages of a new book in 8 days. If he had assistants, how many people would it take to write 420 pages in 12 days?

Solution: We reason that the number of people (writer + assistants) increases with the volume of work if it had to be done in the same amount of time. But how many times? Dividing 420 by 14, we find out that it increases by 30 times. But since, according to the conditions of the task, more time is given for the work, the number of assistants increases not by 30 times, but in this way: x = 1 (writer) * 30 (times): 12/8 (days). Let's transform and find out that x = 20 people will write 420 pages in 12 days.

Let's solve another problem similar to those in our examples.

Problem 3: Two cars set off on the same journey. One was moving at a speed of 70 km/h and covered the same distance in 2 hours as the other took 7 hours. Find the speed of the second car.

Solution: As you remember, the path is determined through speed and time - S = V *t. Since both cars traveled the same distance, we can equate the two expressions: 70*2 = V*7. How do we find that the speed of the second car is V = 70*2/7 = 20 km/h.

And a couple more examples of tasks with functions of direct proportionality. Sometimes problems require finding the coefficient k.

Task 4: Given the functions y = - x/16 and y = 5x/2, determine their proportionality coefficients.

Solution: As you remember, k = y/x. This means that for the first function the coefficient is equal to -1/16, and for the second k = 5/2.

You may also encounter a task like Task 5: Write down direct proportionality with a formula. Its graph and the graph of the function y = -5x + 3 are located in parallel.

Solution: The function that is given to us in the condition is linear. We know that direct proportionality is a special case of a linear function. And we also know that if the coefficients of k functions are equal, their graphs are parallel. This means that all that is required is to calculate the coefficient of a known function and set direct proportionality using the formula familiar to us: y = k *x. Coefficient k = -5, direct proportionality: y = -5*x.

Conclusion

Now you have learned (or remembered, if you have already covered this topic before) what is called direct proportionality, and looked at it examples. We also talked about the direct proportionality function and its graph, and solved several example problems.

If this article was useful and helped you understand the topic, tell us about it in the comments. So that we know if we could benefit you.

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