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:

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Today we will look at what quantities are called inversely proportional, what an inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside of school.

Such different proportions

Proportionality name two quantities that are mutually dependent on each other.

The dependence can be direct and inverse. Consequently, the relationships between quantities are described by direct and inverse proportionality.

Direct proportionality– this is such a relationship between two quantities in which an increase or decrease in one of them leads to an increase or decrease in the other. Those. their attitude does not change.

For example, the more effort you put into studying for exams, the higher your grades. Or the more things you take with you on a hike, the heavier your backpack will be to carry. Those. The amount of effort spent preparing for exams is directly proportional to the grades obtained. And the number of things packed in a backpack is directly proportional to its weight.

Inverse proportionality– this is a functional dependence in which a decrease or increase by several times in an independent value (it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in a dependent value (it is called a function).

Let's illustrate with a simple example. You want to buy apples at the market. The apples on the counter and the amount of money in your wallet are in inverse proportion. Those. The more apples you buy, the less money you will have left.

Function and its graph

The inverse proportionality function can be described as y = k/x. In which x≠ 0 and k≠ 0.

This function has the following properties:

1. Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
2. The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
3. Does not have maximum or minimum values.
4. It is odd and its graph is symmetrical about the origin.
5. Non-periodic.
6. Its graph does not intersect the coordinate axes.
7. Has no zeros.
8. If k> 0 (i.e. the argument increases), the function decreases proportionally on each of its intervals. If k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
9. As the argument increases ( k> 0) negative values ​​of the function are in the interval (-∞; 0), and positive values ​​are in the interval (0; +∞). When the argument decreases ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).

The graph of an inverse proportionality function is called a hyperbola. Shown as follows:

Inverse proportionality problems

To make it clearer, let's look at several tasks. They are not too complicated, and solving them will help you visualize what inverse proportionality is and how this knowledge can be useful in your everyday life.

Task No. 1. A car is moving at a speed of 60 km/h. It took him 6 hours to get to his destination. How long will it take him to cover the same distance if he moves at twice the speed?

We can start by writing down a formula that describes the relationship between time, distance and speed: t = S/V. Agree, it reminds us very much of the inverse proportionality function. And it indicates that the time a car spends on the road and the speed at which it moves are in inverse proportion.

To verify this, let's find V 2, which, according to the condition, is 2 times higher: V 2 = 60 * 2 = 120 km/h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it’s not difficult to find out the time t 2 that is required from us according to the conditions of the problem: t 2 = 360/120 = 3 hours.

As you can see, travel time and speed are indeed inversely proportional: at a speed 2 times higher than the original speed, the car will spend 2 times less time on the road.

The solution to this problem can also be written as a proportion. So let's first create this diagram:

↓ 60 km/h – 6 h

↓120 km/h – x h

Arrows indicate an inversely proportional relationship. They also suggest that when drawing up a proportion, the right side of the record must be turned over: 60/120 = x/6. Where do we get x = 60 * 6/120 = 3 hours.

Task No. 2. The workshop employs 6 workers who can complete a given amount of work in 4 hours. If the number of workers is halved, how long will it take the remaining workers to complete the same amount of work?

Let us write down the conditions of the problem in the form of a visual diagram:

↓ 6 workers – 4 hours

↓ 3 workers – x h

Let's write this as a proportion: 6/3 = x/4. And we get x = 6 * 4/3 = 8 hours. If there are 2 times fewer workers, the remaining ones will spend 2 times more time doing all the work.

Task No. 3. There are two pipes leading into the pool. Through one pipe, water flows at a speed of 2 l/s and fills the pool in 45 minutes. Through another pipe, the pool will fill in 75 minutes. At what speed does water enter the pool through this pipe?

To begin with, let us reduce all the quantities given to us according to the conditions of the problem to the same units of measurement. To do this, we express the speed of filling the pool in liters per minute: 2 l/s = 2 * 60 = 120 l/min.

Since it follows from the condition that the pool fills more slowly through the second pipe, this means that the rate of water flow is lower. The proportionality is inverse. Let us express the unknown speed through x and draw up the following diagram:

↓ 120 l/min – 45 min

↓ x l/min – 75 min

And then we make up the proportion: 120/x = 75/45, from where x = 120 * 45/75 = 72 l/min.

In the problem, the filling rate of the pool is expressed in liters per second; let’s reduce the answer we received to the same form: 72/60 = 1.2 l/s.

Task No. 4. A small private printing house prints business cards. A printing house employee works at a speed of 42 business cards per hour and works a full day - 8 hours. If he worked faster and printed 48 business cards in an hour, how much earlier could he go home?

We follow the proven path and draw up a diagram according to the conditions of the problem, designating the desired value as x:

↓ 42 business cards/hour – 8 hours

↓ 48 business cards/h – x h

We have an inversely proportional relationship: the number of times more business cards an employee of a printing house prints per hour, the same number of times less time he will need to complete the same work. Knowing this, let's create a proportion:

42/48 = x/8, x = 42 * 8/48 = 7 hours.

Thus, having completed the work in 7 hours, the printing house employee could go home an hour earlier.

Conclusion

It seems to us that these inverse proportionality problems are really simple. We hope that now you also think of them that way. And the main thing is that knowledge about the inversely proportional dependence of quantities can really be useful to you more than once.

Not only in math lessons and exams. But even then, when you get ready to go on a trip, go shopping, decide to earn a little extra money during the holidays, etc.

Tell us in the comments what examples of inverse and direct proportional relationships you notice around you. Let it be such a game. You'll see how exciting it is. Don't forget to share this article on social networks so that your friends and classmates can also play.

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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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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.