proportional relationship. Direct and inverse proportionality

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

Proportionality is direct and inverse. In this lesson, we will look at each of them.

Direct proportionality

Suppose a 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 travel a distance equal to fifty kilometers.

Let's plot 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 said to be directly proportional. The relationship between these 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 value decreases by a certain number of times, then the other decreases by the same amount.

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

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

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

But we have increased the time of movement 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 direct proportionality coefficient. It shows how much distance there is per hour of movement. In this case, the coefficient plays the role of the speed of movement, since the speed is the ratio of the distance traveled to the time.

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

Fifty kilometers are related to one hour as one hundred kilometers are related to two hours.

Example 2. The cost and quantity of the purchased goods 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. With the increase in the cost of the purchased goods, its quantity increases by the same amount.

Since the value of a commodity and its quantity are directly proportional, their ratio is always constant.

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

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

Here, the direct proportionality coefficient is the number 30. This coefficient shows how many rubles per kilogram of sweets. AT this example the coefficient plays the role of the price of one kilogram of goods, since the 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 the speed of a motorcyclist was 20 km/h, this means that every hour he traveled a distance equal to 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 see that when the speed changes, the time of movement has changed by the same amount. And it changed in reverse side- that is, the speed increased, and the time, on the contrary, decreased.

Quantities such as speed and time are called inversely proportional. The relationship between these 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 value decreases by a certain number of times, then the other increases by the same amount.

For example, if on the way back the speed of a motorcyclist 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 travel time by the same factor.

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

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

A motorcyclist could cover 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 two quantities are called directly proportional, if when one of them is increased several times, the other is increased by the same amount. Accordingly, when one of them decreases by several times, the other decreases by the same amount.

The relationship between such quantities is a direct proportional relationship. Direct examples proportional dependence:

1) at a constant speed, the distance traveled is directly proportional to time;

2) the perimeter of a square and its side are directly proportional;

3) the cost of a commodity purchased at one price is directly proportional to its quantity.

To distinguish a direct proportional relationship from an inverse one, you can use the proverb: "The farther into the forest, the more firewood."

It is convenient to solve problems for directly proportional quantities using proportions.

1) For the manufacture of 10 parts, 3.5 kg of metal is needed. How much metal will be used to make 12 such parts?

(We argue like this:

1. In the completed column, put the arrow in the direction from more to the smaller one.

2. The more parts, the more metal is needed to make them. So it's a directly proportional relationship.

Let x kg of metal be needed to make 12 parts. We make up the proportion (in the direction from the beginning of the arrow to its end):

12:10=x:3.5

To find , we need to divide the product of the extreme terms by the known middle term:

This means that 4.2 kg of metal will be required.

Answer: 4.2 kg.

2) 1680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?

(1. In the completed column, put the arrow in the direction from the largest number to the smallest.

2. The less fabric you buy, the less you have to pay for it. So it's a directly proportional relationship.

3. Therefore, the second arrow is directed in the same direction as the first).

Let x rubles cost 12 meters of fabric. We make up the proportion (from the beginning of the arrow to its end):

15:12=1680:x

To find the unknown extreme term of the proportion, we divide the product of the middle terms by the known extreme term of the proportion:

So, 12 meters cost 1344 rubles.

Answer: 1344 rubles.

Example

Proportionality factor

The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.

Direct proportionality

Direct proportionality- functional dependence, in which some quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionately, in equal shares, that is, if the argument has changed 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 proportion- 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 .

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

Such different proportions

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

Dependence can be direct and reverse. Therefore, the relationship between quantities describe 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 preparing for exams, the higher your grades will be. Or the more things you take with you on a hike, the harder it is to carry your backpack. Those. the amount of effort spent on preparing for exams is directly proportional to the grades received. 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 of an independent value (it is called an argument) causes a proportional (i.e., by the same amount) increase or decrease in a dependent value (it is called a function).

Illustrate simple example. You want to buy apples in the market. The apples on the counter and the amount of money in your wallet are inversely related. Those. the more apples you buy, the less money you have left.

Function and its graph

The inverse proportionality function can be described as y = k/x. Wherein 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, Besides y= 0. E(y): (-∞; 0) U (0; +∞) .
3. It has no maximum or minimum values.
4. Is odd and its graph is symmetrical about the origin.
5. Non-periodic.
6. Its graph does not cross the coordinate axes.
7. Has no zeros.
8. If a k> 0 (that is, the argument increases), the function decreases proportionally on each of its intervals. If a k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
9. As the argument increases ( k> 0) the negative values ​​of the function are in the interval (-∞; 0), and the positive values ​​are in the interval (0; +∞). When the argument is decreasing ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).

The graph of the inverse proportionality function is called a hyperbola. Depicted as follows:

Inverse Proportional Problems

To make it clearer, let's look at a few tasks. They are not too complicated, and their solution will help you visualize what inverse proportion is and how this knowledge can be useful in your everyday life.

Task number 1. The car is moving at a speed of 60 km/h. It took him 6 hours to reach 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 of time, distance and speed: t = S/V. Agree, it very much reminds us of the inverse proportionality function. And it indicates that the time that the car spends on the road, and the speed with which it moves, are inversely proportional.

To verify this, let's find V 2, which, by condition, is 2 times higher: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is not difficult to find out the time t 2 that is required from us according to the condition of the problem: t 2 = 360/120 = 3 hours.

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

The solution to this problem can also be written as a proportion. Why do we create a diagram like this:

↓ 60 km/h – 6 h

↓120 km/h – x h

Arrows indicate an inverse relationship. And they also suggest that when drawing up the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. Where do we get x \u003d 60 * 6/120 \u003d 3 hours.

Task number 2. The workshop employs 6 workers who cope with a given amount of work in 4 hours. If the number of workers is halved, how long will it take for the remaining workers to complete the same amount of work?

We write the conditions of the problem in the form visual scheme:

↓ 6 workers - 4 hours

↓ 3 workers - x h

Let's write this as a proportion: 6/3 = x/4. And we get x \u003d 6 * 4/3 \u003d 8 hours. If there are 2 times fewer workers, the rest will spend 2 times more time to complete all the work.

Task number 3. Two pipes lead to the pool. Through one pipe, water enters at a rate of 2 l / s and fills the pool in 45 minutes. Through another pipe, the pool will be filled in 75 minutes. How fast does water enter the pool through this pipe?

To begin with, we will bring all the quantities given to us according to the condition of the problem to the same units of measurement. To do this, we express the filling rate of the pool in liters per minute: 2 l / s \u003d 2 * 60 \u003d 120 l / min.

Since it follows from the condition that the pool is filled more slowly through the second pipe, it means that the rate of water inflow is lower. On the face of inverse proportion. Let us express the speed unknown to us in terms of x and draw up the following scheme:

↓ 120 l/min - 45 min

↓ x l/min – 75 min

And then we will make a proportion: 120 / x \u003d 75/45, from where x \u003d 120 * 45/75 \u003d 72 l / min.

In the problem, the filling rate of the pool is expressed in liters per second, let's bring our answer to the same form: 72/60 = 1.2 l/s.

Task number 4. Business cards are printed in a small private printing house. An employee of the printing house works at a speed of 42 business cards per hour and works full time - 8 hours. If he worked faster and printed 48 business cards per hour, how much sooner could he go home?

We go in a proven way and draw up a scheme according to the condition of the problem, denoting the desired value as x:

↓ 42 business cards/h – 8 h

↓ 48 business cards/h – xh

Before us is an inversely proportional relationship: how many times more business cards an employee of a printing house prints per hour, the same amount of time it will take him to complete the same job. Knowing this, we can set up the proportion:

42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 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 consider them so. And most importantly, knowledge of the inversely proportional dependence of quantities can really be useful to you more than once.

Not only in math classes and exams. But even then, when you are going to go on a trip, go shopping, decide to earn some money during the holidays, etc.

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

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