Proportion is proportional and inversely proportional. Practical application of direct and inverse proportionality

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

Let's give examples.

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

Let it be required to allocate a rectangular area for the garden with an area of

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

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

Expressing y in terms of x, we get:

By giving x arbitrary values, we will get the corresponding y values.

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

Let the distance between two cities be 200 km. The faster 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 through x, and the time of movement through y, then the relationship between them will be expressed by the formula:

Definition. The relationship between two quantities, expressed as , where k is a certain number (not equal to zero), is called an inverse relationship.

The number here is also called the coefficient of proportionality.

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

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

And the right one - to a certain number, not zero(by definition), that is, an incorrect equality will be obtained.

2. Graph of inverse proportion.

Let's build a dependency graph

Expressing y in terms of x, we get:

We will give x arbitrary (permissible) values and calculate the corresponding values of y. Let's get a 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 more closely.

For all possible values of x, the corresponding points will be located on two branches of the graph, symmetrical about the origin and passing in the I and III quarters of the coordinate plane (Fig. 29).

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

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

An inversely proportional graph is called a hyperbola.

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

With sufficiently high accuracy, a hyperbola can be drawn using, for example, patterns.

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

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

Dependency Types

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

The dependence of battery life on the time it is charged

Remark 1

This dependency is called straight:

As one value increases, the other also increases. As one value decreases, the other value also decreases.

Let's consider another example.

How more books the student will read less mistakes will do in dictation. Or the higher you climb the mountains, the lower the atmospheric pressure will be.

Remark 2

This dependency is called reverse:

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

Thus, in the case direct dependency both quantities change in the same way (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. With an increase in speed (of the first value) by $2$ times, we will find how the time (second value) that will be spent on the path to a friend will change.

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

Remark 3

This dependency is called proportional:

How many times one value changes, how many times the second will change.

Example 2

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

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

In both examples, proportional dependencies were considered. But in the example with loaves of bread, the values \u200b\u200bchange in one direction, therefore, the dependence is straight. And in the example with a trip to a friend, the relationship between speed and time is reverse. Thus, there is directly proportional relationship and inversely proportional relationship.

Direct proportionality

Consider $2$ proportional quantities: the number of loaves of bread and their cost. Let $2$ loaves of bread cost $80$ rubles. With an increase in the number of rolls by $4$ times ($8$ rolls), their total cost will be $320$ rubles.

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

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

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

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

Definition 1

The equality of two relations is called proportion.

With a directly proportional relationship, a ratio is obtained when the change in the first and second values \u200b\u200bis the same:

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

Definition 2

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

Example 3

The car traveled $180$ km in $2$ hours. Find the time it takes for him to cover $2$ times the distance with the same speed.

Solution.

Time is directly proportional to distance:

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

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

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

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

The car traveled $180$ km - in the time of $2$ hour

The car travels $180 \cdot 2=360$ km - in the time of $x$ hours

The further the car travels, the more time he will need. Therefore, 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:

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

How many times the speed increases, 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 condition of the problem in the form of a table:

The car traveled $60$ km - in the time of $6$ hours

A car travels $120$ km - in a time of $x$ hours

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

Let's make a proportion.

Because proportionality is inverse, we turn the second ratio in proportion:

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

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

Answer: The car will need $3$ hours.

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:

Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
It has no maximum or minimum values.
Is odd and its graph is symmetrical about the origin.
Non-periodic.
Its graph does not cross the coordinate axes.
Has no zeros.
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.
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. They also suggest that when drawing up a proportion right side records must be reversed: 60/120 = 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 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 \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 tasks are inverse proportionality really uncomplicated. 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 in social networks so that your friends and classmates can also play.

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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. Examples of a direct proportional relationship:

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 member of the proportion, we divide the product of the middle terms by the known extreme member of the proportion:

So, 12 meters cost 1344 rubles.

Answer: 1344 rubles.