Direct and inverse proportional relationships. Direct and inverse proportionality

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 two quantities are called directly proportional, if when one of them increases several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.

The relationship between such quantities is a direct proportional relationship. Examples of direct 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 quantities;

3) the cost of a product 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 further into the forest, the more firewood.”

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

1) To make 10 parts you need 3.5 kg of metal. How much metal will go into making 12 of these parts?

(We reason like this:

1. In the filled column, place an arrow in the direction from the largest number to the smallest.

2. The more parts, the more metal needed to make them. This means that this is 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 , you 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) For 15 meters of fabric they paid 1680 rubles. How much does 12 meters of such fabric cost?

(1. In the filled column, place an 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. This means that this is a directly proportional relationship.

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

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

15:12=1680:x

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

This means that 12 meters cost 1344 rubles.

Answer: 1344 rubles.

Example

4 / 5 = 0.8;

5.6 / 7 = 0.8, etc. Proportionality factor A constant relationship of proportional quantities is called

proportionality factor

proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another. 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:) = fMathematically, direct proportionality is written as a formula:,f = xacon

Inverse proportionality

s t

Inverse proportionality

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

Mathematically, inverse proportionality is written as a formula:

Function properties:

Sources

Wikimedia Foundation.

2010.

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 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. 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 reverse As one value increases, the second decreases. As one value decreases, the second value increases. 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 Dependence of battery operating time on the time it is charged. 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.

Example

4 / 5 = 0.8;

5.6 / 7 = 0.8, etc. Proportionality factor A constant relationship of proportional quantities is called

proportionality factor

proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another. 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:) = fMathematically, direct proportionality is written as a formula:,f = xacon

Inverse proportionality

s t

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:

• Newton's second law
• Coulomb barrier

See what “Direct proportionality” is in other dictionaries:

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

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

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

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

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

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