Proportion is proportional and inversely proportional. Direct and inverse proportional relationships

I. Directly proportional quantities.

Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.

Examples.

1 . The quantity of goods purchased and the purchase price (with a fixed price for one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, the more times more they paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer is the path, how many times more time will it take to complete it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than another, then its mass will be 2 times larger)

II. Property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrarily taken values ​​of the first quantity is equal to the ratio of two corresponding values ​​of the second quantity.

Task 1. For raspberry jam we took 12 kg raspberries and 8 kg Sahara. How much sugar will you need if you took it? 9 kg raspberries?

Solution.

We reason like this: let it be necessary x kg sugar for 9 kg raspberries The mass of raspberries and the mass of sugar are directly proportional quantities: how many times less raspberries are, the same number of times less sugar is needed. Therefore, the ratio of raspberries taken (by weight) ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:

12: 9=8: X;

x=9 · 8: 12;

x=6. Answer: on 9 kg raspberries need to be taken 6 kg Sahara.

The solution of the problem It could be done like this:

Let on 9 kg raspberries need to be taken x kg Sahara.

(The arrows in the figure are directed in one direction, and up or down does not matter. Meaning: how many times the number 12 more number 9 , the same number of times 8 more number X, i.e. there is a direct relationship here).

Answer: on 9 kg I need to take some raspberries 6 kg Sahara.

Task 2. Car for 3 hours traveled the distance 264 km. How long will it take him to travel? 440 km, if he drives at the same speed?

Solution.

Let for x hours the car will cover the distance 440 km.

Answer: the car will pass 440 km in 5 hours.

Task 3. Water flows from the pipe into the pool. Behind 2 hours she fills 1/5 swimming pool What part of the pool is filled with water in 5 o'clock?

Solution.

We answer the question of the task: for 5 o'clock will be filled 1/x part of the pool. (The entire pool is taken as one whole).

Example

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:

Sources

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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

- (from Latin proportionalis proportionate, proportional). Proportionality. Dictionary 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

PROPORTIONALITY, proportionality, plural. no, female (book). 1. abstract noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between quantities when they are proportional (see proportional ... Dictionary Ushakova

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

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

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

I. Directly proportional quantities.

Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.

Examples.

1 . The quantity of goods purchased and the purchase price (with a fixed price for one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, the more times more they paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer is the path, how many times more time will it take to complete it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than another, then its mass will be 2 times larger)

II. Property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrarily taken values ​​of the first quantity is equal to the ratio of two corresponding values ​​of the second quantity.

Task 1. For raspberry jam we took 12 kg raspberries and 8 kg Sahara. How much sugar will you need if you took it? 9 kg raspberries?

Solution.

We reason like this: let it be necessary x kg sugar for 9 kg raspberries The mass of raspberries and the mass of sugar are directly proportional quantities: how many times less raspberries are, the same number of times less sugar is needed. Therefore, the ratio of raspberries taken (by weight) ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:

12: 9=8: X;

x=9 · 8: 12;

x=6. Answer: on 9 kg raspberries need to be taken 6 kg Sahara.

The solution of the problem It could be done like this:

Let on 9 kg raspberries need to be taken x kg Sahara.

(The arrows in the figure are directed in one direction, and up or down does not matter. Meaning: how many times the number 12 more number 9 , the same number of times 8 more number X, i.e. there is a direct relationship here).

Answer: on 9 kg I need to take some raspberries 6 kg Sahara.

Task 2. Car for 3 hours traveled the distance 264 km. How long will it take him to travel? 440 km, if he drives at the same speed?

Solution.

Let for x hours the car will cover the distance 440 km.

Answer: the car will pass 440 km in 5 hours.

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 more to less.

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.