What is inverse proportion? Direct and inverse proportional relationships

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.

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 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

This dependence is called reverse:

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

Thus, in case direct dependence both quantities change equally (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. 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 straight. 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

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

Wikimedia Foundation. 2010.

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 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

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 ... Ushakov's Explanatory Dictionary

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

Basic goals:

• introduce the concept of direct and inverse proportional dependence of quantities;
• teach how to solve problems using these dependencies;
• promote the development of problem solving skills;
• consolidate the skill of solving equations using proportions;
• repeat the steps with ordinary and decimal fractions;
• develop students' logical thinking.

DURING THE CLASSES

I. Self-determination for activity(Organizing time)

- Guys! Today in the lesson we will get acquainted with problems solved using proportions.

II. Updating knowledge and recording difficulties in activities

2.1. Oral work (3 min)

– Find the meaning of the expressions and find out the word encrypted in the answers.

14 – s; 0.1 – and; 7 – l; 0.2 – a; 17 – in; 25 – to

– The resulting word is strength. Well done!
– The motto of our lesson today: Power is in knowledge! I'm searching - that means I'm learning!
– Make up a proportion from the resulting numbers. (14:7 = 0.2:0.1 etc.)

2.2. Let's consider the relationship between the quantities we know (7 min)

– the distance covered by the car at a constant speed, and the time of its movement: S = v t ( with increasing speed (time), the distance increases);
– vehicle speed and time spent on the journey: v=S:t(as the time to travel the path increases, the speed decreases);
– the cost of goods purchased at one price and its quantity: C = a · n (with an increase (decrease) in price, the purchase cost increases (decreases));
– price of the product and its quantity: a = C: n (with an increase in quantity, the price decreases)
– area of ​​the rectangle and its length (width): S = a · b (with increasing length (width), the area increases;
– rectangle length and width: a = S: b (as the length increases, the width decreases;
– the number of workers performing some work with the same labor productivity, and the time it takes to complete this work: t = A: n (with an increase in the number of workers, the time spent on performing the work decreases), etc.

We have obtained dependences in which, with an increase in one quantity several times, another immediately increases by the same amount (examples are shown with arrows) and dependences in which, with an increase in one quantity several times, the second quantity decreases by the same number of times.
Such dependencies are called direct and inverse proportionality.
Directly proportional dependence– a relationship in which as one value increases (decreases) several times, the second value increases (decreases) by the same amount.
Inversely proportional relationship– a relationship in which as one value increases (decreases) several times, the second value decreases (increases) by the same amount.

III. Setting a learning task

– What problem is facing us? (Learn to distinguish between direct and inverse dependencies)
- This - target our lesson. Now formulate topic lesson. (Direct and inverse proportional relationship).
- Well done! Write down the topic of the lesson in your notebooks. (The teacher writes the topic on the board.)

IV. "Discovery" of new knowledge(10 min)

Let's look at problem No. 199.

1. The printer prints 27 pages in 4.5 minutes. How long will it take it to print 300 pages?

27 pages – 4.5 min.
300 pages - x?

2. The box contains 48 packs of tea, 250 g each. How many 150g packs of this tea will you get?

48 packs – 250 g.
X? – 150 g.

3. The car drove 310 km, using 25 liters of gasoline. How far can a car travel on a full 40L tank?

310 km – 25 l
X? – 40 l

4. One of the clutch gears has 32 teeth, and the other has 40. How many revolutions will the second gear make while the first one makes 215 revolutions?

32 teeth – 315 rev.
40 teeth – x?

To compile a proportion, one direction of the arrows is necessary; for this, in inverse proportionality, one ratio is replaced by the inverse.

At the board, students find the meaning of quantities; on the spot, students solve one problem of their choice.

– Formulate a rule for solving problems with direct and inverse proportional dependence.

A table appears on the board:

V. Primary consolidation in external speech(10 min)

Worksheet assignments:

1. From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
2. To build the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this site?

VI. Independent work with self-test according to the standard(5 minutes)

Two students complete task No. 225 independently on hidden boards, and the rest - in notebooks. They then check the algorithm's work and compare it with the solution on the board. Errors are corrected and their causes are determined. If the task is completed correctly, then the students put a “+” sign next to them.
Students who make mistakes in independent work can use consultants.

VII. Inclusion in the knowledge system and repetition№ 271, № 270.

Six people work at the board. After 3-4 minutes, students working at the board present their solutions, and the rest check the assignments and participate in their discussion.

VIII. Reflection on activity (lesson summary)

– What new did you learn in the lesson?
-What did they repeat?
– What is the algorithm for solving proportion problems?
– Have we achieved our goal?
– How do you evaluate your work?