Today we will look at what quantities are called inverse proportional, what the inverse proportional 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 call two quantities that are mutually dependent on each other.
The dependence can be direct and inverse. Therefore, the relationship between quantities describes direct and inverse proportionality.
Direct proportionality - this is such a dependence of 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. 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 the exams is directly proportional to the grades received. And the number of things packed in a backpack is directly proportional to its weight.
Inverse proportion - this is a functional dependence, in which a decrease or increase in several times of an independent quantity (called an argument) causes a proportional (i.e., the same amount of time) increase or decrease in a dependent quantity (called a function).
Let's illustrate with a simple example. You want to buy apples at the market. The apples on the counter and the amount of money in your wallet are in inverse proportion. Those. the more apples you buy, the less money you will have left.
Function and its graph
The inverse proportionality function can be described as y \u003d k / x... In which x≠ 0 and k≠ 0.
This function has the following properties:
- Its domain 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; +∞) .
- Has no highest or lowest values.
- It is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If k\u003e 0 (i.e., the argument increases), the function decreases proportionally at each of its intervals. If k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument ( k\u003e 0) negative values \u200b\u200bof the function are in the interval (-∞; 0), and positive values \u200b\u200b- (0; + ∞). As the argument ( 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 proportionality problems
To make it clearer, let's break down a few tasks. They are not too complicated, and their solution will help you to visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Problem 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 for him to cover the same distance if he moves at a speed 2 times higher?
We can start by writing a formula that describes the relationship between time, distance and speed: t \u003d 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 in inverse proportion.
To verify this, let's find V 2, which is 2 times higher by condition: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S \u003d V * t \u003d 60 * 6 \u003d 360 km. Now it is quite easy to find out the time t 2, which is required from us by the problem statement: t 2 \u003d 360/120 \u003d 3 hours.
As you can see, travel time and speed are really inversely proportional: with a speed 2 times higher than the initial one, the car will spend 2 times less time on the road.
The solution to this problem can also be written in the form of proportions. Why, first, let's draw up the following scheme:
↓ 60 km / h - 6 h
↓ 120 km / h - x h
Arrows indicate inversely proportional relationship. And they also suggest that when composing the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. From where we get x \u003d 60 * 6/120 \u003d 3 hours.
Problem number 2. The workshop employs 6 workers who can cope with a given amount of work in 4 hours. If the number of workers is cut in half, how long will it take for those who remain to do the same amount of work?
Let's write down the conditions of the problem in the form of a visual diagram:
↓ 6 workers - 4 hours
↓ 3 workers - x h
Let's write it down as a proportion: 6/3 \u003d x / 4. And we get x \u003d 6 * 4/3 \u003d 8 hours. If the number of workers becomes 2 times less, the rest will spend 2 times more time on doing all the work.
Problem number 3. There are two pipes leading to the pool. Through one pipe, water flows at a rate of 2 l / s and fills the pool in 45 minutes. Another pipe will fill the pool in 75 minutes. At what speed does water enter the pool through this pipe?
To begin with, let us bring all the data to us on the condition of the problem of the value to the same units of measurement. To do this, we express the rate of filling 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. Inverse proportionality is evident. We express the unknown speed 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 the proportion: 120 / x \u003d 75/45, whence x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the rate of filling the pool is expressed in liters per second, we will bring the answer we received to the same form: 72/60 \u003d 1.2 l / s.
Problem 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 in an hour, how early could he go home?
We follow the proven path and draw up a diagram according to the condition of the problem, denoting the desired value as x:
↓ 42 cards / h - 8 h
↓ 48 cards / h - x h
We have before us an inversely proportional relationship: how many times more business cards an employee prints per hour, the same amount of time he will need to complete the same job. Knowing this, let's make the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 7h.
Thus, having completed the work in 7 hours, the employee of the printing house would be able to go home an hour earlier.
Conclusion
It seems to us that these inverse proportionality problems are really simple. We hope you now see them that way too. And the main thing is that knowledge of the inversely proportional relationship of quantities can really be useful for you more than once.
Not only in math lessons and exams. But even then, when you are planning 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 proportional dependence you notice around you. Let it be such a game. You will see how exciting it is. Do not forget to “share” this article on social networks so that your friends and classmates can play too.
blog. site, with full or partial copying of the material, a link to the source is required.
The two quantities are called directly proportional, if when one of them is increased 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 the square and its side are directly proportional values;
3) the cost of a product purchased at one price is directly proportional to its quantity.
To distinguish direct proportional dependence from the inverse, you can use the proverb: "The further into the forest, the more firewood."
It is convenient to solve problems with directly proportional quantities using proportion.
1) To make 10 parts, you need 3.5 kg of metal. How much metal will be used to make 12 of these parts?
(We reason like this:
1. In the filled column, put the arrow in the direction from the largest number to the smallest.
2. The more parts, the more metal is 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 the proportion (in the direction from the beginning of the arrow to its end):
12: 10 \u003d x: 3.5
To find, it is necessary 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) 1,680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?
(1. In the filled column, put the arrow in the direction from the largest number to the smallest.
2. The less fabrics are bought, the less you have to pay for them. This means that this is a directly proportional relationship.
3. Therefore, the second arrow is in the same direction with the first).
Let x rubles cost 12 meters of fabric. We make the proportion (from the beginning of the arrow to its end):
15: 12 \u003d 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:
This means that 12 meters cost 1,344 rubles.
Answer: 1344 rubles.
Today we will look at what quantities are called inverse proportional, what the inverse proportional 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 call two quantities that are mutually dependent on each other.
The dependence can be direct and inverse. Therefore, the relationship between quantities describes direct and inverse proportionality.
Direct proportionality - this is such a dependence of 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. 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 the exams is directly proportional to the grades received. And the number of things packed in a backpack is directly proportional to its weight.
Inverse proportion - this is a functional dependence, in which a decrease or increase in several times of an independent quantity (called an argument) causes a proportional (i.e., the same amount of time) increase or decrease in a dependent quantity (called a function).
Let's illustrate with a simple example. You want to buy apples at the market. The apples on the counter and the amount of money in your wallet are in inverse proportion. Those. the more apples you buy, the less money you will have left.
Function and its graph
The inverse proportionality function can be described as y \u003d k / x... In which x≠ 0 and k≠ 0.
This function has the following properties:
- Its domain 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; +∞) .
- Has no highest or lowest values.
- It is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If k\u003e 0 (i.e., the argument increases), the function decreases proportionally at each of its intervals. If k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument ( k\u003e 0) negative values \u200b\u200bof the function are in the interval (-∞; 0), and positive values \u200b\u200b- (0; + ∞). As the argument ( 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 proportionality problems
To make it clearer, let's break down a few tasks. They are not too complicated, and their solution will help you to visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Problem 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 for him to cover the same distance if he moves at a speed 2 times higher?
We can start by writing a formula that describes the relationship between time, distance and speed: t \u003d 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 in inverse proportion.
To verify this, let's find V 2, which is 2 times higher by condition: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S \u003d V * t \u003d 60 * 6 \u003d 360 km. Now it is quite easy to find out the time t 2, which is required from us by the problem statement: t 2 \u003d 360/120 \u003d 3 hours.
As you can see, travel time and speed are really inversely proportional: with a speed 2 times higher than the initial one, the car will spend 2 times less time on the road.
The solution to this problem can also be written in the form of proportions. Why, first, let's draw up the following scheme:
↓ 60 km / h - 6 h
↓ 120 km / h - x h
Arrows indicate inversely proportional relationship. And they also suggest that when composing the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. From where we get x \u003d 60 * 6/120 \u003d 3 hours.
Problem number 2. The workshop employs 6 workers who can cope with a given amount of work in 4 hours. If the number of workers is cut in half, how long will it take for those who remain to do the same amount of work?
Let's write down the conditions of the problem in the form of a visual diagram:
↓ 6 workers - 4 hours
↓ 3 workers - x h
Let's write it down as a proportion: 6/3 \u003d x / 4. And we get x \u003d 6 * 4/3 \u003d 8 hours. If the number of workers becomes 2 times less, the rest will spend 2 times more time on doing all the work.
Problem number 3. There are two pipes leading to the pool. Through one pipe, water flows at a rate of 2 l / s and fills the pool in 45 minutes. Another pipe will fill the pool in 75 minutes. At what speed does water enter the pool through this pipe?
To begin with, let us bring all the data to us on the condition of the problem of the value to the same units of measurement. To do this, we express the rate of filling 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. Inverse proportionality is evident. We express the unknown speed 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 the proportion: 120 / x \u003d 75/45, whence x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the rate of filling the pool is expressed in liters per second, we will bring the answer we received to the same form: 72/60 \u003d 1.2 l / s.
Problem 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 in an hour, how early could he go home?
We follow the proven path and draw up a diagram according to the condition of the problem, denoting the desired value as x:
↓ 42 cards / h - 8 h
↓ 48 cards / h - x h
We have before us an inversely proportional relationship: how many times more business cards an employee prints per hour, the same amount of time he will need to complete the same job. Knowing this, let's make the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 7h.
Thus, having completed the work in 7 hours, the employee of the printing house would be able to go home an hour earlier.
Conclusion
It seems to us that these inverse proportionality problems are really simple. We hope you now see them that way too. And the main thing is that knowledge of the inversely proportional relationship of quantities can really be useful for you more than once.
Not only in math lessons and exams. But even then, when you are planning 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 proportional dependence you notice around you. Let it be such a game. You will see how exciting it is. Do not forget to “share” this article on social networks so that your friends and classmates can play too.
site, with full or partial copying of the material, a link to the source is required.
Example
1.6 / 2 \u003d 0.8; 4/5 \u003d 0.8; 5.6 / 7 \u003d 0.8, etc.Aspect ratio
The constant ratio of proportional quantities is called aspect ratio... The proportionality coefficient shows how many units of one quantity fall on the 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 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 proportion
Inverse proportionality is a functional dependence in which an increase in the independent quantity (argument) causes a proportional decrease in the dependent quantity (function)
Mathematically, inverse proportionality is written as a formula:
Function properties:
Sources
Wikimedia Foundation. 2010.
- Newton's second law
- Coulomb barrier
See what "Direct proportionality" is in other dictionaries:
direct proportion - - [A.S. Goldberg. The English Russian Energy Dictionary. 2006] Topics energy in general EN direct ratio ... Technical translator's guide
direct proportion - tiesioginis proporcingumas statusas T sritis fizika atitikmenys: angl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f… Fizikos terminų žodynas
PROPORTIONALITY - (from Lat. proportionalis proportional, proportional). Proportionality. Dictionary of foreign words included in the Russian language. Chudinov AN, 1910. PROPORTIONALITY otlat. proportionalis, proportional. Proportionality. Explanation 25000 ... ... Dictionary of foreign words of the Russian language
PROPORTIONALITY - PROPORTIONALITY, proportionality, pl. no, wives. (book). 1.Distract. noun to proportional. Proportionality of parts. The proportionality of the physique. 2. Such a relationship between quantities when they are proportional (see proportional ... Ushakov's Explanatory Dictionary
Proportionality - Two mutually dependent quantities are called proportional if the ratio of their values \u200b\u200bremains unchanged .. Contents 1 Example 2 Proportionality coefficient ... Wikipedia
PROPORTIONALITY - PROPORTIONALITY, and, wives. 1. see proportional. 2. In mathematics: such a relationship between quantities, when a swarm of one of them increases, the other is changed by the same amount. Straight p. (With a swarm with an increase in one value ... ... Ozhegov's Explanatory Dictionary
proportionality - and; g. 1. to Proportional (1 digit); proportionality. P. parts. P. physique. P. representation in parliament. 2. Mat. The relationship between proportionally varying quantities. Aspect ratio. Straight p. (In which with ... ... encyclopedic Dictionary
Dependency types
Consider battery charging. As the first value, we take the time it takes to charge. The second value is the time 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 the battery life on the time it is charged
Remark 1
This dependence is called straight:
As one value increases, the second increases. As one value decreases, the second value decreases.
Let's look at another example.
The more books the student reads, the fewer mistakes he will make in the dictation. Or the higher you climb the mountains, the lower the atmospheric pressure will be.
Remark 2
This dependence is called reverse:
As one value increases, the second decreases. As one value decreases, the second increases.
Thus, in the case direct dependenceboth quantities change in the same way (both either increase or decrease), and in the case inverse relationship - the opposite (one increases and the other decreases, or vice versa).
Determination of dependencies between quantities
Example 1
The time taken to visit a friend is $ 20 $ minutes. With an increase in the speed (the first value) by $ 2 $ times, we will find how the time (the second value) will change, which will be spent on the path to the friend.
Obviously, the time will decrease by $ 2 $ times.
Remark 3
This dependence is called proportional:
How many times one value changes, the second will change as many times.
Example 2
For $ 2 a loaf of bread in the store you need to pay 80 rubles. If you need to buy $ 4 $ loaves of bread (the amount of bread increases by $ 2 $ times), how many times will you have to pay more?
Obviously, the cost will also increase by $ 2 $ times. We have an example of proportional dependence.
In both examples, proportional relationships 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 - reverse... So 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. If the number of buns is increased by $ 4 times ($ 8 buns), their total cost will be $ 320 rubles.
The ratio of the number of buns: $ \\ frac (8) (2) \u003d 4 $.
Loaf value ratio: $ \\ frac (320) (80) \u003d $ 4.
As you can see, these relationships are equal:
$ \\ frac (8) (2) \u003d \\ frac (320) (80) $.
Definition 1
The equality of two relations is called proportion.
With a directly proportional relationship, the ratio is obtained when the change in the first and second quantities coincides:
$ \\ frac (A_2) (A_1) \u003d \\ frac (B_2) (B_1) $.
Definition 2
The two quantities are called directly proportionalif, when changing (increasing or decreasing) one of them, the other value changes (increases or decreases, respectively) by the same amount.
Example 3
The car traveled $ 180 $ km in $ 2 $ hour. Find the time it takes it to travel $ 2 $ times the distance with the same speed.
Decision.
Time is directly proportional to distance:
$ t \u003d \\ frac (S) (v) $.
How many times the distance increases, at a constant speed, the same amount of time will increase:
$ \\ frac (2S) (v) \u003d 2t $;
$ \\ frac (3S) (v) \u003d 3t $.
The car drove $ 180 $ km - for $ 2 $ hour
The car will travel $ 180 \\ cdot 2 \u003d 360 $ km - in $ x $ hours
The more distance the car travels, the longer it will take. Consequently, the relationship between the quantities is directly proportional.
Let's make the proportion:
$ \\ frac (180) (360) \u003d \\ frac (2) (x) $;
$ x \u003d \\ frac (360 \\ cdot 2) (180) $;
Answer: the car will need $ 4 $ hour.
Inverse proportion
Definition 3
Decision.
Time is inversely proportional to speed:
$ t \u003d \\ frac (S) (v) $.
How many times the speed increases, with the same path, the same amount of time decreases:
$ \\ frac (S) (2v) \u003d \\ frac (t) (2) $;
$ \\ frac (S) (3v) \u003d \\ frac (t) (3) $.
Let's write down the condition of the problem in the form of a table:
The car traveled $ 60 $ km - for $ 6 $ hours
The car will travel $ 120 $ km - in $ x $ hours
The higher the speed of the car, 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 ratio in proportion is inverted:
$ \\ frac (60) (120) \u003d \\ frac (x) (6) $;
$ x \u003d \\ frac (60 \\ cdot 6) (120) $;
Answer: the car will take $ 3 $ hour.