How to understand direct and inverse proportionality. Direct and inverse proportional dependencies
Example
1.6 / 2 = 0.8; 4/5 = 0.8; 5.6 / 7 = 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 of
Wikimedia Foundation. 2010.
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
- (from Lat. proportionalis proportional, proportional). Proportionality. Dictionary 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, pl. no, wives. (book). 1.Distract. noun to proportional. Proportionality of parts. The proportionality of the physique. 2. Such a relationship between the 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, wives. 1. see proportional. 2. In mathematics: such a relationship between quantities, when a swarm of one of them increases, the other changes by the same amount. Straight p. (With a swarm with an increase in one value ... ... Ozhegov's Explanatory Dictionary
AND; f. 1. to Proportional (1 digit); proportionality. P. parts. P. physique. P. representation in parliament. 2. Mat. Relationship between proportionally varying quantities. Aspect ratio. Straight p. (In which with ... ... encyclopedic Dictionary
Today we will consider 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. Consequently, 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 by several times in 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 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 = 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 and 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> 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> 0) negative values of the function are in the interval (-∞; 0), and positive ones - (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 proportionality 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 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 way, 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 = 60 * 2 = 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is quite easy to find out the time t 2 that is required from us according to the problem statement: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are really inversely proportional: with a speed 2 times higher than the original, 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 relationships. And they also suggest that when composing the proportion right side records must be turned over: 60/120 = x / 6. From where we get x = 60 * 6/120 = 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 = x / 4. And we get x = 6 * 4/3 = 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 the water enter the pool through this pipe?
To begin with, let us bring all the data to us according to 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 = 2 * 60 = 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 = 75/45, whence x = 120 * 45/75 = 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 = 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 soon 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 of the printing house 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 = x / 8, x = 42 * 8/48 = 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 the knowledge of the back proportional relationship values can really be useful to 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 in in social networks so that your friends and classmates can play too.
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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:
With an increase in one value, the second also increases. With a decrease in one value, the second value also decreases.
Let's look at 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 dependence is called reverse:
With an increase in one value, the second decreases. As one value decreases, the second increases.
Thus, in the case direct dependence both 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).
Determining 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 change 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. With an increase in the number of buns by $ 4 times ($ 8 buns), their total cost will be $ 320 rubles.
The ratio of the number of buns: $ \ frac (8) (2) = 4 $.
Loaf value ratio: $ \ frac (320) (80) = $ 4.
As you can see, these relationships are equal:
$ \ frac (8) (2) = \ frac (320) (80) $.
Definition 1
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) = \ 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, respectively) by the same amount.
Example 3
The car traveled $ 180 $ km in $ 2 $ hour. Find the time during which it will travel $ 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 same amount of time will increase the time:
$ \ frac (2S) (v) = 2t $;
$ \ frac (3S) (v) = 3t $.
The car traveled $ 180 $ km - for $ 2 $ hour
The car will travel $ 180 \ cdot 2 = 360 $ km - in $ x $ hours
The more distance the car travels, the more longer time he will need it. Consequently, the relationship between the quantities is directly proportional.
Let's make the proportion:
$ \ frac (180) (360) = \ frac (2) (x) $;
$ x = \ frac (360 \ cdot 2) (180) $;
Answer: the car will need $ 4 $ hour.
Inverse proportion
Definition 3
Solution.
Time is inversely proportional to speed:
$ t = \ frac (S) (v) $.
How many times the speed increases, with the same path, the same amount of time decreases:
$ \ frac (S) (2v) = \ frac (t) (2) $;
$ \ frac (S) (3v) = \ 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, we turn the second ratio in proportion:
$ \ frac (60) (120) = \ frac (x) (6) $;
$ x = \ frac (60 \ cdot 6) (120) $;
Answer: the car will take $ 3 $ hour.
Proportionality is the relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.
Proportionality is direct and inverse. In this tutorial, we'll cover each of them.
Lesson contentDirect proportionality
Suppose the car is traveling at 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 travel a distance equal to fifty kilometers.
Let's depict in the picture the distance covered by the car in 1 hour
Let the car drive another hour at the same speed equal to fifty kilometers per hour. Then it turns out that the car will travel 100 km
As you can see from the example, doubling the time led to an increase in the distance covered 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 value decreases by a certain number of times, then the other decreases by the same number.
Suppose that it was originally planned to drive 100 km in 2 hours, but after driving 50 km, the driver decided to take a break. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, a decrease in 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 considered example, the distance was initially 50 km, and the time was one hour. The ratio of distance to time is 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 direct proportionality coefficient... It shows how much distance falls on an hour of movement. In this case, the coefficient plays the role of the speed of movement, since the speed is the ratio of the distance traveled to time.
Proportions can be made from directly proportional quantities. For example, relationships are proportional:
Fifty kilometers are related to one hour as one hundred kilometers are related to two hours.
Example 2... The cost and quantity of the purchased goods 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. With an increase in the value of the purchased product, its quantity increases by the same amount.
Since the value of a commodity and its quantity are directly proportional, 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 per kilogram of sweets. V this example the coefficient plays the role of the price of one kilogram of the product, since the price is the ratio of the value of the product to its quantity.
Inverse proportion
Consider the following example. The distance between the two cities is 80 km. The motorcyclist left the first city and reached the second city at a speed of 20 km / h in 4 hours.
If the motorcyclist's speed was 20 km / h, this means that every hour he traveled a distance equal to 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 see that when changing the speed, the travel time changed by the same amount. Moreover, it has changed in reverse side- that is, the speed has increased, while the time, on the contrary, has decreased.
Quantities such as speed and time are called inversely proportional. And the relationship between such quantities is called inverse proportion.
Inverse proportionality is the relationship between two values, in which an increase in one of them entails a decrease in the other by the same amount.
and vice versa, if one value decreases by a certain number of times, then the other increases by the same number.
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 you can see from the example, a decrease in speed led to an increase in the travel time by the same amount.
The peculiarity of inverse proportions is that their product is always constant. That is, when the values of inversely proportional quantities change, their product remains unchanged.
In the considered example, the distance between the cities was 80 km. When changing the speed and time of movement of the motorcyclist, 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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