Dependency Types
Consider battery charging. As the first value, let's take the time it takes to charge. The second value is the time that 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 battery life on the time it is charged
Remark 1
This dependency is called straight:
As one value increases, the other also increases. As one value decreases, the other value also decreases.
Let's consider 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 dependency is called reverse:
As one value increases, the other decreases. As one value decreases, the other value increases.
Thus, in the case direct dependency both quantities change in the same way (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. With an increase in speed (of the first value) by $2$ times, we will find how the time (second value) that will be spent on the path to a friend will change.
Obviously, the time will decrease by $2$ times.
Remark 3
This dependency is called proportional:
How many times one value changes, how many times the second will change.
Example 2
For a $2 loaf of bread in a store, you have to pay 80 rubles. If you need to buy $4$ loaves of bread (the amount of bread increases $2$ times), how much more will you have to pay?
Obviously, the cost will also increase by $2$ times. We have an example of proportional dependence.
In both examples, proportional dependencies were considered. But in the example with loaves of bread, the 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 is reverse. Thus, 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 rolls by $4$ times ($8$ rolls), their total cost will be $320$ rubles.
The ratio of the number of rolls: $\frac(8)(2)=4$.
Roll cost ratio: $\frac(320)(80)=4$.
As you can see, these ratios are equal to each other:
$\frac(8)(2)=\frac(320)(80)$.
Definition 1
The equality of two relations is called proportion.
With a directly proportional relationship, a ratio is obtained when the change in the first and second values \u200b\u200bis the same:
$\frac(A_2)(A_1)=\frac(B_2)(B_1)$.
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 accordingly) by the same amount.
Example 3
The car traveled $180$ km in $2$ hours. Find the time it takes for him to cover $2$ 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 time will increase by the same amount:
$\frac(2S)(v)=2t$;
$\frac(3S)(v)=3t$.
The car traveled $180$ km - in the time of $2$ hour
The car travels $180 \cdot 2=360$ km - in the time of $x$ hours
The more distance the car travels, the more time it will take. Therefore, 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:
$t=\frac(S)(v)$.
How many times the speed increases, 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 condition of the problem in the form of a table:
The car traveled $60$ km - in the time of $6$ hours
A car travels $120$ km - in a time of $x$ hours
The faster the car, the less time it will take. Therefore, the relationship between the quantities is inversely proportional.
Let's make a proportion.
Because 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 need $3$ hours.
Today we will look at what quantities are called inversely proportional, what the inverse proportionality 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 name two quantities that are mutually dependent on each other.
Dependence can be direct and reverse. Therefore, the relationship between quantities describe direct and inverse proportionality.
Direct proportionality- this is such a relationship between 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 will be. 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 exams is directly proportional to the grades received. And the number of things packed in a backpack is directly proportional to its weight.
Inverse proportionality- this is a functional dependence, in which a decrease or increase by several times of an independent value (it is called an argument) causes a proportional (i.e., by the same amount) increase or decrease in a dependent value (it is called a function).
Let's illustrate with a simple example. You want to buy apples in the market. The apples on the counter and the amount of money in your wallet are inversely related. Those. the more apples you buy, the less money you have left.
Function and its graph
The inverse proportionality function can be described as y = k/x. Wherein x≠ 0 and k≠ 0.
This function has the following properties:
- Its domain of definition 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; +∞) .
- It has no maximum or minimum values.
- Is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If a k> 0 (that is, the argument increases), the function decreases proportionally on each of its intervals. If a k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument increases ( k> 0) the negative values of the function are in the interval (-∞; 0), and the positive values are in the interval (0; +∞). When the argument is decreasing ( 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 Proportional Problems
To make it clearer, let's look at a few tasks. They are not too complicated, and their solution will help you visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Task 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 him to cover the same distance if he moves at twice the speed?
We can start by writing down 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 road, and the speed with which it moves, are inversely proportional.
To verify this, let's find V 2, which, by condition, is 2 times higher: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is not difficult to find out the time t 2 that is required from us according to the condition of the problem: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are indeed inversely proportional: with a speed 2 times higher than the original one, the car will spend 2 times less time on the road.
The solution to this problem can also be written as a proportion. Why do we create a diagram like this:
↓ 60 km/h – 6 h
↓120 km/h – x h
Arrows indicate an inverse relationship. And they also suggest that when drawing up the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. Where do we get x \u003d 60 * 6/120 \u003d 3 hours.
Task number 2. The workshop employs 6 workers who cope with a given amount of work in 4 hours. If the number of workers is halved, how long will it take for the remaining workers to complete the same amount of work?
We write the conditions of the problem in the form of a visual diagram:
↓ 6 workers - 4 hours
↓ 3 workers - x h
Let's write this as a proportion: 6/3 = x/4. And we get x \u003d 6 * 4/3 \u003d 8 hours. If there are 2 times fewer workers, the rest will spend 2 times more time to complete all the work.
Task number 3. Two pipes lead to the pool. Through one pipe, water enters at a rate of 2 l / s and fills the pool in 45 minutes. Through another pipe, the pool will be filled in 75 minutes. How fast does water enter the pool through this pipe?
To begin with, we will bring all the quantities given to us according to the condition of the problem to the same units of measurement. To do this, we express the filling rate of 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. On the face of inverse proportion. Let us express the speed unknown to us 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 a proportion: 120 / x \u003d 75/45, from where x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the filling rate of the pool is expressed in liters per second, let's bring our answer to the same form: 72/60 = 1.2 l/s.
Task 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 per hour, how much sooner could he go home?
We go in a proven way and draw up a scheme according to the condition of the problem, denoting the desired value as x:
↓ 42 business cards/h – 8 h
↓ 48 business cards/h – xh
Before us is an inversely proportional relationship: how many times more business cards an employee of a printing house prints per hour, the same amount of time it will take him to complete the same job. Knowing this, we can set up the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 7 hours.
Thus, having completed the work in 7 hours, the printing house employee could go home an hour earlier.
Conclusion
It seems to us that these inverse proportionality problems are really simple. We hope that now you also consider them so. And most importantly, knowledge of the inversely proportional dependence of quantities can really be useful to you more than once.
Not only in math classes and exams. But even then, when you are about 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 proportionality you notice around you. Let this be a game. You'll see how exciting it is. Do not forget to "share" this article on social networks so that your friends and classmates can also play.
site, with full or partial copying of the material, a link to the source is required.
Completed by: Chepkasov Rodion
student of 6 "B" class
MBOU "Secondary School No. 53"
Barnaul
Head: Bulykina O.G.
mathematic teacher
MBOU "Secondary School No. 53"
Barnaul
Introduction. one
Relationships and proportions. 3
Direct and inverse proportions. four
Application of direct and inverse proportionality 6
dependencies in solving various problems.
Conclusion. eleven
Literature. 12
Introduction.
The word proportion comes from the Latin word proportion, meaning in general proportionality, evenness of parts (a certain ratio of parts to each other). In ancient times, the doctrine of proportions was held in high esteem by the Pythagoreans. With proportions, they connected thoughts about order and beauty in nature, about consonant chords in music and harmony in the universe. Some types of proportions they called musical or harmonic.
Even in ancient times, man discovered that all phenomena in nature are connected with each other, that everything is in constant motion, change, and, when expressed in numbers, reveals amazing patterns.
The Pythagoreans and their followers were looking for a numerical expression for everything that exists in the world. They found; that mathematical proportions underlie music (the ratio of string length to pitch, the relationship between intervals, the ratio of sounds in chords that give a harmonic sound). The Pythagoreans tried to mathematically substantiate the idea of the unity of the world, they argued that the basis of the universe is symmetrical geometric shapes. The Pythagoreans were looking for a mathematical justification for beauty.
Following the Pythagoreans, the medieval scholar Augustine called beauty "numerical equality." The scholastic philosopher Bonaventure wrote: "There is no beauty and pleasure without proportionality, while proportionality primarily exists in numbers. It is necessary that everything be calculable." Leonardo da Vinci wrote about the use of proportion in art in his treatise on painting: "The painter embodies in the form of proportion the same laws lurking in nature that the scientist knows in the form of a numerical law."
Proportions were used in solving various problems both in antiquity and in the Middle Ages. Certain types of problems are now easily and quickly solved using proportions. Proportions and proportionality have been and are used not only in mathematics, but also in architecture and art. Proportionality in architecture and art means the observance of certain ratios between the sizes of different parts of a building, figure, sculpture or other work of art. Proportionality in such cases is a condition for the correct and beautiful construction and image
In my work, I tried to consider the use of direct and inverse proportional relationships in various areas of the surrounding life, to trace the connection with academic subjects through tasks.
Relationships and proportions.
The quotient of two numbers is called attitude these numbers.
Attitude Shows, how many times the first number is greater than the second, or what part the first number is from the second.
A task.
2.4 tons of pears and 3.6 tons of apples were brought to the store. What part of the imported fruits are pears?
Solution . Find how much fruit was brought in total: 2.4 + 3.6 = 6 (t). To find what part of the brought fruits are pears, we will make the ratio 2.4:6 =. The answer can also be written as a decimal or as a percentage: = 0.4 = 40%.
mutually inverse called numbers, whose products are equal to 1. Therefore the relationship is called the inverse relationship.
Consider two equal ratios: 4.5:3 and 6:4. Let's put an equal sign between them and get the proportion: 4.5:3=6:4.
Proportion is the equality of two relations: a : b =c :d or = 
, where a and d are extreme terms of proportion, c and b middle terms(all terms of the proportion are non-zero).
Basic property of proportion:
in the right proportion, the product of the extreme terms is equal to the product of the middle terms.
By applying the commutative property of multiplication, we get that in the right proportion, you can swap the extreme terms or the middle terms. The resulting proportions will also be correct.
Using the basic property of a proportion, one can find its unknown member if all other members are known.
To find the unknown extreme term of the proportion, it is necessary to multiply the middle terms and divide by the known extreme term. x : b = c : d , x = 
To find the unknown middle term of the proportion, one must multiply the extreme terms and divide by the known middle term. a : b = x : d , x = 
.
Direct and inverse proportions.
The values of two different quantities can mutually depend on each other. So, the area of a square depends on the length of its side, and vice versa - the length of the side of a square depends on its area.
Two quantities are said to be proportional if, with increasing
(reduction) of one of them by several times, the other increases (decreases) by the same amount.
If two quantities are directly proportional, then the ratios of the corresponding values of these quantities are equal.
Example direct proportional relationship .
At the gas station 2 liters of gasoline weigh 1.6 kg. How much will they weigh 5 liters of gasoline?
Solution:
The weight of kerosene is proportional to its volume.
2l - 1.6 kg
5l - x kg
2:5=1.6:x,
x \u003d 5 * 1.6 x \u003d 4
Answer: 4 kg.
Here the ratio of weight to volume remains unchanged.
Two quantities are called inversely proportional if, when one of them increases (decreases) several times, the other decreases (increases) by the same amount.
If quantities are inversely proportional, then the ratio of the values of one quantity is equal to the inverse ratio of the corresponding values of the other quantity.
P exampleinverse proportional relationship.
The two rectangles have the same area. The length of the first rectangle is 3.6 m and the width is 2.4 m. The length of the second rectangle is 4.8 m. Find the width of the second rectangle.
Solution:
1 rectangle 3.6 m 2.4 m
2 rectangle 4.8 m x m
3.6 m x m
4.8 m 2.4 m
x \u003d 3.6 * 2.4 \u003d 1.8 m
Answer: 1.8 m.
As you can see, problems with proportional quantities can be solved using proportions.
Not every two quantities are directly proportional or inversely proportional. For example, the height of a child increases with increasing age, but these values are not proportional, since when the age is doubled, the height of the child does not double.
Practical application of direct and inverse proportionality.
Task #1
The school library has 210 mathematics textbooks, which is 15% of the entire library stock. How many books are in the library stock?
Solution:
Total textbooks - ? - 100%
Mathematicians - 210 -15%
15% 210 accounts
X \u003d 100 * 210 \u003d 1400 textbooks
100% x account. fifteen
Answer: 1400 textbooks.
Task #2
A cyclist travels 75 km in 3 hours. How long will it take the cyclist to travel 125 km at the same speed?
Solution:
3 h – 75 km
H - 125 km
Time and distance are directly proportional, so
3: x = 75: 125,
x= 
,
x=5.
Answer: 5 hours.
Task #3
8 identical pipes fill the pool in 25 minutes. How many minutes will it take 10 such pipes to fill the pool?
Solution:
8 pipes - 25 minutes
10 pipes - ? minutes
The number of pipes is inversely proportional to time, so
8:10 = x:25,
x = 
x = 20
Answer: 20 minutes.
Task #4
A team of 8 workers completes the task in 15 days. How many workers can complete the task in 10 days, working at the same productivity?
Solution:
8 working - 15 days
Working - 10 days
The number of workers is inversely proportional to the number of days, so
x: 8 = 15: 10,
x= 
,
x=12.
Answer: 12 workers.
Task number 5
From 5.6 kg of tomatoes, 2 liters of sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes?
Solution:
5.6 kg - 2 l
54 kg - ? l
The number of kilograms of tomatoes is directly proportional to the amount of sauce obtained, therefore
5.6: 54 = 2: x,
x = 
,
x = 19 .
Answer: 19 l.
Task number 6
For heating the school building, coal was harvested for 180 days at a consumption rate
0.6 tons of coal per day. How many days will this reserve last if it is consumed daily by 0.5 tons?
Solution:
Number of days
Consumption rate
The number of days is inversely proportional to the coal consumption rate, so
180: x = 0.5: 0.6,
x \u003d 180 * 0.6: 0.5,
x = 216.
Answer: 216 days.
Task number 7
In iron ore, 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in an ore that contains 73.5 tons of iron?
Solution:
Number of pieces
Weight
Iron
73,5
impurities
The number of parts is directly proportional to the mass, so
7: 73.5 = 3: x.
x \u003d 73.5 * 3: 7,
x = 31.5.
Answer: 31.5 tons
Task number 8
The car drove 500 km, having spent 35 liters of gasoline. How many liters of gasoline do you need to travel 420 km?
Solution:
Distance, km
Gasoline, l
The distance is directly proportional to the consumption of gasoline, so
500: 35 = 420: x,
x \u003d 35 * 420: 500,
x = 29.4.
Answer: 29.4 liters
Task number 9
In 2 hours we caught 12 crucians. How many carp will be caught in 3 hours?
Solution:
The number of crucians does not depend on time. These quantities are neither directly proportional nor inversely proportional.
Answer: There is no answer.
Task number 10
A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles per one. How many of these cars can the company buy if the price for one car becomes 15,000 rubles?
Solution:
Number of cars, pcs.
Price, thousand rubles
The number of cars is inversely proportional to the cost, so
5:x=15:12,
x= 5*12:15,
x=4.
Answer: 4 cars.
Task number 11
In the town N, there is a shop on square P, the owner of which is so strict that he deducts 70 rubles from wages for being late for 1 lateness per day. Two girls Yulia and Natasha work in one department. Their wages depend on the number of working days. Julia received 4,100 rubles in 20 days, and Natasha should have received more in 21 days, but she was late for 3 days in a row. How many rubles will Natasha get?
Solution:
Working day
Salary, rub.
Julia
4100
Natasha
Salary is directly proportional to the number of working days, therefore
20: 21 = 4100: x,
x= 4305.
4305 rub. Natasha should have.
4305 - 3 * 70 = 4095 (rub.)
Answer: Natasha will receive 4095 rubles.
Task number 12
The distance between two cities on the map is 6 cm. Find the distance between these cities on the ground if the map scale is 1: 250000.
Solution:
Let's denote the distance between cities on the ground through x (in centimeters) and find the ratio of the length of the segment on the map to the distance on the ground, which will be equal to the scale of the map: 6: x \u003d 1: 250000,
x \u003d 6 * 250000,
x = 1500000.
1500000 cm = 15 km
Answer: 15 km.
Task number 13
4000 g of solution contains 80 g of salt. What is the concentration of salt in this solution?
Solution:
Weight, g
Concentration, %
Solution
4000
Salt
4000: 80 = 100: x,
x = 
,
x = 2.
Answer: The concentration of salt is 2%.
Task number 14
The bank gives a loan at 10% per annum. You received a loan of 50,000 rubles. How much do you have to pay back to the bank in a year?
Solution:
50 000 rub.
100%
x rub.
50000: x = 100: 10,
x= 50000*10:100,
x=5000.
5000 rub. is 10%.
50,000 + 5000=55,000 (rubles)
Answer: in a year, 55,000 rubles will be returned to the bank.
Conclusion.
As we can see from the above examples, direct and inverse proportional relationships are applicable in various areas of life:
Economy,
trade,
in manufacturing and industry,
school life,
cooking,
Construction and architecture.
sports,
animal husbandry,
topography,
physicists,
Chemistry, etc.
In Russian, there are also proverbs and sayings that establish direct and inverse relationships:
As it comes around, so it will respond.
The higher the stump, the higher the shadow.
The more people, the less oxygen.
And ready, yes stupidly.
Mathematics is one of the oldest sciences; it arose on the basis of the needs and needs of mankind. Having gone through the history of formation since ancient Greece, it still remains relevant and necessary in the daily life of any person. The concept of direct and inverse proportionality has been known since ancient times, since it was the laws of proportion that moved architects during any construction or creation of any sculpture.
Knowledge of proportions is widely used in all spheres of human life and activity - one cannot do without them when painting pictures (landscapes, still lifes, portraits, etc.), they are also widespread among architects and engineers - in general, it is hard to imagine the creation of anything anything without the use of knowledge about proportions and their relationship.
Literature.
Mathematics-6, N.Ya. Vilenkin and others.
Algebra -7, G.V. Dorofeev and others.
Mathematics-9, GIA-9, edited by F.F. Lysenko, S.Yu. Kulabukhov
Mathematics-6, didactic materials, P.V. Chulkov, A.B. Uedinov
Tasks in mathematics for grades 4-5, I.V. Baranova et al., M. "Enlightenment" 1988
Collection of tasks and examples in mathematics grade 5-6, N.A. Tereshin,
T.N. Tereshina, M. "Aquarium" 1997
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.Proportionality factor
The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.
Direct proportionality
Direct proportionality- functional dependence, in which some 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 proportionality
Inverse proportion- 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 .
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8 etc.Proportionality factor
The constant ratio of proportional quantities is called coefficient of proportionality. The proportionality coefficient shows how many units of one quantity fall on a unit of another.
Direct proportionality
Direct proportionality- functional dependence, in which some 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 proportionality
Inverse proportion- 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 .
