Today we will look at what quantities are called inversely proportional, what an inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside of school.
Such different proportions
Proportionality name two quantities that are mutually dependent on each other.
The dependence can be direct and inverse. Consequently, the relationships between quantities are described by 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 studying for exams, the higher your grades. Or the more things you take with you on a hike, the heavier your backpack will be to carry. Those. The amount of effort spent preparing for exams is directly proportional to the grades obtained. And the number of things packed in a backpack is directly proportional to its weight.
Inverse proportionality- This functional dependence, in which a decrease or increase by several times in an independent quantity (it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in a dependent quantity (it is 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 = k/x. In which 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; +∞) .
- Does not have maximum or minimum values.
- It is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not intersect the coordinate axes.
- Has no zeros.
- If k> 0 (i.e. the argument increases), the function decreases proportionally on each of its intervals. If k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument increases ( k> 0) negative values of the function are in the interval (-∞; 0), and positive values are in the interval (0; +∞). When the argument decreases ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).
The graph of an inverse proportionality function is called a hyperbola. Shown as follows:
Inverse proportionality problems
To make it clearer, let's look at several tasks. They are not too complicated, and solving them will help you visualize what inverse proportionality is and how this knowledge can be useful in your everyday life.
Task No. 1. A car is moving at a speed of 60 km/h. It took him 6 hours to get to 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 between time, distance and speed: t = S/V. Agree, it reminds us very much of the inverse proportionality function. And it indicates that the time a car spends on the road and the speed at which it moves are in inverse proportion.
To verify this, let's find V 2, which, according to the condition, is 2 times higher: V 2 = 60 * 2 = 120 km/h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it’s not difficult to find out the time t 2 that is required from us according to the conditions of the problem: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are indeed inversely proportional: at a speed 2 times higher than the original speed, the car will spend 2 times less time on the road.
The solution to this problem can also be written as a proportion. So let's first create this diagram:
↓ 60 km/h – 6 h
↓120 km/h – x h
Arrows indicate an inversely proportional relationship. They also suggest that when drawing up proportions right side the records must be turned over: 60/120 = x/6. Where do we get x = 60 * 6/120 = 3 hours.
Task No. 2. The workshop employs 6 workers who can complete a given amount of work in 4 hours. If the number of workers is halved, how long will it take the remaining workers to complete the same amount of work?
Let us write the conditions of the problem in the form 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 = 6 * 4/3 = 8 hours. If there are 2 times fewer workers, the remaining ones will spend 2 times more time doing all the work.
Task No. 3. There are two pipes leading into the pool. Through one pipe, water flows at a speed of 2 l/s and fills the pool in 45 minutes. Through another pipe, the pool will fill in 75 minutes. At what speed does water enter the pool through this pipe?
To begin with, let us reduce all the quantities given to us according to the conditions of the problem to the same units of measurement. To do this, we express the speed 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 fills more slowly through the second pipe, this means that the rate of water flow is lower. The proportionality is inverse. Let us express the unknown speed through x and draw up the following diagram:
↓ 120 l/min – 45 min
↓ x l/min – 75 min
And then we make up the proportion: 120/x = 75/45, from where x = 120 * 45/75 = 72 l/min.
In the problem, the filling rate of the pool is expressed in liters per second; let’s reduce the answer we received to the same form: 72/60 = 1.2 l/s.
Task No. 4. A small private printing house prints business cards. A printing house employee works at a speed of 42 business cards per hour and works a full day - 8 hours. If he worked faster and printed 48 business cards in an hour, how much earlier could he go home?
We follow the proven path and draw up a diagram according to the conditions of the problem, designating the desired value as x:
↓ 42 business cards/hour – 8 hours
↓ 48 business cards/h – x h
We have an inversely proportional relationship: the number of times more business cards an employee of a printing house prints per hour, the same number of times less time he will need to complete the same work. Knowing this, let's create a proportion:
42/48 = x/8, x = 42 * 8/48 = 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 think of them that way. And the main thing is that knowledge about the inversely proportional dependence of quantities can really be useful to you more than once.
Not only in math lessons and exams. But even then, when you get ready to go on a trip, go shopping, decide to earn a little extra money during the holidays, etc.
Tell us in the comments what examples of inverse and direct proportional relationships you notice around you. Let it be such a game. You'll see how exciting it is. Don't forget to share this article on in social networks so that your friends and classmates can also play.
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Completed by: Chepkasov Rodion
6th grade student
MBOU "Secondary School No. 53"
Barnaul
Head: Bulykina O.G.
mathematic teacher
MBOU "Secondary School No. 53"
Barnaul
Introduction. 1
Relationships and proportions. 3
Direct and reverse proportional dependencies. 4
Application of direct and inverse proportional 6
dependencies when solving various problems.
Conclusion. eleven
Literature. 12
Introduction.
The word proportion comes from the Latin word proportion, which generally means proportionality, alignment 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 associated thoughts about order and beauty in nature, about consonant chords in music and harmony in the universe. They called some types of proportions musical or harmonic.
Even in ancient times, man discovered that all phenomena in nature are connected with each other, that everything is in continuous movement, change, and, when expressed in numbers, reveals amazing patterns.
The Pythagoreans and their followers sought a numerical expression for everything in the world. They discovered; What mathematical proportions lie at the basis of music (the ratio of the length of the string to the 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 was symmetrical geometric shapes. The Pythagoreans sought a mathematical basis for beauty.
Following the Pythagoreans, the medieval scientist Augustine called beauty “numerical equality.” The scholastic philosopher Bonaventure wrote: “There is no beauty and pleasure without proportionality, and proportionality exists primarily in numbers. It is necessary that everything be countable.” 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 patterns hidden in nature that the scientist knows in the form of the numerical law.”
Proportions were used to solve various problems in both antiquity and the Middle Ages. Certain types of problems are now easily and quickly solved using proportions. Proportions and proportionality were and are used not only in mathematics, but also in architecture and art. Proportion in architecture and art means maintaining certain relationships between sizes different parts building, figure, sculpture or other work of art. Proportionality in such cases is a condition for correct and beautiful construction and depiction
In my work, I tried to consider the use of direct and inverse proportional relationships in various areas of 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 of the second.
Task.
2.4 tons of pears and 3.6 tons of apples were brought to the store. What proportion of the fruits brought are pears?
Solution . Let's find how much fruit they brought: 2.4+3.6=6(t). To find what part of the brought fruits are pears, we make the ratio 2.4:6=. The answer can also be written as a decimal fraction or as a percentage: = 0.4 = 40%.
Mutually inverse called numbers, whose products are equal to 1. Therefore the relationship is called the inverse of the 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 – average members(all terms of the proportion are different from zero).
Basic property of proportion:
in the correct proportion, the product of the extreme terms is equal to the product of the middle terms.
Applying the commutative property of multiplication, we find that in the correct proportion the extreme terms or middle terms can be interchanged. The resulting proportions will also be correct.
Using the basic property of proportion, you can find its unknown term if all other terms are known.
To find the unknown extreme term of the proportion, you need to multiply the average terms and divide by the known extreme term. x : b = c : d , x = 
To find the unknown middle term of a proportion, you need to multiply the extreme terms and divide by the known middle term. a : b =x : d , x = 
.
Direct and inverse proportional relationships.
The values of two different quantities can be mutually dependent 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
(decrease) one of them several times, the other increases (decreases) the same number of times.
If two quantities are directly proportional, then the ratios of the corresponding values of these quantities are equal.
Example direct proportional dependence .
At a 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=5*1.6 x=4
Answer: 4 kg.
Here the weight to volume ratio 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 another quantity.
P exampleinversely proportional relationship.
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 = 3.6*2.4 = 1.8 m
Answer: 1.8 m.
As you can see, problems involving proportional quantities can be solved using proportions.
Not every two quantities are directly proportional or inversely proportional. For example, a child’s height increases as his age increases, but these values are not proportional, since when the age doubles, the child’s height does not double.
Practical use direct and inverse proportional dependence.
Task No. 1
IN school library 210 mathematics textbooks, which is 15% of the entire library collection. How many books are there in the library collection?
Solution:
Total textbooks - ? - 100%
Mathematicians - 210 -15%
15% 210 academic.
X = 100* 210 = 1400 textbooks
100% x account. 15
Answer: 1400 textbooks.
Problem No. 2
A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed?
Solution:
3 h – 75 km
H – 125 km
Time and distance are directly proportional quantities, therefore
3: x = 75: 125,
x= 
,
x=5.
Answer: in 5 hours.
Problem No. 3
8 identical pipes fills the pool in 25 minutes. How many minutes will it take to fill a pool with 10 such pipes?
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: in 20 minutes.
Problem No. 4
A team of 8 workers completes the task in 15 days. How many workers can complete the task in 10 days while working at the same productivity?
Solution:
8 working days – 15 days
Workers - 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.
Problem No. 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.
Problem No. 6
To heat the school building, coal was stored for 180 days at the consumption rate
0.6 tons of coal per day. How many days will this supply last if 0.5 tons are spent daily?
Solution:
Number of days
Consumption rate
The number of days is inversely proportional to the rate of coal consumption, therefore
180: x = 0.5: 0.6,
x = 180*0.6:0.5,
x = 216.
Answer: 216 days.
Problem No. 7
IN iron ore For 7 parts of iron there are 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
Solution:
Number of parts
Weight
Iron
73,5
Impurities
The number of parts is directly proportional to the mass, therefore
7: 73.5 = 3: x.
x = 73.5 * 3:7,
x = 31.5.
Answer: 31.5 t
Problem No. 8
The car traveled 500 km, using 35 liters of gasoline. How many liters of gasoline will be needed to travel 420 km?
Solution:
Distance, km
Gasoline, l
The distance is directly proportional to gasoline consumption, so
500:35 = 420:x,
x = 35*420:500,
x = 29.4.
Answer: 29.4 l
Problem No. 9
In 2 hours we caught 12 crucian carp. How many crucian carp will be caught in 3 hours?
Solution:
The number of crucian carp does not depend on time. These quantities are neither directly proportional nor inversely proportional.
Answer: There is no answer.
Problem No. 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 machines can an enterprise buy if the price for one machine becomes 15 thousand 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.
Problem No. 11
In the city N on square P there is a store whose owner is so strict that for lateness he deducts 70 rubles from the salary for 1 lateness per day. Two girls, Yulia and Natasha, work in one department. Their wage depends on the number of working days. Yulia 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 receive?
Solution:
Work days
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 received it.
4305 – 3 * 70 = 4095 (rub.)
Answer: Natasha will receive 4095 rubles.
Problem No. 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 us denote the distance between cities on the ground by 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 map scale: 6: x = 1: 250000,
x = 6*250000,
x = 1500000.
1500000 cm = 15 km
Answer: 15 km.
Problem No. 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 salt concentration is 2%.
Problem No. 14
The bank gives a loan at 10% per annum. You received a loan of 50,000 rubles. How much should you return 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 (rub.)
Answer: in a year the bank will get 55,000 rubles back.
Conclusion.
As we can see from the examples given, direct and inverse proportional relationships are applicable in various areas of life:
Economics,
Trade,
In production and industry,
Cooking,
Construction and architecture.
Sports,
Animal husbandry,
Topographies,
Physicists,
Chemistry, etc.
In the Russian language there are also proverbs and sayings that establish direct and inverse relationships:
As it comes back, so will it respond.
The higher the stump, the higher the shadow.
The more people, the less oxygen.
And it’s ready, but stupid.
Mathematics is one of the oldest sciences; it arose on the basis of the needs and wants of mankind. Having gone through the history of formation since Ancient Greece, it still remains relevant and necessary in Everyday life any person. The concept of direct and inverse proportionality has been known since ancient times, since it was the laws of proportion that motivated architects during any construction or creation of any sculpture.
Knowledge about proportions is widely used in all spheres of human life and activity - one cannot do without it when painting (landscapes, still lifes, portraits, etc.), it is also widespread among architects and engineers - in general, it is difficult to imagine creating anything something without using knowledge about proportions and their relationships.
Literature.
Mathematics-6, N.Ya. Vilenkin et al.
Algebra -7, G.V. Dorofeev and others.
Mathematics-9, GIA-9, edited by F.F. Lysenko, S.Yu. Kulabukhova
Mathematics-6, didactic materials, P.V. Chulkov, A.B. Uedinov
Problems in mathematics for grades 4-5, I.V. Baranova et al., M. "Prosveshchenie" 1988
Collection of problems and examples in mathematics grades 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
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.
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 decimals;
- develop logical thinking students.
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 – c; 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 straight lines 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:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
- 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 against standard(5 minutes)
Two students complete tasks No. 225 independently on hidden boards, and the rest are 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?
§ 129. Preliminary clarifications.
A person constantly deals with a wide variety of quantities. An employee and a worker are trying to get to work by a certain time, a pedestrian is in a hurry to get to a certain place by the shortest route, a stoker steam heating worries that the temperature in the boiler is slowly rising, the business manager makes plans to reduce the cost of production, etc.
One could give any number of such examples. Time, distance, temperature, cost - all these are various quantities. In the first and second parts of this book, we became acquainted with some particularly common quantities: area, volume, weight. We encounter many quantities when studying physics and other sciences.
Imagine that you are traveling on a train. Every now and then you look at your watch and notice how long you've been on the road. You say, for example, that 2, 3, 5, 10, 15 hours have passed since your train departed, etc. These numbers represent different periods of time; they are called the values of this quantity (time). Or you look out the window and follow the road posts to see the distance your train travels. The numbers 110, 111, 112, 113, 114 km flash in front of you. These numbers represent the different distances the train has traveled from its departure point. They are also called values, this time of a different magnitude (path or distance between two points). Thus, one quantity, for example time, distance, temperature, can take on as many different meanings.
Please note that a person almost never considers only one quantity, but always connects it with some other quantities. He has to simultaneously deal with two, three or more quantities. Imagine that you need to get to school by 9 o'clock. You look at your watch and see that you have 20 minutes. Then you quickly figure out whether you should take the tram or whether you can walk to school. After thinking, you decide to walk. Notice that while you were thinking, you were solving some problem. This task has become simple and familiar, since you solve such problems every day. In it you quickly compared several quantities. It was you who looked at the clock, which means you took into account the time, then you mentally imagined the distance from your home to the school; Finally, you compared two values: the speed of your step and the speed of the tram, and concluded that in a given time (20 minutes) you will have time to walk. From this simple example you see that in our practice some quantities are interconnected, that is, they depend on each other
Chapter twelve talked about the relationship of homogeneous quantities. For example, if one segment is 12 m and the other is 4 m, then the ratio of these segments will be 12: 4.
We said that this is the ratio of two homogeneous quantities. Another way to say this is that it is the ratio of two numbers one name.
Now that we are more familiar with quantities and have introduced the concept of the value of a quantity, we can express the definition of a ratio in a new way. In fact, when we considered two segments 12 m and 4 m, we were talking about one value - length, and 12 m and 4 m were only two different meanings this value.
Therefore, in the future, when we start talking about ratios, we will consider two values of one quantity, and the ratio of one value of a quantity to another value of the same quantity will be called the quotient of dividing the first value by the second.
§ 130. Values are directly proportional.
Let's consider a problem whose condition includes two quantities: distance and time.
Task 1. A body moving rectilinearly and uniformly travels 12 cm every second. Determine the distance traveled by the body in 2, 3, 4, ..., 10 seconds.
Let's create a table that can be used to track changes in time and distance.
The table gives us the opportunity to compare these two series of values. We see from it that when the values of the first quantity (time) gradually increase by 2, 3,..., 10 times, then the values of the second quantity (distance) also increase by 2, 3,..., 10 times. Thus, when the values of one quantity increase several times, the values of another quantity increase by the same amount, and when the values of one quantity decrease several times, the values of another quantity decrease by the same number.
Let us now consider a problem that involves two such quantities: the amount of matter and its cost.
Task 2. 15 m of fabric costs 120 rubles. Calculate the cost of this fabric for several other quantities of meters indicated in the table.
Using this table, we can trace how the cost of a product gradually increases depending on the increase in its quantity. Despite the fact that this problem involves completely different quantities (in the first problem - time and distance, and here - the quantity of goods and its value), nevertheless, great similarities can be found in the behavior of these quantities.
In fact, in the top line of the table there are numbers indicating the number of meters of fabric; under each of them there is a number expressing the cost of the corresponding quantity of goods. Even a quick glance at this table shows that the numbers in both the top and bottom rows are increasing; upon closer examination of the table and when comparing individual columns, it is discovered that in all cases the values of the second quantity increase by the same number of times as the values of the first increase, i.e. if the value of the first quantity increases, say, 10 times, then the value of the second quantity also increased 10 times.
If we look through the table from right to left, we will find that specified values values will decrease by same number once. In this sense, there is an unconditional similarity between the first task and the second.
The pairs of quantities that we encountered in the first and second problems are called directly proportional.
Thus, if two quantities are related to each other in such a way that as the value of one of them increases (decreases) several times, the value of the other increases (decreases) by the same amount, then such quantities are called directly proportional.
Such quantities are also said to be related to each other by a directly proportional relationship.
There are many similar quantities found in nature and in the life around us. Here are some examples:
1. Time work (day, two days, three days, etc.) and earnings, received during this time with daily wages.
2. Volume any object made of a homogeneous material, and weight this item.
§ 131. Property of directly proportional quantities.
Let's take a problem that involves the following two quantities: work time and earnings. If daily earnings are 20 rubles, then earnings for 2 days will be 40 rubles, etc. It is most convenient to create a table in which a certain number days will correspond to a certain income.
Looking at this table, we see that both quantities took 10 different values. Each value of the first value corresponds to a certain value of the second value, for example, 2 days correspond to 40 rubles; 5 days correspond to 100 rubles. In the table these numbers are written one below the other.
We already know that if two quantities are directly proportional, then each of them, in the process of its change, increases as many times as the other increases. It immediately follows from this: if we take the ratio of any two values of the first quantity, then it will be equal to the ratio of the two corresponding values of the second quantity. Indeed:
Why is this happening? But because these values are directly proportional, i.e. when one of them (time) increased by 3 times, then the other (earnings) increased by 3 times.
We have therefore come to the following conclusion: if we take two values of the first quantity and divide them one by the other, and then divide by one the corresponding values of the second quantity, then in both cases we will get the same number, i.e. i.e. the same relationship. This means that the two relations that we wrote above can be connected with an equal sign, i.e.
There is no doubt that if we took not these relations, but others, and not in that order, but in the opposite order, we would also obtain equality of relations. In fact, we will consider the values of our quantities from left to right and take the third and ninth values:
60:180 = 1 / 3 .
So we can write:
This leads to the following conclusion: if two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.
§ 132. Formula of direct proportionality.
Let's make a table of the cost of various quantities of sweets, if 1 kg of them costs 10.4 rubles.
Now let's do it this way. Take any number in the second line and divide it by the corresponding number in the first line. For example:
You see that in the quotient the same number is obtained all the time. Consequently, for a given pair of directly proportional quantities, the quotient of dividing any value of one quantity by the corresponding value of another quantity is a constant number (i.e., not changing). In our example, this quotient is 10.4. This constant number is called the proportionality factor. IN in this case it expresses the price of a unit of measurement, i.e. one kilogram of goods.
How to find or calculate the proportionality coefficient? To do this, you need to take any value of one quantity and divide it by the corresponding value of the other.
Let us denote this arbitrary value of one quantity by the letter at , and the corresponding value of another quantity - the letter X , then the proportionality coefficient (we denote it TO) we find by division:
In this equality at - divisible, X - divisor and TO- quotient, and since, by the property of division, the dividend is equal to the divisor multiplied by the quotient, we can write:
y = K x
The resulting equality is called formula of direct proportionality. Using this formula, we can calculate any number of values of one of the directly proportional quantities if we know the corresponding values of the other quantity and the coefficient of proportionality.
Example. From physics we know that weight R of any body is equal to its specific gravity d , multiplied by the volume of this body V, i.e. R = d V.
Let's take five iron bars of different volumes; knowing specific gravity iron (7.8), we can calculate the weights of these blanks using the formula:
R = 7,8 V.
Comparing this formula with the formula at = TO X , we see that y = R, x = V, and the proportionality coefficient TO= 7.8. The formula is the same, only the letters are different.
Using this formula, let's make a table: let the volume of the 1st blank be equal to 8 cubic meters. cm, then its weight is 7.8 8 = 62.4 (g). The volume of the 2nd blank is 27 cubic meters. cm. Its weight is 7.8 27 = 210.6 (g). The table will look like this:
Calculate the numbers missing in this table using the formula R= d V.
§ 133. Other methods of solving problems with directly proportional quantities.
In the previous paragraph, we solved a problem whose condition included directly proportional quantities. For this purpose, we first derived the direct proportionality formula and then applied this formula. Now we will show two other ways to solve similar problems.
Let's create a problem using the numerical data given in the table in the previous paragraph.
Task. Blank with a volume of 8 cubic meters. cm weighs 62.4 g. How much will a blank with a volume of 64 cubic meters weigh? cm?
Solution. The weight of iron, as is known, is proportional to its volume. If 8 cu. cm weigh 62.4 g, then 1 cu. cm will weigh 8 times less, i.e.
62.4:8 = 7.8 (g).
Blank with a volume of 64 cubic meters. cm will weigh 64 times more than a 1 cubic meter blank. cm, i.e.
7.8 64 = 499.2(g).
We solved our problem by reducing to unity. The meaning of this name is justified by the fact that to solve it we had to find the weight of a unit of volume in the first question.
2. Method of proportion. Let's solve the same problem using the proportion method.
Since the weight of iron and its volume are directly proportional quantities, the ratio of two values of one quantity (volume) is equal to the ratio of two corresponding values of another quantity (weight), i.e.
(letter R we designated the unknown weight of the blank). From here:
(G).
The problem was solved using the method of proportions. This means that to solve it, a proportion was compiled from the numbers included in the condition.
§ 134. Values are inversely proportional.
Consider the following problem: “Five masons can add brick walls at home in 168 days. Determine in how many days 10, 8, 6, etc. masons could complete the same work.”
If 5 masons laid the walls of a house in 168 days, then (with the same labor productivity) 10 masons could do it in half the time, since on average 10 people do twice as much work as 5 people.
Let's draw up a table by which we could monitor changes in the number of workers and working hours.
For example, to find out how many days it takes 6 workers, you must first calculate how many days it takes one worker (168 5 = 840), and then how many days it takes six workers (840: 6 = 140). Looking at this table, we see that both quantities took on six different values. Each value of the first quantity corresponds to a specific one; the value of the second value, for example, 10 corresponds to 84, the number 8 corresponds to the number 105, etc.
If we consider the values of both quantities from left to right, we will see that the values of the upper quantity increase, and the values of the lower quantity decrease. The increase and decrease are subject to the following law: the values of the number of workers increase by the same times as the values of the spent working time decrease. This idea can be expressed even more simply as follows: the more workers are involved in any task, the less time they need to complete certain work. The two quantities we encountered in this problem are called inversely proportional.
Thus, if two quantities are related to each other in such a way that as the value of one of them increases (decreases) several times, the value of the other decreases (increases) by the same amount, then such quantities are called inversely proportional.
There are many similar quantities in life. Let's give examples.
1. If for 150 rubles. If you need to buy several kilograms of sweets, the number of sweets will depend on the price of one kilogram. The higher the price, the less goods you can buy with this money; this can be seen from the table:
As the price of candy increases several times, the number of kilograms of candy that can be bought for 150 rubles decreases by the same amount. In this case, two quantities (the weight of the product and its price) are inversely proportional.
2. If the distance between two cities is 1,200 km, then it can be covered in different times depending on the speed of movement. Exist different ways transportation: on foot, on horseback, by bicycle, by boat, in a car, by train, by plane. The lower the speed, the more time it takes to move. This can be seen from the table:
With an increase in speed several times, the travel time decreases by the same amount. This means that under these conditions, speed and time are inversely proportional quantities.
§ 135. Property of inversely proportional quantities.
Let's take the second example, which we looked at in the previous paragraph. There we dealt with two quantities - speed and time. If we look at the table of values of these quantities from left to right, we will see that the values of the first quantity (speed) increase, and the values of the second (time) decrease, and the speed increases by the same amount as the time decreases. It is not difficult to understand that if you write the ratio of some values of one quantity, then it will not be equal to the ratio of the corresponding values of another quantity. In fact, if we take the ratio of the fourth value of the upper value to the seventh value (40: 80), then it will not be equal to the ratio of the fourth and seventh values of the lower value (30: 15). It can be written like this:
40:80 is not equal to 30:15, or 40:80 =/=30:15.
But if instead of one of these relations we take the opposite, then we get equality, i.e., from these relations it will be possible to create a proportion. For example:
80: 40 = 30: 15,
40: 80 = 15: 30."
Based on the foregoing, we can draw the following conclusion: if two quantities are inversely proportional, then the ratio of two arbitrarily taken values of one quantity is equal to the inverse ratio of the corresponding values of another quantity.
§ 136. Inverse proportionality formula.
Consider the problem: “There are 6 pieces of silk fabric different sizes And different varieties. All pieces cost the same. One piece contains 100 m of fabric, priced at 20 rubles. per meter How many meters are in each of the other five pieces, if a meter of fabric in these pieces costs 25, 40, 50, 80, 100 rubles, respectively?” To solve this problem, let's create a table:
We need to fill in the empty cells in the top row of this table. Let's first try to determine how many meters there are in the second piece. This can be done as follows. From the conditions of the problem it is known that the cost of all pieces is the same. The cost of the first piece is easy to determine: it contains 100 meters and each meter costs 20 rubles, which means that the first piece of silk is worth 2,000 rubles. Since the second piece of silk contains the same amount of rubles, then, dividing 2,000 rubles. for the price of one meter, i.e. 25, we find the size of the second piece: 2,000: 25 = 80 (m). In the same way we will find the size of all other pieces. The table will look like:
It is easy to see that there is an inversely proportional relationship between the number of meters and the price.
If you do the necessary calculations yourself, you will notice that each time you have to divide the number 2,000 by the price of 1 m. On the contrary, if you now start multiplying the size of the piece in meters by the price of 1 m, you will always get the number 2,000. This and it was necessary to wait, since each piece costs 2,000 rubles.
From here we can draw the following conclusion: for a given pair of inversely proportional quantities, the product of any value of one quantity by the corresponding value of another quantity is a constant number (i.e., not changing).
In our problem, this product is equal to 2,000. Check that in the previous problem, which talked about the speed of movement and the time required to move from one city to another, there was also a constant number for that problem (1,200).
Taking everything into account, it is easy to derive the inverse proportionality formula. Let us denote a certain value of one quantity by the letter X , and the corresponding value of another quantity is represented by the letter at . Then, based on the above, the work X on at must be equal to some constant value, which we denote by the letter TO, i.e.
x y = TO.
In this equality X - multiplicand at - multiplier and K- work. According to the property of multiplication, the multiplier is equal to the product divided by the multiplicand. Means,
This is the inverse proportionality formula. Using it, we can calculate any number of values of one of the inversely proportional quantities, knowing the values of the other and the constant number TO.
Let's consider another problem: “The author of one essay calculated that if his book is in a regular format, then it will have 96 pages, but if it is a pocket format, then it will have 300 pages. He tried different variants, started with 96 pages, and then he had 2,500 letters per page. Then he took the page numbers shown in the table below and again calculated how many letters there would be on the page.”
Let's try to calculate how many letters there will be on a page if the book has 100 pages.
There are 240,000 letters in the entire book, since 2,500 96 = 240,000.
Taking this into account, we use the inverse proportionality formula ( at - number of letters on the page, X - number of pages):
In our example TO= 240,000 therefore
So there are 2,400 letters on the page.
Similarly, we learn that if a book has 120 pages, then the number of letters on the page will be:
Our table will look like:
Fill in the remaining cells yourself.
§ 137. Other methods of solving problems with inversely proportional quantities.
In the previous paragraph, we solved problems whose conditions included inversely proportional quantities. We first derived the inverse proportionality formula and then applied this formula. We will now show two other solutions for such problems.
1. Method of reduction to unity.
Task. 5 turners can do some work in 16 days. In how many days can 8 turners complete this work?
Solution. There is an inverse relationship between the number of turners and working hours. If 5 turners do the job in 16 days, then one person will need 5 times more time for this, i.e.
5 turners complete the job in 16 days,
1 turner will complete it in 16 5 = 80 days.
The problem asks how many days it will take 8 turners to complete the job. Obviously, they will cope with the work 8 times faster than 1 turner, i.e. in
80: 8 = 10 (days).
This is the solution to the problem by reducing it to unity. Here it was necessary first of all to determine the time required to complete the work by one worker.
2. Method of proportion. Let's solve the same problem in the second way.
Since there is an inversely proportional relationship between the number of workers and working time, we can write: duration of work of 5 turners new number of turners (8) duration of work of 8 turners previous number of turners (5) Let us denote the required duration of work by the letter X and substitute it into the proportion expressed in words, required numbers:
The same problem is solved by the method of proportions. To solve it, we had to create a proportion from the numbers included in the problem statement.
Note. In the previous paragraphs we examined the issue of direct and inverse proportionality. Nature and life give us many examples of direct and inverse proportional dependence of quantities. However, it should be noted that these two types of dependence are only the simplest. Along with them, there are other, more complex dependencies between quantities. In addition, one should not think that if any two quantities increase simultaneously, then there is necessarily a direct proportionality between them. This is far from true. For example, tolls for railway increases depending on the distance: the further we travel, the more we pay, but this does not mean that the payment is proportional to the distance.