Direct and inverse proportionality
If t is the time the pedestrian is moving (in hours), s is the distance traveled (in kilometers), and he moves uniformly at a speed of 4 km/h, then the relationship between these quantities can be expressed by the formula s = 4t. Since each value of t corresponds to a unique value of s, we can say that a function is given using the formula s = 4t. It is called direct proportionality and is defined as follows.
Definition. Direct proportionality is a function that can be specified using the formula y \u003d kx, where k is a non-zero real number.
The name of the function y \u003d k x is due to the fact that in the formula y \u003d kx there are variables x and y, which can be values of quantities. And if the ratio of two values \u200b\u200bis equal to some number other than zero, they are called directly proportional . In our case = k (k≠0). This number is called proportionality factor.
The function y \u003d k x is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described above. Another example: if there are 2 kg of flour in one package, and x such packages are bought, then the entire mass of the purchased flour (we denote it by y) can be represented as a formula y \u003d 2x, i.e. the relationship between the number of packages and the total mass of purchased flour is directly proportional with the coefficient k=2.
Recall some properties of direct proportionality, which are studied in the school course of mathematics.
1. The domain of the function y \u003d k x and the domain of its values is the set of real numbers.
2. The graph of direct proportionality is a straight line passing through the origin. Therefore, to construct a graph of direct proportionality, it is enough to find only one point that belongs to it and does not coincide with the origin, and then draw a straight line through this point and the origin.
For example, to plot the function y = 2x, it is enough to have a point with coordinates (1, 2), and then draw a straight line through it and the origin (Fig. 7).
3. For k > 0, the function y = kx increases over the entire domain of definition; for k< 0 - убывает на всей области определения.
4. If the function f is a direct proportionality and (x 1, y 1), (x 2, y 2) - pairs of corresponding values of the variables x and y, and x 2 ≠ 0 then.
Indeed, if the function f is a direct proportionality, then it can be given by the formula y \u003d kx, and then y 1 \u003d kx 1, y 2 \u003d kx 2. Since at x 2 ≠0 and k≠0, then y 2 ≠0. That's why 
and means .
If the values of the variables x and y are positive real numbers, then the proved property of direct proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y increases (decreases) by the same amount.
This property is inherent only in direct proportionality, and it can be used in solving word problems in which directly proportional quantities are considered.
Task 1. In 8 hours, the turner made 16 parts. How many hours will it take a turner to make 48 parts if he works at the same productivity?
Solution. The problem considers the quantities - the turner's working time, the number of parts made by him and productivity (i.e., the number of parts manufactured by the turner in 1 hour), the latter value being constant, and the other two taking different values. In addition, the number of parts made and the time of work are directly proportional, since their ratio is equal to a certain number that is not equal to zero, namely, the number of parts made by a turner in 1 hour. If the number of parts made is denoted by the letter y, the work time is x, and performance - k, then we get that = k or y = kx, i.e. the mathematical model of the situation presented in the problem is direct proportionality.
The problem can be solved in two arithmetic ways:
1 way: 2 way:
1) 16:8 = 2 (children) 1) 48:16 = 3 (times)
2) 48:2 = 24(h) 2) 8-3 = 24(h)
Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 2, and then, knowing that y \u003d 2x, we found the value of x, provided that y \u003d 48.
When solving the problem in the second way, we used the property of direct proportionality: how many times the number of parts made by a turner increases, the amount of time for their manufacture increases by the same amount.
Let us now turn to the consideration of a function called inverse proportionality.
If t is the time of the pedestrian's movement (in hours), v is his speed (in km/h) and he walked 12 km, then the relationship between these values can be expressed by the formula v∙t = 20 or v = .
Since each value of t (t ≠ 0) corresponds to a single value of velocity v, we can say that a function is given using the formula v = . It is called inverse proportionality and is defined as follows.
Definition. Inverse proportionality is a function that can be specified using the formula y \u003d, where k is a non-zero real number.
The name of this function comes from the fact that y= there are variables x and y, which can be values of quantities. And if the product of two quantities is equal to some number other than zero, then they are called inversely proportional. In our case, xy = k(k ≠ 0). This number k is called the coefficient of proportionality.
Function y= is a mathematical model of many real situations considered already in the initial course of mathematics. One of them is described before the definition of inverse proportionality. Another example: if you bought 12 kg of flour and put it in l: cans of y kg each, then the relationship between these quantities can be represented as x-y= 12, i.e. it is inversely proportional with the coefficient k=12.
Recall some properties of inverse proportionality, known from the school course of mathematics.
1. Function scope y= and its range x is the set of non-zero real numbers.
2. The inverse proportionality graph is a hyperbola.
3. For k > 0, the branches of the hyperbola are located in the 1st and 3rd quadrants and the function y= is decreasing on the entire domain of x (Fig. 8).
Rice. 8 Fig.9
When k< 0 ветви гиперболы расположены во 2-й и 4-й четвертях и функция y= is increasing over the entire domain of x (Fig. 9).
4. If the function f is inversely proportional and (x 1, y 1), (x 2, y 2) are pairs of corresponding values of the variables x and y, then.
Indeed, if the function f is inversely proportional, then it can be given by the formula y=
,and then 
. Since x 1 ≠0, x 2 ≠0, x 3 ≠0, then 
If the values of the variables x and y are positive real numbers, then this property of inverse proportionality can be formulated as follows: with an increase (decrease) in the value of the variable x several times, the corresponding value of the variable y decreases (increases) by the same amount.
This property is inherent only in inverse proportionality, and it can be used in solving word problems in which inversely proportional quantities are considered.
Problem 2. A cyclist, moving at a speed of 10 km/h, covered the distance from A to B in 6 hours.
Solution. The problem considers the following quantities: the speed of the cyclist, the time of movement and the distance from A to B, the latter value being constant, and the other two taking different values. In addition, the speed and time of movement are inversely proportional, since their product is equal to a certain number, namely the distance traveled. If the time of the cyclist's movement is denoted by the letter y, the speed is x, and the distance AB is k, then we get that xy \u003d k or y \u003d, i.e. the mathematical model of the situation presented in the problem is inverse proportionality.
You can solve the problem in two ways:
1 way: 2 way:
1) 10-6 = 60 (km) 1) 20:10 = 2 (times)
2) 60:20 = 3(4) 2) 6:2 = 3(h)
Solving the problem in the first way, we first found the proportionality coefficient k, it is equal to 60, and then, knowing that y \u003d, we found the value of y, provided that x \u003d 20.
When solving the problem in the second way, we used the inverse proportionality property: how many times the speed of movement increases, the time to travel the same distance decreases by the same amount.
Note that when solving specific problems with inversely proportional or directly proportional quantities, some restrictions are imposed on x and y, in particular, they can be considered not on the entire set of real numbers, but on its subsets.
Problem 3. Lena bought x pencils, and Katya bought 2 times more. Denote the number of pencils Katya bought as y, express y in terms of x, and plot the established correspondence graph, provided that x ≤ 5. Is this match a function? What is its domain of definition and range of values?
Solution. Katya bought u = 2 pencils. When plotting the function y=2x, it must be taken into account that the variable x denotes the number of pencils and x≤5, which means that it can only take on the values 0, 1, 2, 3, 4, 5. This will be the domain of this function. To get the range of this function, you need to multiply each value x from the domain of definition by 2, i.e. it will be a set (0, 2, 4, 6, 8, 10). Therefore, the graph of the function y \u003d 2x with the domain of definition (0, 1, 2, 3, 4, 5) will be the set of points shown in Figure 10. All these points belong to the line y \u003d 2x.
Basic goals:
- introduce the concept of direct and inverse proportional dependence 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 to activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with the problems solved using proportions.
II. Updating knowledge and fixing difficulties in activities
2.1. oral work (3 min)
- Find the meaning of expressions and find out the word encrypted in the answers.
14 - s; 0.1 - and; 7 - l; 0.2 - a; 17 - in; 25 - to
- The word came out - strength. Well done!
- The motto of our lesson today: Power is in knowledge! I'm looking - so I'm learning!
- Make a proportion of the resulting numbers. (14:7=0.2:0.1 etc.)
2.2. Consider the relationship between known quantities (7 min)
- the path traveled by the car at a constant speed, and the time of its movement: S = v t ( with an increase in speed (time), the path increases);
- the speed of the car and the time spent on the road: v=S:t(with an increase in the time to travel the path, the speed decreases);
–
the cost of goods purchased at one price and its quantity:
C \u003d a n (with an increase (decrease) in price, the cost of purchase increases (decreases);
- the price of the product and its quantity: a \u003d C: n (with an increase in quantity, the price decreases)
- the area of the rectangle and its length (width): S = a b (with an increase in the length (width), the area increases;
- the length of the rectangle and the width: a = S: b (with an increase in the length, 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 \u003d A: n (with an increase in the number of workers, the time spent on doing work decreases), etc.
We have obtained dependencies in which, with an increase in one value several times, the other immediately increases by the same amount (shown with arrows for examples) and dependencies in which, with an increase in one value several times, the second value decreases by the same number of times.
Such relationships are called direct and inverse proportions.
Directly proportional dependence- a dependence in which with an increase (decrease) in one value several times, the second value increases (decreases) by the same amount.
Inverse proportional relationship- a dependence in which with an increase (decrease) in one value several times, the second value decreases (increases) by the same amount.
III. Statement of the learning task
What is the problem we are facing? (Learn to distinguish between direct and inverse relationships)
- It - goal our lesson. Now formulate topic lesson. (Direct and inverse proportionality).
- Well done! Write the topic of the lesson in your notebooks. (The teacher writes the topic on the blackboard.)
IV. "Discovery" of new knowledge(10 min)
Let's analyze problems number 199.
1. The printer prints 27 pages in 4.5 minutes. How long will it take to print 300 pages?
27 pages - 4.5 min.
300 pp. - x?
2. There are 48 packs of tea in a box, 250 g each. How many packs of 150g will come out of this tea?
48 packs - 250 g.
X? - 150 g.
3. The car drove 310 km, having spent 25 liters of gasoline. How far can a car travel on a full tank of 40 liters?
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 will make 215 revolutions?
32 teeth - 315 rpm
40 teeth - x?
To draw up a proportion, one direction of the arrows is necessary, for this, in inverse proportion, one ratio is replaced by the inverse.
At the blackboard, students find the value of the quantities, in the field, students solve one problem of their choice.
– Formulate a rule for solving problems with direct and inverse proportionality.
A table appears on the board:
v. Primary fastening in external speech(10 min)
Tasks on the sheets:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
- For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this area?
VI. Independent work with self-test according to the standard(5 minutes)
Two students complete assignments No. 225 on their own on hidden boards, and the rest in notebooks. Then they check the work according to the algorithm and compare it with the solution on the board. Errors are corrected, their causes are clarified. If the task is completed, right, then next to the students put a “+” sign for themselves.
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 blackboard. After 3–4 minutes, the students who worked at the blackboard present their solutions, and the rest check the tasks and participate in their discussion.
VIII. Reflection of activity (the result of the lesson)
- What new did you learn at the lesson?
- What did you repeat?
What is the algorithm for solving proportion problems?
Have we reached our goal?
- How do you rate your work?
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).
Illustrate 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 going 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. Don't forget to share this article in social networks so that your friends and classmates can also play.
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>>Math: Direct proportionality and its graph
Direct proportionality and its graph
Among the linear functions y = kx + m, the case when m = 0 is highlighted; in this case takes the form y = kx and it is called direct proportionality. This name is explained by the fact that two quantities y and x are called directly proportional if their ratio is equal to a specific
a number other than zero. Here , this number k is called the coefficient of proportionality.
Many real situations are modeled using direct proportionality.
For example, the path s and time t at a constant speed, 20 km/h, are related by the dependence s = 20t; this is a direct proportionality, with k = 20.
Another example:
the cost y and the number x of loaves of bread at the price of 5 rubles. per loaf are linked by the dependence y = 5x; this is a direct proportionality, where k = 5.
Proof. Let's do it in two stages.
1. y \u003d kx - special case linear function, and the graph of a linear function is a straight line; let's denote it by I.
2. The pair x \u003d 0, y \u003d 0 satisfies the equation y - kx, and therefore the point (0; 0) belongs to the graph of the equation y \u003d kx, that is, the line I.
Therefore, the line I passes through the origin. The theorem has been proven.
One must be able to move not only from the analytical model y \u003d kx to the geometric one (direct proportionality graph), but also from the geometric models to analytical. Consider, for example, a straight line on the xOy coordinate plane shown in Figure 50. It is a direct proportionality graph, you just need to find the value of the coefficient k. Since y, it is enough to take any point on the line and find the ratio of the ordinate of this point to its abscissa. The straight line passes through the point P (3; 6), and for this point we have: Hence, k = 2, and therefore the given straight line serves as a graph of direct proportionality y \u003d 2x.
As a result, the coefficient k in the notation of the linear function y \u003d kx + m is also called the slope. If k>0, then the line y \u003d kx + m forms an acute angle with the positive direction of the x axis (Fig. 49, a), and if k< О, - тупой угол (рис. 49, б).
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