§ 129. Preliminary clarifications.
Man constantly deals with a wide variety of quantities. The employee and the worker try to get to the service, to work by a certain time, the pedestrian hurries to reach a certain place by the shortest route, the 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.
Any number of such examples could be cited. Time, distance, temperature, cost - all these are various quantities. In the first and second parts of this book, we got acquainted with some especially common quantities: area, volume, weight. We encounter many quantities in the study of physics and other sciences.
Imagine that you are on a train. From time to time you look at your watch and notice how long you have already been on the road. You say, for example, that 2, 3, 5, 10, 15 hours, etc. have elapsed since the departure of your train. These numbers indicate various periods of time; they are called values of this quantity (time). Or you look out the window and follow the road poles for the distance your train travels. The numbers 110, 111, 112, 113, 114 km flash before you. These numbers represent various distances passed by the train from the point of departure. They are also called values, this time with a different value (path or distance between two points). Thus, one value, for example, time, distance, temperature, can take on any different meanings.
Pay attention to the fact that a person almost never considers only one value, but always connects it with some other values. He has to deal simultaneously with two, three and 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 decide whether you should take the tram or you will have time to walk to the school. After thinking, you decide to walk. Note that at the time you were thinking, you were solving some problem. This task has become simple and familiar, as you solve such problems every day. In it, you quickly compared several values. 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 school; finally, you compared two quantities: the speed of your step and the speed of the tram, and concluded that for given time(20 min.) You will have time to walk. From this simple example, you can see that in our practice, some quantities are interconnected, that is, they depend on each other
In chapter twelve, it was told about the ratio of homogeneous quantities. For example, if one segment is 12 m and the other 4 m, then the ratio of these segments will be 12: 4.
We said that it is the ratio of two homogeneous quantities. In other words, it is the ratio of two numbers one name.
Now that we have become more familiar with quantities and have introduced the concept of the value of a quantity, we can state the definition of a relation in a new way. Indeed, when we considered two segments 12 m and 4 m, we were talking about one value - length, and 12 m and 4 m - these were only two different meanings this value.
Therefore, in the future, when we start talking about a ratio, we will consider two values of one of some quantities, 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. Quantities are directly proportional.
Consider a problem whose condition includes two quantities: distance and time.
Task 1. A body moving in a straight line and uniformly passes 12 cm in every second. Determine the path traveled by the body in 2, 3, 4, ..., 10 seconds.
Let's make a table by which it would be possible to monitor the change 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 the other quantity decrease by the same amount.
Consider now a problem that includes two such quantities: the amount of matter and its cost.
Task 2. 15 m of fabric cost 120 rubles. Calculate the cost of this fabric for several other quantities of meters indicated in the table.
From this table, we can see how the value of a commodity gradually increases, depending on the increase in its quantity. Despite the fact that completely different quantities appear in this problem (in the first problem - time and distance, and here - the quantity of goods and its cost), nevertheless, a great similarity can be found in the behavior of these quantities.
Indeed, in the top line of the table are numbers indicating the number of meters of fabric, under each of them is written a number expressing the cost of the corresponding quantity of goods. Even a cursory glance at this table shows that the numbers in both the top and bottom rows are increasing; a more careful examination of the table and a comparison of individual columns reveals that in all cases the values of the second quantity increase as much as the values of the first increase, i.e. if the value of the first quantity has increased, say, 10 times, then the value of the second value also increased 10 times.
If we look at the table from right to left, we will find that the indicated values \u200b\u200bof the quantities will decrease by the same number of times. In this sense, there is an unconditional similarity between the first task and the second.
The pairs of quantities that we met in the first and second problems are called directly proportional.
Thus, if two quantities are interconnected so that with an increase (decrease) in the value of one of them several times, the value of the other increases (decreases) by the same amount, then such quantities are called directly proportional.
They also say about such quantities that they are interconnected by a directly proportional dependence.
In nature and in the life around us, there are many such quantities. Here are some examples:
1. Time work (a day, two days, three days, etc.) and earnings received during this time at day wages.
2. Volume any object made of a homogeneous material, and the weight this item.
§ 131. The property of directly proportional quantities.
Let's take a problem that includes the following two quantities: working time and earnings. If the daily earnings are 20 rubles, then the earnings for 2 days will be 40 rubles, etc. It is most convenient to draw up a table in which a certain earnings will correspond to a certain number of days.
Looking at this table, we see that both quantities have taken 10 different values. Each value of the first value corresponds to a certain value of the second value, for example, 40 rubles correspond to 2 days; 5 days correspond to 100 rubles. In the table, these numbers are written one under the other.
We already know that if two quantities are directly proportional, then each of them, in the process of its change, increases by the same amount 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, that is, 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 any two values of the first magnitude and divide them one by the other, and then divide one by the other the values of the second magnitude corresponding to them, then in both cases one and the same number will be obtained, i.e. e. the same relation. 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 direction, we would also obtain equality of relations. Indeed, 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 implies 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.
Create a cost table various quantities sweets, if 1 kg of them costs 10.4 rubles.
Now let's do it this way. Let's take any number of the second row and divide it by the corresponding number of the first row. For example:
You see that in the quotient the same number is obtained all the time. Therefore, 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 (that is, not changing). In our example, this quotient is 10.4. This constant number is called the proportionality factor. AT this case it expresses the price of a unit of measure, that is, one kilogram of a commodity.
How to find or calculate the proportionality factor? To do this, you need to take any value of one quantity and divide it by the corresponding value of another.
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 coefficient of proportionality (we denote it To) find by dividing:
In this equality at - divisible X - divider 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 \u200b\u200bof the other quantity and the coefficient of proportionality.
Example. From physics we know that the 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.
Take five iron ingots of various sizes; 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 coefficient of proportionality 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 8 cubic meters. cm, then its weight is 7.8 8 \u003d 62.4 (g). The volume of the 2nd blank is 27 cubic meters. cm. Its weight is 7.8 27 \u003d 210.6 (g). The table will look like this:
Calculate the numbers missing in this table yourself using the formula R= d V.
§ 133. Other ways of solving problems with directly proportional quantities.
In the previous paragraph, we solved the problem, the condition of which included directly proportional quantities. For this purpose, we previously derived the direct proportionality formula and then applied this formula. Now we will show two other ways to solve similar problems.
Let's make a problem according to the numerical data given in the table of the previous paragraph.
A 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 you know, 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).
A blank with a volume of 64 cubic meters. cm will weigh 64 times more than a blank of 1 cu. 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 in order to solve it, we had to find the weight of a unit 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 denoted the unknown weight of the blank). From here:
(G).
The problem is solved by the method of proportions. This means that in order to solve it, a proportion was made up of the numbers included in the condition.
§ 134. Quantities are inversely proportional.
Consider the following problem: "Five masons can add brick walls at home at 168 days. Determine in how many days 10, 8, 6, etc. masons could do the same work.
If 5 masons laid down the walls of a house in 168 days, then (with the same labor productivity) 10 masons could do it twice as fast, since on average 10 people do twice as much work as 5 people.
Let's make a table according to which it would be possible to monitor the change in the number of working hours 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 six workers (840: 6 = 140). Looking at this table, we see that both quantities have taken six different values. Each value of the first magnitude corresponds more definitely; the value of the second value, for example, 10 corresponds to 84, the number 8 - the number 105, etc.
If we consider the values of both values from left to right, we will see that the values of the upper value increase and the values of the lower value decrease. The increase and decrease is subject to the following law: the values of the number of workers increase as many times as the values of the spent working time decrease. Even more simply, this idea can be expressed as follows: the more workers are employed in any business, the less time they need to do a certain job. The two quantities we encountered in this problem are called inversely proportional.
Thus, if two quantities are interconnected in such a way that with an increase (decrease) in the value of one of them several times, the value of the other decreases (increases) by the same amount, then such quantities are called inversely proportional.
There are many such things in life. Let's give examples.
1. If for 150 rubles. you need to buy several kilograms of sweets, then the number of sweets will depend on the price of one kilogram. The higher the price, the less goods can be bought with this money; this can be seen from the table:
With an increase in the price of sweets several times, the number of kilograms of sweets that can be bought for 150 rubles decreases by the same amount. In this case, the 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 at different times depending on the speed of movement. Exist different ways transportation: on foot, on horseback, by bicycle, by boat, by 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 time of movement decreases by the same amount. Hence, under given conditions, speed and time are inversely proportional.
§ 135. The property of inversely proportional quantities.
Let's take the second example, which we considered in the previous paragraph. There we were dealing with two quantities - the speed of movement and time. If we consider the values of these quantities from left to right in the table, we will see that the values of the first quantity (speed) increase, and the values of the second (time) decrease, and speed increases by the same factor as time decreases. It is easy to figure out 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. Indeed, 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 ratios we take the opposite, then we get equality, i.e., from these ratios it will be possible to make 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 the other quantity.
§ 136. Inverse proportionality formula.
Consider the problem: “There are 6 pieces of silk fabric different sizes and different varieties. All pieces are the same price. In one piece 100 m of fabric at a price of 20 rubles. per metre. How many meters are in each of the remaining five pieces, if a meter of fabric in these pieces costs 25, 40, 50, 80, 100 rubles, respectively? Let's create a table to solve this problem:
We need to fill in the empty cells in the top row of this table. Let's first try to determine how many meters are in the second piece. This can be done in the following way. It is known from the condition of the problem that the cost of all pieces is the same. The cost of the first piece is easy to determine: it has 100 m and each meter costs 20 rubles, which means that in the first piece of silk for 2,000 rubles. Since the second piece of silk contains the same number of rubles, then, dividing 2,000 rubles. at the price of one meter, that is, at 25, we find the value 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 inverse 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. Conversely, if you now start multiplying the size of a piece in meters by the price of 1 m, you will always get the number 2,000. and it was to be expected, since each piece costs 2,000 rubles.
From this 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, where it was said 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 into account all that has been said, it is easy to derive the inverse proportionality formula. Denote some value of one quantity by the letter X , and the corresponding value of another value - the letter at . Then, on the basis of the above work X on the at must be equal to some constant value, which we denote by the letter To, i.e.
x y = To.
In this equality X - multiplier, at - multiplier and K- work. By 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 \u200b\u200bof the other and a constant number To.
Consider another problem: “The author of one essay calculated that if his book was in the usual format, then it would have 96 pages, but if it was a pocket format, then it would have 300 pages. He tried different variants, started with 96 pages, and then he got 2,500 letters per page. Then he took the number of pages indicated in the table below, and again calculated how many letters would be on the page.
Let's try and calculate how many letters there will be on a page if the book has 100 pages.
There are 240,000 letters in the whole book, since 2,500 96 = 240,000.
Taking this into account, we use the inverse proportionality formula ( at - number of letters per page X - number of pages):
In our example To= 240,000, therefore,
So, there are 2,400 letters on a page.
Similarly, we learn that if the book has 120 pages, then the number of letters on the page will be:
Our table will look like:
Fill in the rest of the cells yourself.
§ 137. Other ways of solving problems with inversely proportional quantities.
In the previous paragraph, we solved problems that included inversely proportional quantities. We previously derived the inverse proportionality formula and then applied this formula. Now we will show two other ways of solving such problems.
1. Method of reduction to unity.
A 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 time. If 5 turners do the work in 16 days, then one person will need 5 times more time for this, i.e.
5 turners do the work in 16 days,
1 turner will complete it in 16 5 = 80 days.
The problem asks, in how many days will 8 turners complete the work. Obviously, they will do the job 8 times faster than 1 turner, i.e. for
80: 8 = 10 (days).
This is the solution of the problem by the method of reduction to unity. Here, first of all, it was necessary to determine the time for the performance of 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: the duration of the work of 5 turners the new number of turners (8) the duration of the work of 8 turners the former number of turners (5) Let us denote the desired duration of work by the letter X and substitute in the proportion expressed in words the necessary numbers:
The same problem is solved by the method of proportions. To solve it, we had to make a proportion of the numbers included in the condition of the problem.
Note. In the previous paragraphs, we considered the question of direct and inverse proportionality. Nature and life give us many examples of direct and inverse proportions 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 relationships 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, the fare for railway increases with distance: the farther we go, the more we pay, but this does not mean that the payment is proportional to the distance.
The two quantities are called directly proportional, if when one of them is increased several times, the other is increased by the same amount. Accordingly, when one of them decreases by several times, the other decreases by the same amount.
The relationship between such quantities is a direct proportional relationship. Examples of a direct proportional relationship:
1) at a constant speed, the distance traveled is directly proportional to time;
2) the perimeter of a square and its side are directly proportional;
3) the cost of a commodity purchased at one price is directly proportional to its quantity.
To distinguish a direct proportional relationship from an inverse one, you can use the proverb: "The farther into the forest, the more firewood."
It is convenient to solve problems for directly proportional quantities using proportions.
1) For the manufacture of 10 parts, 3.5 kg of metal is needed. How much metal will be used to make 12 such parts?
(We argue like this:
1. In the completed column, put the arrow in the direction from more to the smaller one.
2. The more parts, the more metal is needed to make them. So it's a directly proportional relationship.
Let x kg of metal be needed to make 12 parts. We make up the proportion (in the direction from the beginning of the arrow to its end):
12:10=x:3.5
To find , we need to divide the product of the extreme terms by the known middle term:
This means that 4.2 kg of metal will be required.
Answer: 4.2 kg.
2) 1680 rubles were paid for 15 meters of fabric. How much does 12 meters of such fabric cost?
(1. In the completed column, put the arrow in the direction from the largest number to the smallest.
2. The less fabric you buy, the less you have to pay for it. So it's a directly proportional relationship.
3. Therefore, the second arrow is directed in the same direction as the first).
Let x rubles cost 12 meters of fabric. We make up the proportion (from the beginning of the arrow to its end):
15:12=1680:x
To find the unknown extreme member of the proportion, we divide the product of the middle terms by the known extreme member of the proportion:
So, 12 meters cost 1344 rubles.
Answer: 1344 rubles.
You can talk endlessly about the advantages of learning with the help of video lessons. First, they express thoughts clearly and understandably, consistently and structured. Secondly, they take a certain fixed time, are not, often stretched and tedious. Thirdly, they are more exciting for students than the usual lessons to which they are accustomed. You can view them in a relaxed atmosphere.
In many tasks from the mathematics course, students in grade 6 will encounter direct and inverse proportionality. Before starting the study of this topic, it is worth remembering what proportions are and what basic property they have.
The topic “Proportions” is devoted to the previous video lesson. This one is a logical continuation. It is worth noting that the topic is quite important and often encountered. It should be properly understood once and for all.
To show the importance of the topic, the video tutorial starts with a task. The condition appears on the screen and is announced by the announcer. The data recording is given in the form of a diagram so that the student viewing the video recording can understand it as best as possible. It would be better if for the first time he adheres to this form of recording.
The unknown, as is customary in most cases, is denoted by the Latin letter x. To find it, you must first multiply the values crosswise. Thus, the equality of the two ratios will be obtained. This suggests that it has to do with proportions and it is worth remembering their main property. Please note that all values are given in the same unit of measure. Otherwise, it was necessary to bring them to the same dimension.
After viewing the solution method in the video, there should not be any difficulties in such tasks. The announcer comments on each move, explains all the actions, recalls the studied material that is used.
Immediately after watching the first part of the video lesson “Direct and inverse proportional relationships”, you can offer the student to solve the same problem without the help of prompts. After that, an alternative task can be proposed.
Depending on the mental capacity student, you can gradually increase the complexity of subsequent tasks.
After the first considered problem, the definition of directly proportional quantities is given. The definition is read out by the announcer. The main concept is highlighted in red.
Next, another problem is demonstrated, on the basis of which the inverse proportional relationship is explained. It is best for the student to write these concepts in a notebook. If necessary before control work, the student can easily find all the rules and definitions and reread.
After watching this video, a 6th grader will understand how to use proportions in certain tasks. This is an important topic that should not be missed in any case. If the student is not adapted to perceive the material presented by the teacher during the lesson among other students, then such learning resources will be a great salvation!
The concept of direct proportionality
Imagine that you are thinking of buying your favorite candy (or whatever you really like). The sweets in the store have their own price. Suppose 300 rubles per kilogram. The more candies you buy, the more money pay. That is, if you want 2 kilograms - pay 600 rubles, and if you want 3 kilos - give 900 rubles. Everything seems to be clear with this, right?
If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the ratio of two quantities that depend on each other. And the ratio of these quantities remains unchanged and constant: by how many parts one of them increases or decreases, by the same number of parts the second increases or decreases proportionally.
Direct proportionality can be described by the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our candy example, the price is a constant, a constant. It does not increase or decrease, no matter how many sweets you decide to buy. The independent variable (argument) x is how many kilograms of sweets you are going to buy. And the dependent variable f(x) (function) is how much money you end up paying for your purchase. So we can substitute the numbers in the formula and get: 600 r. = 300 r. * 2 kg.
The intermediate conclusion is this: if the argument increases, the function also increases, if the argument decreases, the function also decreases
Function and its properties
Direct proportional function is a special case of a linear function. If the linear function is y = k*x + b, then for direct proportionality it looks like this: y = k*x, where k is called the proportionality factor, and this is always a non-zero number. Calculating k is easy - it is found as a quotient of a function and an argument: k = y/x.
To make it clearer, let's take another example. Imagine that a car is moving from point A to point B. Its speed is 60 km/h. If we assume that the speed of movement remains constant, then it can be taken as a constant. And then we write the conditions in the form: S \u003d 60 * t, and this formula is similar to the direct proportionality function y \u003d k * x. Let's draw a parallel further: if k \u003d y / x, then the speed of the car can be calculated, knowing the distance between A and B and the time spent on the road: V \u003d S / t.
And now, from the applied application of knowledge about direct proportionality, let's return back to its function. The properties of which include:
its domain of definition is the set of all real numbers (as well as its subset);
the function is odd;
the change in variables is directly proportional to the entire length of the number line.
Direct proportionality and its graph
A graph of a direct proportional function is a straight line that intersects the origin point. To build it, it is enough to mark only one more point. And connect it and the origin of the line.
In the case of a graph, this is slope. If the slope is less than zero (k< 0), то угол между графиком функции прямой пропорциональности и осью абсцисс тупой, а функция убывающая. Если угловой коэффициент больше нуля (k >0), the graph and the x-axis form an acute angle, and the function is increasing.
And one more property of the graph of the direct proportionality function is directly related to the slope k. Suppose we have two non-identical functions and, accordingly, two graphs. So, if the coefficients k of these functions are equal, their graphs are parallel on the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.
Task examples
Let's decide a couple direct proportionality problems
Let's start simple.
Task 1: Imagine that 5 hens laid 5 eggs in 5 days. And if there are 20 hens, how many eggs will they lay in 20 days?
Solution: Denote the unknown as x. And we will argue as follows: how many times have there been more chickens? Divide 20 by 5 and find out that 4 times. And how many times more eggs 20 chickens will be laid in the same 5 days? Also 4 times more. So, we find ours like this: 5 * 4 * 4 \u003d 80 eggs will be laid by 20 hens in 20 days.
Now the example is a little more complicated, let's rephrase the problem from Newton's "General Arithmetic". Task 2: A writer can write 14 pages of a new book in 8 days. If he had assistants, how many people would it take to write 420 pages in 12 days?
Solution: We reason that the number of people (writer + assistants) increases with the increase in the amount of work if it had to be done in the same amount of time. But how many times? Dividing 420 by 14, we find out that it increases by 30 times. But since, according to the condition of the task, more time is given for work, the number of assistants does not increase by 30 times, but in this way: x \u003d 1 (writer) * 30 (times): 12/8 (days). Let's transform and find out that x = 20 people will write 420 pages in 12 days.
Let's solve another problem similar to those that we had in the examples.
Task 3: Two cars set off on the same journey. One was moving at a speed of 70 km/h and covered the same distance in 2 hours as the other in 7 hours. Find the speed of the second car.
Solution: As you remember, the path is determined through speed and time - S = V *t. Since both cars traveled the same way, we can equate the two expressions: 70*2 = V*7. Where do we find that the speed of the second car is V = 70*2/7 = 20 km/h.
And a couple more examples of tasks with direct proportionality functions. Sometimes in problems it is required to find the coefficient k.
Task 4: Given the functions y \u003d - x / 16 and y \u003d 5x / 2, determine their proportionality coefficients.
Solution: As you remember, k = y/x. Hence, for the first function, the coefficient is -1/16, and for the second, k = 5/2.
And you may also come across a task like Task 5: Write down the direct proportionality formula. Its graph and the graph of the function y \u003d -5x + 3 are located in parallel.
Solution: The function that is given to us in the condition is linear. We know that direct proportionality is a special case of a linear function. And we also know that if the coefficients of k functions are equal, their graphs are parallel. This means that all that is required is to calculate the coefficient of a known function and set direct proportionality using the familiar formula: y \u003d k * x. Coefficient k \u003d -5, direct proportionality: y \u003d -5 * x.
Conclusion
Now you have learned (or remembered, if you have already covered this topic before), what is called direct proportionality, and considered it examples. We also talked about the direct proportionality function and its graph, solved a few problems for example.
If this article was useful and helped to understand the topic, tell us about it in the comments. So that we know if we could benefit you.
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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 visual scheme:
↓ 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 tasks are inverse proportionality really uncomplicated. 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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