§ 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 different distances which the train passed from the point of departure. 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 quantities: the speed of your step and the speed of the tram, and concluded that 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
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 the indicated values of 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 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 of days will correspond to a certain earnings.
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 by one another, 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 create a cost table various quantities sweets, if 1 kg 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 quantity, 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 engaged in any task, the less time they need to complete a certain job. 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 work 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 the necessary numbers into the proportion expressed in words:
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.
The two quantities are called directly proportional, if when one of them increases several times, the other increases by the same amount. Accordingly, when one of them decreases several times, the other decreases by the same amount.
The relationship between such quantities is a direct proportional relationship. Examples of direct proportional dependence:
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 quantities;
3) the cost of a product 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 further into the forest, the more firewood.”
It is convenient to solve problems involving directly proportional quantities using proportions.
1) To make 10 parts you need 3.5 kg of metal. How much metal will go into making 12 such parts?
(We reason like this:
1. In the filled column, place an arrow in the direction from more to less.
2. The more parts, the more metal needed to make them. This means that this is 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 , you 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) For 15 meters of fabric they paid 1680 rubles. How much does 12 meters of such fabric cost?
(1. In the filled column, place an 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. This means that this is a directly proportional relationship.
3. Therefore, the second arrow is in the same direction as the first).
Let x rubles cost 12 meters of fabric. We make a proportion (from the beginning of the arrow to its end):
15:12=1680:x
To find the unknown extreme term of the proportion, divide the product of the middle terms by the known extreme term of the proportion:
This means that 12 meters cost 1344 rubles.
Answer: 1344 rubles.
We can talk endlessly about the advantages of learning using video lessons. Firstly, they present their thoughts clearly and understandably, consistently and in a structured manner. Secondly, they take a certain fixed time and are not often drawn out and tedious. Thirdly, they are more exciting for students than the regular lessons they are used to. You can view them in a calm environment.
In many problems from the mathematics course, 6th grade students will be faced with direct and inverse proportional relationships. Before you start studying this topic, it is worth remembering what proportions are and what basic properties they have.
The previous video lesson is devoted to the topic “Proportions”. This one is a logical continuation. It is worth noting that the topic is quite important and frequently encountered. It is worth understanding properly once and for all.
To show the importance of the topic, the video lesson begins with a task. The condition appears on the screen and is announced by the announcer. The data recording is given in the form of some kind of diagram so that the student watching the video recording can understand as best as possible. It would be better if at first 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 indicated in the same unit of measurement. Otherwise, it was necessary to reduce them to one dimension.
After watching the solution method in the video, you should not have any difficulties with such problems. The announcer comments on each move, explains all the actions, and recalls the studied material that is used.
Immediately after watching the first part of the video lesson “Direct and inverse proportional dependencies”, you can ask the student to solve the same problem without the help of hints. Afterwards, you can offer an alternative task.
Depending on the mental abilities student, you can gradually increase the complexity of subsequent tasks.
After the first problem considered, 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 down these concepts in a notebook. If necessary, before tests, the student can easily find all the rules and definitions and re-read.
After watching this video, a 6th grader will understand how to use proportions in certain tasks. This is a fairly important topic that should not be missed under any circumstances. If a student is not able to perceive the material presented by the teacher during a lesson among other students, then such educational resources will be a great salvation!
The concept of direct proportionality
Imagine that you are planning to buy your favorite candies (or anything that you really like). Sweets in the store have their own price. Let's say 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 kilograms, pay 900 rubles. This seems to be all clear, right?
If yes, then it is now clear to you what direct proportionality is - this is a concept that describes the relationship of two quantities dependent 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 with the following formula: f(x) = a*x, and a in this formula is a constant value (a = const). In our example about candy, the price is a constant value, a constant. It does not increase or decrease, no matter how many candies you decide to buy. The independent variable (argument)x is how many kilograms of candy 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 into the formula and get: 600 rubles. = 300 rub. * 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 coefficient, and it is always a non-zero number. It is easy to calculate k - 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 = 60*t, and this formula is similar to the function of direct proportionality y = k *x. Let's draw a parallel further: if k = 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 = 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 subsets);
function is odd;
the change in variables is directly proportional along the entire length of the number line.
Direct proportionality and its graph
The graph of a direct proportionality function is a straight line that intersects the origin. To build it, it is enough to mark only one more point. And connect it and the origin of coordinates with a straight line.
In the case of a graph, k 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 located parallel to the coordinate axis. And if the coefficients k are not equal to each other, the graphs intersect.
Sample problems
Now let's solve a couple direct proportionality problems
Let's start with something simple.
Problem 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: Let's denote the unknown by kx. And we will reason as follows: how many times more chickens have there become? Divide 20 by 5 and find out that it is 4 times. How many times more eggs will 20 hens lay eggs in the same 5 days? Also 4 times more. So, we find ours like this: 5*4*4 = 80 eggs will be laid by 20 hens in 20 days.
Now the example is a little more complicated, let’s paraphrase the problem from Newton’s “General Arithmetic”. Problem 2: A writer can compose 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 volume 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 conditions of the task, more time is given for the work, the number of assistants increases not by 30 times, but in this way: x = 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 in our examples.
Problem 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 took 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 distance, we can equate the two expressions: 70*2 = V*7. How 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 functions of direct proportionality. Sometimes problems require finding the coefficient k.
Task 4: Given the functions y = - x/16 and y = 5x/2, determine their proportionality coefficients.
Solution: As you remember, k = y/x. This means that for the first function the coefficient is equal to -1/16, and for the second k = 5/2.
You may also encounter a task like Task 5: Write direct proportionality with a formula. Its graph and the graph of the function y = -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 formula familiar to us: y = k *x. Coefficient k = -5, direct proportionality: y = -5*x.
Conclusion
Now you have learned (or remembered, if you have already covered this topic before) what is called direct proportionality, and looked at it examples. We also talked about the direct proportionality function and its graph, and solved several example problems.
If this article was useful and helped you understand the topic, tell us about it in the comments. So that we know if we could benefit you.
blog.site, when copying material in full or in part, a link to the original source is required.
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 is a functional dependence in which a decrease or increase by several times in an independent value (it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in a dependent value (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 a proportion, the right side of the record 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 the condition implies 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 tasks are inverse proportionality 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.
website, when copying material in full or in part, a link to the original source is required.
Loading...