Examples of direct and inverse proportionality. Direct and inverse proportional dependencies
I. Directly proportional quantities.
Let the value y depends on the size X. If with an increase X several times the size at increases by the same factor, then such values X and at are called directly proportional.
Examples.
1 . The quantity of the purchased goods and the cost of the purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, so many times more and paid.
2 . The distance traveled and the time spent on it (at constant speed). How many times longer the path, how many times more time we will spend on it.
3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than the other, then its mass will be 2 times larger)
II. The property of direct proportionality of quantities.
If two quantities are directly proportional, then the ratio of two arbitrary values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.
Task 1. For raspberry jam 12 kg raspberries and 8 kg Sahara. How much sugar will be required if taken 9 kg raspberries?
Decision.
We argue like this: let it be necessary x kg sugar on 9 kg raspberries. The mass of raspberries and the mass of sugar are directly proportional: how many times less raspberries, the same amount of sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:
12: 9=8: X;
x=9 · 8: 12;
x=6. Answer: on the 9 kg raspberries to take 6 kg Sahara.
The solution of the problem could have been done like this:
Let on 9 kg raspberries to take x kg Sahara.
(The arrows in the figure are directed in one direction, and it does not matter up or down. Meaning: how many times the number 12 more number 9 , the same number 8 more number X, i.e., there is a direct dependence here).
Answer: on the 9 kg raspberries to take 6 kg Sahara.
Task 2. car for 3 hours traveled distance 264 km. How long will it take him 440 km if it travels at the same speed?
Decision.
Let for x hours the car will cover the distance 440 km.
Answer: the car will pass 440 km in 5 hours.
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 you 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 as follows: 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, k is the 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 reason 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 will 20 hens lay 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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I. Directly proportional quantities.
Let the value y depends on the size X. If with an increase X several times the size at increases by the same factor, then such values X and at are called directly proportional.
Examples.
1 . The quantity of the purchased goods and the cost of the purchase (at a fixed price of one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, so many times more and paid.
2 . The distance traveled and the time spent on it (at constant speed). How many times longer the path, how many times more time we will spend on it.
3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than the other, then its mass will be 2 times larger)
II. The property of direct proportionality of quantities.
If two quantities are directly proportional, then the ratio of two arbitrary values of the first quantity is equal to the ratio of the two corresponding values of the second quantity.
Task 1. For raspberry jam 12 kg raspberries and 8 kg Sahara. How much sugar will be required if taken 9 kg raspberries?
Decision.
We argue like this: let it be necessary x kg sugar on 9 kg raspberries. The mass of raspberries and the mass of sugar are directly proportional: how many times less raspberries, the same amount of sugar is needed. Therefore, the ratio of taken (by weight) raspberries ( 12:9 ) will be equal to the ratio of sugar taken ( 8:x). We get the proportion:
12: 9=8: X;
x=9 · 8: 12;
x=6. Answer: on the 9 kg raspberries to take 6 kg Sahara.
The solution of the problem could have been done like this:
Let on 9 kg raspberries to take x kg Sahara.
(The arrows in the figure are directed in one direction, and it does not matter up or down. Meaning: how many times the number 12 more number 9 , the same number 8 more number X, i.e., there is a direct dependence here).
Answer: on the 9 kg raspberries to take 6 kg Sahara.
Task 2. car for 3 hours traveled distance 264 km. How long will it take him 440 km if it travels at the same speed?
Decision.
Let for x hours the car will cover the distance 440 km.
Answer: the car will pass 440 km in 5 hours.
Task 3. Water enters the pool from the pipe. Behind 2 hours she fills 1/5 pool. What part of the pool is filled with water for 5 o'clock?
Decision.
We answer the question of the task: for 5 o'clock fill up 1/x part of the pool. (The whole pool is taken as one whole).