Direct and inverse proportionality. Direct and inverse proportional relationships

I. Directly proportional quantities.

Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.

Examples.

1 . The quantity of goods purchased and the purchase price (with a fixed price for one unit of goods - 1 piece or 1 kg, etc.) How many times more goods were bought, the more times more they paid.

2 . The distance traveled and the time spent on it (at constant speed). How many times longer is the path, how many times more time will it take to complete it.

3 . The volume of a body and its mass. ( If one watermelon is 2 times larger than another, then its mass will be 2 times larger)

II. Property of direct proportionality of quantities.

If two quantities are directly proportional, then the ratio of two arbitrarily taken values ​​of the first quantity is equal to the ratio of two corresponding values ​​of the second quantity.

Task 1. For raspberry jam we took 12 kg raspberries and 8 kg Sahara. How much sugar will you need if you took it? 9 kg raspberries?

Solution.

We reason like this: let it be necessary x kg sugar for 9 kg raspberries The mass of raspberries and the mass of sugar are directly proportional quantities: how many times less raspberries are, the same number of times less sugar is needed. Therefore, the ratio of raspberries taken (by weight) ( 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 9 kg raspberries need to be taken 6 kg Sahara.

The solution of the problem It could be done like this:

Let on 9 kg raspberries need to be taken x kg Sahara.

(The arrows in the figure are directed in one direction, and up or down does not matter. Meaning: how many times the number 12 more number 9 , the same number of times 8 more number X, i.e. there is a direct relationship here).

Answer: on 9 kg I need to take some raspberries 6 kg Sahara.

Task 2. Car for 3 hours traveled the distance 264 km. How long will it take him to travel? 440 km, if he drives at the same speed?

Solution.

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 flows from the pipe into the pool. Behind 2 hours she fills 1/5 swimming pool What part of the pool is filled with water in 5 o'clock?

Solution.

We answer the question of the task: for 5 o'clock will be filled 1/x part of the pool. (The entire pool is taken as one whole).

Along with directly proportional quantities in arithmetic, inversely proportional quantities were also considered.

Let's give examples.

1) The length of the base and the height of a rectangle with a constant area.

Suppose you need to allocate a rectangular plot of land with an area of

We “can arbitrarily set, for example, the length of the section. But then the width of the area will depend on what length we have chosen. The different (possible) lengths and widths are shown in the table.

In general, if we denote the length of the section by x and the width by y, then the relationship between them can be expressed by the formula:

Expressing y through x, we get:

Giving x arbitrary values, we will obtain the corresponding y values.

2) Time and speed of uniform motion at a certain distance.

Let the distance between two cities be 200 km. The higher the speed, the less time it will take to cover a given distance. This can be seen from the following table:

In general, if we denote the speed by x, and the time of movement by y, then the relationship between them will be expressed by the formula:

Definition. The relationship between two quantities expressed by the equality , where k is a certain number (not equal to zero), is called an inversely proportional relationship.

The number here is also called the proportionality coefficient.

Just as in the case of direct proportionality, in equality the quantities x and y in the general case can take on positive and negative values.

But in all cases of inverse proportionality, none of the quantities can be equal to zero. In fact, if at least one of the quantities x or y is equal to zero, then the left side of the equality will be equal to

And the right one - to some number that is not equal to zero (by definition), that is, the result will be an incorrect equality.

2. Graph of inverse proportionality.

Let's build a dependence graph

Expressing y through x, we get:

We will give x arbitrary (valid) values ​​and calculate the corresponding y values. We get the table:

Let's construct the corresponding points (Fig. 28).

If we take the values ​​of x at smaller intervals, then the points will be located closer together.

For all possible values ​​of x, the corresponding points will be located on two branches of the graph, symmetrical with respect to the origin of coordinates and passing in the first and third quarters of the coordinate plane (Fig. 29).

So, we see that the graph of inverse proportionality is a curved line. This line consists of two branches.

One branch will be obtained for positive, the other - for negative values ​​of x.

The graph of an inversely proportional relationship is called a hyperbola.

To get a more accurate graph, you need to build as many points as possible.

A hyperbole can be drawn with fairly high accuracy using, for example, patterns.

In drawing 30, a graph of an inversely proportional relationship with a negative coefficient is plotted. For example, by creating a table like this:

we obtain a hyperbola, the branches of which are located in the II and IV quarters.

Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Shvartsburd for 6th grade in mathematics on the topic:

Chapter I. Ordinary fractions.
§ 4. Relations and proportions:
22. Direct and inverse proportional relationships

1 For 3.2 kg of goods they paid 115.2 rubles. How much should you pay for 1.5 kg of this product?
SOLUTION

2 Two rectangles have the same area. The length of the first rectangle is 3.6 m and the width is 2.4 m. The length of the second is 4.8 m. Find its width.
SOLUTION

782 Determine whether the relationship between the quantities is direct, inverse, or not proportional: the distance covered by the car at a constant speed and the time of its movement; the cost of goods purchased at one price and its quantity; the area of ​​the square and the length of its side; the mass of the steel bar and its volume; the number of workers performing some work with the same labor productivity, and the time of completion; the cost of the product and its quantity purchased for a certain amount of money; the age of the person and the size of his shoes; the volume of the cube and the length of its edge; the perimeter of the square and the length of its side; a fraction and its denominator, if the numerator does not change; a fraction and its numerator if the denominator does not change.
SOLUTION

783 A steel ball with a volume of 6 cm3 has a mass of 46.8 g. What is the mass of a ball made of the same steel if its volume is 2.5 cm3?
SOLUTION

784 From 21 kg of cotton seed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
SOLUTION

785 For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long will it take 7 bulldozers to clear this site?
SOLUTION

786 To transport the cargo, 24 vehicles with a carrying capacity of 7.5 tons were required. How many vehicles with a carrying capacity of 4.5 tons are needed to transport the same cargo?
SOLUTION

787 To determine the germination of seeds, peas were sown. Of the 200 peas sown, 170 sprouted. What percentage of the peas sprouted (germinated)?
SOLUTION

788 During the city greening Sunday, linden trees were planted on the street. 95% of all planted linden trees were accepted. How many of them were planted if 57 linden trees were planted?
SOLUTION

789 There are 80 students in the ski section. Among them are 32 girls. What percentage of section participants are girls and boys?
SOLUTION

790 According to the plan, the plant was supposed to smelt 980 tons of steel in a month. But the plan was fulfilled by 115%. How many tons of steel did the plant produce?
SOLUTION

791 In 8 months, the worker completed 96% of the annual plan. What percentage of the annual plan will the worker complete in 12 months if he works with the same productivity?
SOLUTION

792 In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of the beets if you work at the same productivity?
SOLUTION

793 In iron ore, for every 7 parts of iron there are 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
SOLUTION

794 To prepare borscht, for every 100 g of meat you need to take 60 g of beets. How many beets should you take for 650 g of meat?
SOLUTION

796 Express each of the following fractions as the sum of two fractions with numerator 1.
SOLUTION

797 From the numbers 3, 7, 9 and 21, form two correct proportions.
SOLUTION

798 The middle terms of the proportion are 6 and 10. What can the extreme terms be? Give examples.
SOLUTION

799 At what value of x is the proportion correct.
SOLUTION

800 Find the ratio of 2 min to 10 sec; 0.3 m2 to 0.1 dm2; 0.1 kg to 0.1 g; 4 hours to 1 day; 3 dm3 to 0.6 m3
SOLUTION

801 Where on the coordinate ray should the number c be located for the proportion to be correct.
SOLUTION

802 Cover the table with a sheet of paper. Open the first line for a few seconds and then, closing it, try to repeat or write down the three numbers of that line. If you have reproduced all the numbers correctly, move on to the second row of the table. If there is an error in any line, write several sets of the same number of two-digit numbers yourself and practice memorizing. If you can reproduce at least five two-digit numbers without errors, you have a good memory.
SOLUTION

804 Is it possible to formulate the correct proportion from the following numbers?
SOLUTION

805 From the equality of the products 3 · 24 = 8 · 9, form three correct proportions.
SOLUTION

806 The length of segment AB is 8 dm, and the length of segment CD is 2 cm. Find the ratio of the lengths AB and CD. What part of AB is the length CD?
SOLUTION

807 A trip to the sanatorium costs 460 rubles. The trade union pays 70% of the cost of the trip. How much will a vacationer pay for a trip?
SOLUTION

808 Find the meaning of the expression.
SOLUTION

809 1) When processing a casting part weighing 40 kg, 3.2 kg was wasted. What percentage is the mass of the part from the casting? 2) When sorting grain from 1750 kg, 105 kg went to waste. What percentage of grain is left?

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 you 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 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 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 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 down 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.

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