Direct and inverse proportionality. Direct proportional dependence

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:

1. Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
2. The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
3. Does not have maximum or minimum values.
4. It is odd and its graph is symmetrical about the origin.
5. Non-periodic.
6. Its graph does not intersect the coordinate axes.
7. Has no zeros.
8. 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.
9. 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 down the conditions of the problem in the form of a visual diagram:

↓ 6 workers – 4 hours

↓ 3 workers – x h

Let's write this as a proportion: 6/3 = x/4. And we get x = 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 inverse proportionality problems are 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 social networks so that your friends and classmates can also play.

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Proportionality is a relationship between two quantities, in which a change in one of them entails a change in the other by the same amount.

Proportionality can be direct or inverse. In this lesson we will look at each of them.

Direct proportionality

Let's assume that the car is moving at a speed of 50 km/h. We remember that speed is the distance traveled per unit of time (1 hour, 1 minute or 1 second). In our example, the car is moving at a speed of 50 km/h, that is, in one hour it will cover a distance of fifty kilometers.

Let us depict in the figure the distance traveled by the car in 1 hour.

Let the car drive for another hour at the same speed of fifty kilometers per hour. Then it turns out that the car will travel 100 km

As can be seen from the example, doubling the time led to an increase in the distance traveled by the same amount, that is, twice.

Quantities such as time and distance are called directly proportional. And the relationship between such quantities is called direct proportionality.

Direct proportionality is the relationship between two quantities in which an increase in one of them entails an increase in the other by the same amount.

and vice versa, if one quantity decreases by a certain number of times, then the other decreases by the same number of times.

Let's assume that the original plan was to drive a car 100 km in 2 hours, but after driving 50 km, the driver decided to rest. Then it turns out that by reducing the distance by half, the time will decrease by the same amount. In other words, reducing the distance traveled will lead to a decrease in time by the same amount.

An interesting feature of directly proportional quantities is that their ratio is always constant. That is, when the values ​​of directly proportional quantities change, their ratio remains unchanged.

In the example considered, the distance was initially 50 km and the time was one hour. The ratio of distance to time is the number 50.

But we increased the travel time by 2 times, making it equal to two hours. As a result, the distance traveled increased by the same amount, that is, it became equal to 100 km. The ratio of one hundred kilometers to two hours is again the number 50

The number 50 is called coefficient of direct proportionality. It shows how much distance there is per hour of movement. In this case, the coefficient plays the role of movement speed, since speed is the ratio of the distance traveled to the time.

Proportions can be made from directly proportional quantities. For example, the ratios make up the proportion:

Fifty kilometers is to one hour as one hundred kilometers is to two hours.

Example 2. The cost and quantity of goods purchased are directly proportional. If 1 kg of sweets costs 30 rubles, then 2 kg of the same sweets will cost 60 rubles, 3 kg 90 rubles. As the cost of a purchased product increases, its quantity increases by the same amount.

Since the cost of a product and its quantity are directly proportional quantities, their ratio is always constant.

Let's write down what is the ratio of thirty rubles to one kilogram

Now let’s write down what the ratio of sixty rubles to two kilograms is. This ratio will again be equal to thirty:

Here the coefficient of direct proportionality is the number 30. This coefficient shows how many rubles are per kilogram of sweets. In this example, the coefficient plays the role of the price of one kilogram of goods, since price is the ratio of the cost of the goods to its quantity.

Inverse proportionality

Consider the following example. The distance between the two cities is 80 km. The motorcyclist left the first city and, at a speed of 20 km/h, reached the second city in 4 hours.

If a motorcyclist's speed was 20 km/h, this means that every hour he covered a distance of twenty kilometers. Let us depict in the figure the distance traveled by the motorcyclist and the time of his movement:

On the way back, the motorcyclist's speed was 40 km/h, and he spent 2 hours on the same journey.

It is easy to notice that when the speed changes, the time of movement changes by the same amount. Moreover, it changed in the opposite direction - that is, the speed increased, but the time, on the contrary, decreased.

Quantities such as speed and time are called inversely proportional. And the relationship between such quantities is called inverse proportionality.

Inverse proportionality is the relationship between two quantities in which an increase in one of them entails a decrease in the other by the same amount.

and vice versa, if one quantity decreases by a certain number of times, then the other increases by the same number of times.

For example, if on the way back the motorcyclist’s speed was 10 km/h, then he would cover the same 80 km in 8 hours:

As can be seen from the example, a decrease in speed led to an increase in movement time by the same amount.

The peculiarity of inversely proportional quantities is that their product is always constant. That is, when the values ​​of inversely proportional quantities change, their product remains unchanged.

In the example considered, the distance between cities was 80 km. When the speed and time of movement of the motorcyclist changed, this distance always remained unchanged

A motorcyclist could travel this distance at a speed of 20 km/h in 4 hours, and at a speed of 40 km/h in 2 hours, and at a speed of 10 km/h in 8 hours. In all cases, the product of speed and time was equal to 80 km

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Dependency Types

Let's look at charging the battery. As the first quantity, let's take the time it takes to charge. The second value is the time it will work after charging. The longer you charge the battery, the longer it will last. The process will continue until the battery is fully charged.

Dependence of battery operating time on the time it is charged

Note 1

This dependence is called straight:

As one value increases, so does the second. As one value decreases, the second value also decreases.

Let's look at another example.

The more books a student reads, the fewer mistakes he will make in the dictation. Or the higher you rise in the mountains, the lower the atmospheric pressure will be.

Note 2

This dependence is called reverse:

As one value increases, the second decreases. As one value decreases, the second value increases.

Thus, in case direct dependence both quantities change equally (both either increase or decrease), and in the case inverse relationship– opposite (one increases and the other decreases, or vice versa).

Determining dependencies between quantities

Example 1

The time it takes to visit a friend is $20$ minutes. If the speed (first value) increases by $2$ times, we will find how the time (second value) that will be spent on the path to a friend changes.

Obviously, the time will decrease by $2$ times.

Note 3

This dependence is called proportional:

The number of times one quantity changes, the number of times the second quantity changes.

Example 2

For $2$ loaves of bread in the store you need to pay 80 rubles. If you need to buy $4$ loaves of bread (the quantity of bread increases by $2$ times), how many times more will you have to pay?

Obviously, the cost will also increase $2$ times. We have an example of proportional dependence.

In both examples, proportional dependencies were considered. But in the example with loaves of bread, the quantities change in one direction, therefore, the dependence is straight. And in the example of going to a friend’s house, the relationship between speed and time is reverse. Thus there is directly proportional relationship And inversely proportional relationship.

Direct proportionality

Let's consider $2$ proportional quantities: the number of loaves of bread and their cost. Let $2$ loaves of bread cost $80$ rubles. If the number of buns increases by $4$ times ($8$ buns), their total cost will be $320$ rubles.

The ratio of the number of buns: $\frac(8)(2)=4$.

Bun cost ratio: $\frac(320)(80)=$4.

As you can see, these relations are equal to each other:

$\frac(8)(2)=\frac(320)(80)$.

Definition 1

The equality of two ratios is called proportion.

With a directly proportional dependence, a relationship is obtained when the change in the first and second quantities coincides:

$\frac(A_2)(A_1)=\frac(B_2)(B_1)$.

Definition 2

The two quantities are called directly proportional, if when one of them changes (increases or decreases), the other value also changes (increases or decreases, respectively) by the same amount.

Example 3

The car traveled $180$ km in $2$ hours. Find the time during which he will cover $2$ times the distance at the same speed.

Solution.

Time is directly proportional to distance:

$t=\frac(S)(v)$.

How many times will the distance increase, at a constant speed, by the same amount will the time increase:

$\frac(2S)(v)=2t$;

$\frac(3S)(v)=3t$.

The car traveled $180$ km in $2$ hours

The car will travel $180 \cdot 2=360$ km - in $x$ hours

The further the car travels, the longer it will take. Consequently, the relationship between the quantities is directly proportional.

Let's make a proportion:

$\frac(180)(360)=\frac(2)(x)$;

$x=\frac(360 \cdot 2)(180)$;

Answer: The car will need $4$ hours.

Inverse proportionality

Definition 3

Solution.

Time is inversely proportional to speed:

$t=\frac(S)(v)$.

By how many times does the speed increase, with the same path, the time decreases by the same amount:

$\frac(S)(2v)=\frac(t)(2)$;

$\frac(S)(3v)=\frac(t)(3)$.

Let's write the problem condition in the form of a table:

The car traveled $60$ km - in $6$ hours

The car will travel $120$ km – in $x$ hours

The faster the car speeds, the less time it will take. Consequently, the relationship between the quantities is inversely proportional.

Let's make a proportion.

Because the proportionality is inverse, the second relation in the proportion is reversed:

$\frac(60)(120)=\frac(x)(6)$;

$x=\frac(60 \cdot 6)(120)$;

Answer: The car will need $3$ hours.

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 student’s mental abilities, the difficulty of subsequent tasks can be gradually increased.

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 them.

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!