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.
Lesson contentDirect 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 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
Let's consider next 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. When the number of buns increases by $4$ times ($8$ buns), they 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.
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 the equality of the values of x and y in general case can take 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 turn out when positive, the other - when negative values 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.
Example
1.6 / 2 = 0.8; 4 / 5 = 0.8; 5.6 / 7 = 0.8, etc.Proportionality factor
A constant relationship of proportional quantities is called proportionality factor. The proportionality coefficient shows how many units of one quantity are per unit of another.
Direct proportionality
Direct proportionality- functional dependence, in which a certain quantity depends on another quantity in such a way that their ratio remains constant. In other words, these variables change proportionally, in equal shares, that is, if the argument changes twice in any direction, then the function also changes twice in the same direction.
Mathematically, direct proportionality is written as a formula:
f(x) = ax,a = const
Inverse proportionality
Inverse proportionality- this is a functional dependence, in which an increase in the independent value (argument) causes a proportional decrease in the dependent value (function).
Mathematically, inverse proportionality is written as a formula:
Function properties:
Sources
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