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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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 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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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.
Problem solution 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. For 2 hours she fills 1/5 swimming pool What part of the pool is filled with water in 5 hours?
Solution.
We answer the question of the task: for 5 hours will be filled 1/x part of the pool. (The entire pool is taken as one whole).
