Directly and inversely proportional quantities. Direct and inverse proportional relationships
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
Wikimedia Foundation. 2010.
- Newton's second law
- Coulomb barrier
See what “Direct proportionality” is in other dictionaries:
direct proportionality- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN direct ratio ... Technical Translator's Guide
direct proportionality- tiesioginis proporcingumas statusas T sritis fizika atitikmenys: engl. direct proportionality vok. direkte Proportionalität, f rus. direct proportionality, f pranc. proportionnalité directe, f … Fizikos terminų žodynas
PROPORTIONALITY- (from Latin proportionalis proportionate, proportional). Proportionality. Dictionary foreign words, included in the Russian language. Chudinov A.N., 1910. PROPORTIONALITY lat. proportionalis, proportional. Proportionality. Explanation 25000... ... Dictionary of foreign words of the Russian language
PROPORTIONALITY- PROPORTIONALITY, proportionality, plural. no, female (book). 1. abstract noun to proportional. Proportionality of parts. Body proportionality. 2. Such a relationship between quantities when they are proportional (see proportional ... Dictionary Ushakova
Proportionality- Two mutually dependent quantities are called proportional if the ratio of their values remains unchanged. Contents 1 Example 2 Proportionality coefficient ... Wikipedia
PROPORTIONALITY- PROPORTIONALITY, and, female. 1. see proportional. 2. In mathematics: such a relationship between quantities in which an increase in one of them entails a change in the other by the same amount. Straight line (with a cut with an increase in one value... ... Ozhegov's Explanatory Dictionary
proportionality- And; and. 1. to Proportional (1 value); proportionality. P. parts. P. physique. P. representation in parliament. 2. Math. Dependence between proportionally changing quantities. Proportionality factor. Direct line (in which with... ... encyclopedic Dictionary
Basic goals:
- introduce the concept of direct and inverse proportional dependence of quantities;
- teach how to solve problems using these dependencies;
- promote the development of problem solving skills;
- consolidate the skill of solving equations using proportions;
- repeat the steps with ordinary and decimals;
- develop students' logical thinking.
DURING THE CLASSES
I. Self-determination for activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with problems solved using proportions.
II. Updating knowledge and recording difficulties in activities
2.1. Oral work (3 min)
– Find the meaning of the expressions and find out the word encrypted in the answers.
14 – s; 0.1 – and; 7 – l; 0.2 – a; 17 – c; 25 – to
– The resulting word is strength. Well done!
– The motto of our lesson today: Power is in knowledge! I'm searching - that means I'm learning!
– Make up a proportion from the resulting numbers. (14:7 = 0.2:0.1 etc.)
2.2. Let's consider the relationship between the quantities we know (7 min)
– the distance covered by the car at a constant speed, and the time of its movement: S = v t ( with increasing speed (time), the distance increases);
– vehicle speed and time spent on the journey: v=S:t(as the time to travel the path increases, the speed decreases);
–
the cost of goods purchased at one price and its quantity:
C = a · n (with an increase (decrease) in price, the purchase cost increases (decreases));
– price of the product and its quantity: a = C: n (with an increase in quantity, the price decreases)
– area of the rectangle and its length (width): S = a · b (with increasing length (width), the area increases;
– rectangle length and width: a = S: b (as the length increases, the width decreases;
– the number of workers performing some work with the same labor productivity, and the time it takes to complete this work: t = A: n (with an increase in the number of workers, the time spent on performing the work decreases), etc.
We have obtained dependences in which, with an increase in one quantity several times, another immediately increases by the same amount (examples are shown with arrows) and dependences in which, with an increase in one quantity several times, the second quantity decreases by the same number of times.
Such dependencies are called direct and inverse proportionality.
Directly proportional dependence– a relationship in which as one value increases (decreases) several times, the second value increases (decreases) by the same amount.
Inversely proportional relationship– a relationship in which as one value increases (decreases) several times, the second value decreases (increases) by the same amount.
III. Setting a learning task
– What problem is facing us? (Learn to distinguish between direct and inverse dependencies)
- This - target our lesson. Now formulate topic lesson. (Direct and inverse proportional relationship).
- Well done! Write down the topic of the lesson in your notebooks. (The teacher writes the topic on the board.)
IV. "Discovery" of new knowledge(10 min)
Let's look at problem No. 199.
1. The printer prints 27 pages in 4.5 minutes. How long will it take it to print 300 pages?
27 pages – 4.5 min.
300 pages - x?
2. The box contains 48 packs of tea, 250 g each. How many 150g packs of this tea will you get?
48 packs – 250 g.
X? – 150 g.
3. The car drove 310 km, using 25 liters of gasoline. How far can a car travel on a full 40L tank?
310 km – 25 l
X? – 40 l
4. One of the clutch gears has 32 teeth, and the other has 40. How many revolutions will the second gear make while the first one makes 215 revolutions?
32 teeth – 315 rev.
40 teeth – x?
To compile a proportion, one direction of the arrows is necessary; for this, in inverse proportionality, one ratio is replaced by the inverse.
At the board, students find the meaning of quantities; on the spot, students solve one problem of their choice.
– Formulate a rule for solving problems with direct and inverse proportional dependence.
A table appears on the board:
V. Primary consolidation in external speech(10 min)
Worksheet assignments:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
- To build the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this site?
VI. Independent work with self-test against standard(5 minutes)
Two students complete task No. 225 independently on hidden boards, and the rest - in notebooks. They then check the algorithm's work and compare it with the solution on the board. Errors are corrected and their causes are determined. If the task is completed correctly, then the students put a “+” sign next to them.
Students who make mistakes in independent work can use consultants.
VII. Inclusion in the knowledge system and repetition№ 271, № 270.
Six people work at the board. After 3-4 minutes, students working at the board present their solutions, and the rest check the assignments and participate in their discussion.
VIII. Reflection on activity (lesson summary)
– What new did you learn in the lesson?
-What did they repeat?
– What is the algorithm for solving proportion problems?
– Have we achieved our goal?
– How do you evaluate your work?
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.
How more books read the student, 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 when constant speed, the time will increase by the same amount:
$\frac(2S)(v)=2t$;
$\frac(3S)(v)=3t$.
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.
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 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:
- Its domain of definition is the set of all real numbers except x = 0. D(y): (-∞; 0) U (0; +∞).
- The range is all real numbers except y= 0. E(y): (-∞; 0) U (0; +∞) .
- Does not have maximum or minimum values.
- It is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not intersect the coordinate axes.
- Has no zeros.
- 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.
- As the argument increases ( k> 0) negative values functions are in the interval (-∞; 0), and positive ones are (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 proportions right side the records 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 it follows from the condition 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 tasks are inverse proportionality 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 in social networks so that your friends and classmates can also play.
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