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Today we will look at what quantities are called inversely proportional, what the inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside the school walls.
Such different proportions
Proportionality name two quantities that are mutually dependent on each other.
Dependence can be direct and reverse. Therefore, the relationship between quantities describe 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 preparing for exams, the higher your grades will be. Or the more things you take with you on a hike, the harder it is to carry your backpack. Those. the amount of effort spent on preparing for exams is directly proportional to the grades received. 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 independent value(it is called an argument) causes a proportional (i.e., the same number of times) increase or decrease in the dependent value (it is called a function).
Illustrate simple example. You want to buy apples in the market. The apples on the counter and the amount of money in your wallet are inversely related. 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. Wherein 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; +∞) .
- It has no maximum or minimum values.
- Is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If a k> 0 (that is, the argument increases), the function decreases proportionally on each of its intervals. If a k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument increases ( k> 0) the negative values of the function are in the interval (-∞; 0), and the positive values are in the interval (0; +∞). When the argument is decreasing ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).
The graph of the inverse proportionality function is called a hyperbola. Depicted as follows:
Inverse Proportional Problems
To make it clearer, let's look at a few tasks. They are not too complicated, and their solution will help you visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Task number 1. The car is moving at a speed of 60 km/h. It took him 6 hours to reach 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 of time, distance and speed: t = S/V. Agree, it very much reminds us of the inverse proportionality function. And it indicates that the time that the car spends on the road, and the speed with which it moves, are inversely proportional.
To verify this, let's find V 2, which, by condition, is 2 times higher: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is not difficult to find out the time t 2 that is required from us according to the condition of the problem: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are indeed inversely proportional: with a speed 2 times higher than the original one, the car will spend 2 times less time on the road.
The solution to this problem can also be written as a proportion. Why do we create a diagram like this:
↓ 60 km/h – 6 h
↓120 km/h – x h
Arrows indicate an inverse relationship. And they also suggest that when drawing up the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. Where do we get x \u003d 60 * 6/120 \u003d 3 hours.
Task number 2. The workshop employs 6 workers who cope with a given amount of work in 4 hours. If the number of workers is halved, how long will it take for the remaining workers to complete the same amount of work?
We write 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 \u003d 6 * 4/3 \u003d 8 hours. If there are 2 times fewer workers, the rest will spend 2 times more time to complete all the work.
Task number 3. Two pipes lead to the pool. Through one pipe, water enters at a rate of 2 l / s and fills the pool in 45 minutes. Through another pipe, the pool will be filled in 75 minutes. How fast does water enter the pool through this pipe?
To begin with, we will bring all the quantities given to us according to the condition of the problem to the same units of measurement. To do this, we express the filling rate of the pool in liters per minute: 2 l / s \u003d 2 * 60 \u003d 120 l / min.
Since it follows from the condition that the pool is filled more slowly through the second pipe, it means that the rate of water inflow is lower. On the face of inverse proportion. Let us express the speed unknown to us in terms of x and draw up the following scheme:
↓ 120 l/min - 45 min
↓ x l/min – 75 min
And then we will make a proportion: 120 / x \u003d 75/45, from where x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the filling rate of the pool is expressed in liters per second, let's bring our answer to the same form: 72/60 = 1.2 l/s.
Task number 4. Business cards are printed in a small private printing house. An employee of the printing house works at a speed of 42 business cards per hour and works full time - 8 hours. If he worked faster and printed 48 business cards per hour, how much sooner could he go home?
We go in a proven way and draw up a scheme according to the condition of the problem, denoting the desired value as x:
↓ 42 business cards/h – 8 h
↓ 48 business cards/h – xh
Before us back proportional dependence: how many times more business cards an employee of a printing house prints per hour, the same amount of time it will take him to complete the same job. Knowing this, we can set up the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 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 consider them so. And most importantly, knowledge of the inversely proportional dependence of quantities can really be useful to you more than once.
Not only in math classes and exams. But even then, when you are going to go on a trip, go shopping, decide to earn some money during the holidays, etc.
Tell us in the comments what examples of inverse and direct proportionality you notice around you. Let this be a game. You'll see how exciting it is. Don't forget to share this article in social networks so that your friends and classmates can also play.
blog.site, with full or partial copying of the material, a link to the source is required.
You can talk endlessly about the advantages of learning with the help of video lessons. First, they express thoughts clearly and understandably, consistently and structured. Secondly, they take a certain fixed time, are not, often stretched and tedious. Thirdly, they are more exciting for students than the usual lessons to which they are accustomed. You can view them in a relaxed atmosphere.
In many tasks from the mathematics course, students in grade 6 will encounter direct and inverse proportionality. Before starting the study of this topic, it is worth remembering what proportions are and what basic property they have.
The topic “Proportions” is devoted to the previous video lesson. This one is a logical continuation. It is worth noting that the topic is quite important and often encountered. It should be properly understood once and for all.
To show the importance of the topic, the video tutorial starts with a task. The condition appears on the screen and is announced by the announcer. The data recording is given in the form of a diagram so that the student viewing the video recording can understand it as best as possible. It would be better if for the first time 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 given in the same unit of measure. Otherwise, it was necessary to bring them to the same dimension.
After viewing the solution method in the video, there should not be any difficulties in such tasks. The announcer comments on each move, explains all the actions, recalls the studied material that is used.
Immediately after watching the first part of the video lesson “Direct and inverse proportional relationships”, you can offer the student to solve the same problem without the help of prompts. After that, an alternative task can be proposed.
Depending on the mental abilities student, you can gradually increase the complexity of subsequent tasks.
After the first problem considered, the definition is given directly proportional values. 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 these concepts in a notebook. If necessary before control work, the student can easily find all the rules and definitions and reread.
After watching this video, a 6th grader will understand how to use proportions in certain tasks. This is an important topic that should not be missed in any case. If the student is not adapted to perceive the material presented by the teacher during the lesson among other students, then such learning resources will be a great salvation!
g) the age of the person and the size of his shoes;
h) the volume of the cube and the length of its edge;
i) the perimeter of the square and the length of its side;
j) a fraction and its denominator, if the numerator does not change;
k) a fraction and its numerator, if the denominator does not change.
Solve problems 767-778 by compiling .
767. A steel ball with a volume of 6 cm 3 has a mass of 46.8 g. What is the mass of a ball of the same steel if its volume is 2.5 cm 3?
768. 5.1 kg of oil were obtained from 21 kg of cottonseed. How much oil will be obtained from 7 kg of cottonseed?
769. 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?
770. For the transportation of cargo, 24 cars with a lifting capacity of 7.5 tons were required. How many cars with a carrying capacity of 4.5 tons are needed to transport the same cargo?
771. To determine the germination of seeds, peas were sown. Out of 200 peas sown, 170 sprouted. What percentage of peas germinated (germination rate)?
772. Linden trees were planted on the street during Sunday Sunday for the landscaping of the city. 95% of all planted lindens were accepted. How many lindens were planted if 57 lindens were taken?
773. There are 80 students in the ski section. Among them, 32 girls. Which members of the section are girls and which are boys?
774. According to the plan, the collective farm is to sow 980 hectares with corn. But the plan was fulfilled by 115%. How many hectares of corn did the collective farm sow?
775. For 8 months, the worker completed 96% of the annual plan. What percentage of the annual plan will the worker fulfill in 12 months if he works with the same productivity?
776. In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of all beetroot if you work at the same capacity?
777. In iron ore 7 parts of iron account for 3 parts of impurities. How many tons of impurities are in an ore that contains 73.5 tons of iron?
778. To prepare borsch for every 100 g of meat, you need to take 60 g of beets. How many beets should be taken for 650 g of meat?
P 779. Calculate orally:
780. Express as the sum of two fractions with numerator 1 each of the following fractions: 
.
781. From the numbers 3, 7, 9 and 21 make two correct proportions.
782. Middle terms of proportion 6 and 10. What can be extreme terms? Give examples.
783. At what value of x is the proportion true:
784. Find the relation:
a) 2 min to 10 s; c) 0.1 kg to 0.1 g; e) 3 dm 3 to 0.6 m 3.
b) 0.3 m 2 to 0.1 dm 2; d) 4 hours to 1 day;
1) 6,0008:2,6 + 4,23 0,4;
2) 2,91 1,2 + 12,6288:3,6.
D 795. From 20 kg of apples, 16 kg of applesauce is obtained. ^^ How much applesauce will be made from 45 kg of apples?
796. Three painters can finish the job in 5 days. To speed up the work, two more painters were added. How long will it take them to finish the job, assuming that all the painters will work with the same productivity?
797. For 2.5 kg of lamb they paid 4.75 rubles. How much lamb can be bought for the same price for 6.65 rubles?
798. Sugar beets contain 18.5% sugar. How much sugar is contained in 38.5 tons of sugar beets? Round your answer to tenths of a ton.
799. Sunflower seeds of a new variety contain 49.5% oil. How many kilograms of such seeds should be taken to contain 29.7 kg of oil?
800. 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such potatoes.
801. Flax seeds contain 47% oil. How much oil is in 80 kg of flax seeds?
802. Rice contains 75% starch and barley 60%. How much barley should be taken so that it contains as much starch as 5 kg of rice contains?
803. Find the value of the expression:
a) 203.81: (141 -136.42) + 38.4: 0.7 5;
b) 96:7.5 + 288.51:(80 - 76.74).
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for Grade 6, Textbook for high school
Today we will look at what quantities are called inversely proportional, what the inverse proportionality graph looks like, and how all this can be useful to you not only in mathematics lessons, but also outside the school walls.
Such different proportions
Proportionality name two quantities that are mutually dependent on each other.
Dependence can be direct and reverse. Therefore, the relationship between quantities describe 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 preparing for exams, the higher your grades will be. Or the more things you take with you on a hike, the harder it is to carry your backpack. Those. the amount of effort spent on preparing for exams is directly proportional to the grades received. 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 of an independent value (it is called an argument) causes a proportional (i.e., by the same amount) increase or decrease in a dependent value (it is called a function).
Let's illustrate with a simple example. You want to buy apples in the market. The apples on the counter and the amount of money in your wallet are inversely related. Those. the more apples you buy, the less money you have left.
Function and its graph
The inverse proportionality function can be described as y = k/x. Wherein 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; +∞) .
- It has no maximum or minimum values.
- Is odd and its graph is symmetrical about the origin.
- Non-periodic.
- Its graph does not cross the coordinate axes.
- Has no zeros.
- If a k> 0 (that is, the argument increases), the function decreases proportionally on each of its intervals. If a k< 0 (т.е. аргумент убывает), функция пропорционально возрастает на каждом из своих промежутков.
- As the argument increases ( k> 0) the negative values of the function are in the interval (-∞; 0), and the positive values are in the interval (0; +∞). When the argument is decreasing ( k< 0) отрицательные значения расположены на промежутке (0; +∞), положительные – (-∞; 0).
The graph of the inverse proportionality function is called a hyperbola. Depicted as follows:
Inverse Proportional Problems
To make it clearer, let's look at a few tasks. They are not too complicated, and their solution will help you visualize what inverse proportion is and how this knowledge can be useful in your everyday life.
Task number 1. The car is moving at a speed of 60 km/h. It took him 6 hours to reach 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 of time, distance and speed: t = S/V. Agree, it very much reminds us of the inverse proportionality function. And it indicates that the time that the car spends on the road, and the speed with which it moves, are inversely proportional.
To verify this, let's find V 2, which, by condition, is 2 times higher: V 2 \u003d 60 * 2 \u003d 120 km / h. Then we calculate the distance using the formula S = V * t = 60 * 6 = 360 km. Now it is not difficult to find out the time t 2 that is required from us according to the condition of the problem: t 2 = 360/120 = 3 hours.
As you can see, travel time and speed are indeed inversely proportional: with a speed 2 times higher than the original one, the car will spend 2 times less time on the road.
The solution to this problem can also be written as a proportion. Why do we create a diagram like this:
↓ 60 km/h – 6 h
↓120 km/h – x h
Arrows indicate an inverse relationship. And they also suggest that when drawing up the proportion, the right side of the record must be turned over: 60/120 \u003d x / 6. Where do we get x \u003d 60 * 6/120 \u003d 3 hours.
Task number 2. The workshop employs 6 workers who cope with a given amount of work in 4 hours. If the number of workers is halved, how long will it take for the remaining workers to complete the same amount of work?
We write 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 \u003d 6 * 4/3 \u003d 8 hours. If there are 2 times fewer workers, the rest will spend 2 times more time to complete all the work.
Task number 3. Two pipes lead to the pool. Through one pipe, water enters at a rate of 2 l / s and fills the pool in 45 minutes. Through another pipe, the pool will be filled in 75 minutes. How fast does water enter the pool through this pipe?
To begin with, we will bring all the quantities given to us according to the condition of the problem to the same units of measurement. To do this, we express the filling rate of the pool in liters per minute: 2 l / s \u003d 2 * 60 \u003d 120 l / min.
Since it follows from the condition that the pool is filled more slowly through the second pipe, it means that the rate of water inflow is lower. On the face of inverse proportion. Let us express the speed unknown to us in terms of x and draw up the following scheme:
↓ 120 l/min - 45 min
↓ x l/min – 75 min
And then we will make a proportion: 120 / x \u003d 75/45, from where x \u003d 120 * 45/75 \u003d 72 l / min.
In the problem, the filling rate of the pool is expressed in liters per second, let's bring our answer to the same form: 72/60 = 1.2 l/s.
Task number 4. Business cards are printed in a small private printing house. An employee of the printing house works at a speed of 42 business cards per hour and works full time - 8 hours. If he worked faster and printed 48 business cards per hour, how much sooner could he go home?
We go in a proven way and draw up a scheme according to the condition of the problem, denoting the desired value as x:
↓ 42 business cards/h – 8 h
↓ 48 business cards/h – xh
Before us is an inversely proportional relationship: how many times more business cards an employee of a printing house prints per hour, the same amount of time it will take him to complete the same job. Knowing this, we can set up the proportion:
42/48 \u003d x / 8, x \u003d 42 * 8/48 \u003d 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 consider them so. And most importantly, knowledge of the inversely proportional dependence of quantities can really be useful to you more than once.
Not only in math classes and exams. But even then, when you are going to go on a trip, go shopping, decide to earn some money during the holidays, etc.
Tell us in the comments what examples of inverse and direct proportionality you notice around you. Let this be a game. You'll see how exciting it is. Do not forget to "share" this article on social networks so that your friends and classmates can also play.
site, with full or partial copying of the material, a link to the source is required.
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 to activity(Organizing time)
- Guys! Today in the lesson we will get acquainted with the problems solved using proportions.
II. Updating knowledge and fixing difficulties in activities
2.1. oral work (3 min)
- Find the meaning of expressions and find out the word encrypted in the answers.
14 - s; 0.1 - and; 7 - l; 0.2 - a; 17 - in; 25 - to
- The word came out - strength. Well done!
- The motto of our lesson today: Power is in knowledge! I'm looking - so I'm learning!
- Make a proportion of the resulting numbers. (14:7=0.2:0.1 etc.)
2.2. Consider the relationship between known quantities (7 min)
- the path traveled by the car at a constant speed, and the time of its movement: S = v t ( with an increase in speed (time), the path increases);
- the speed of the car and the time spent on the road: v=S:t(with an increase in the time to travel the path, the speed decreases);
–
the cost of goods purchased at one price and its quantity:
C \u003d a n (with an increase (decrease) in price, the cost of purchase increases (decreases);
- the price of the product and its quantity: a \u003d C: n (with an increase in quantity, the price decreases)
- the area of the rectangle and its length (width): S = a b (with an increase in the length (width), the area increases;
- the length of the rectangle and the width: a = S: b (with an increase in the length, 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 \u003d A: n (with an increase in the number of workers, the time spent on doing work decreases), etc.
We have obtained dependencies in which, with an increase in one value several times, the other immediately increases by the same amount (shown with arrows for examples) and dependencies in which, with an increase in one value several times, the second value decreases by the same number of times.
Such relationships are called direct and inverse proportions.
Directly proportional dependence- a dependence in which with an increase (decrease) in one value several times, the second value increases (decreases) by the same amount.
Inverse proportional relationship- a dependence in which with an increase (decrease) in one value several times, the second value decreases (increases) by the same amount.
III. Statement of the learning task
What is the problem we are facing? (Learn to distinguish between direct and inverse relationships)
- It - goal our lesson. Now formulate topic lesson. (Direct and inverse proportionality).
- Well done! Write the topic of the lesson in your notebooks. (The teacher writes the topic on the blackboard.)
IV. "Discovery" of new knowledge(10 min)
Let's analyze problems number 199.
1. The printer prints 27 pages in 4.5 minutes. How long will it take to print 300 pages?
27 pages - 4.5 min.
300 pp. - x?
2. There are 48 packs of tea in a box, 250 g each. How many packs of 150g will come out of this tea?
48 packs - 250 g.
X? - 150 g.
3. The car drove 310 km, having spent 25 liters of gasoline. How far can a car travel on a full tank of 40 liters?
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 will make 215 revolutions?
32 teeth - 315 rpm
40 teeth - x?
To draw up a proportion, one direction of the arrows is necessary, for this, in inverse proportion, one ratio is replaced by the inverse.
At the blackboard, students find the value of the quantities, in the field, students solve one problem of their choice.
– Formulate a rule for solving problems with direct and inverse proportionality.
A table appears on the board:
V. Primary consolidation in external speech(10 min)
Tasks on the sheets:
- From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
- For the construction of the stadium, 5 bulldozers cleared the site in 210 minutes. How long would it take 7 bulldozers to clear this area?
VI. Independent work with self-test according to the standard(5 minutes)
Two students complete assignments No. 225 on their own on hidden boards, and the rest in notebooks. Then they check the work according to the algorithm and compare it with the solution on the board. Errors are corrected, their causes are clarified. If the task is completed, right, then next to the students put a “+” sign for themselves.
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 blackboard. After 3–4 minutes, the students who worked at the blackboard present their solutions, and the rest check the tasks and participate in their discussion.
VIII. Reflection of activity (the result of the lesson)
- What new did you learn at the lesson?
- What did you repeat?
What is the algorithm for solving proportion problems?
Have we reached our goal?
- How do you rate your work?
