Class: 6
In my work, I use different forms and methods of teaching, I try to use a variety of organizational techniques. learning activities to keep students interested in learning. Only in this case, the cognitive activity of students increases, thinking begins to work more productively and creatively. One of the means of increasing interest in the subject is the use of information technology.
The use of computer technology in the classroom allows you to continuously change the forms of work, constantly alternate oral and written exercises, implement different approaches to solving mathematical problems, and this constantly creates and maintains the intellectual tension of students, forms their steady interest in studying this subject.
Group work in the classroom stimulates the cognitive activity of students, promotes their involvement in creative activities and communication. In the process of individual work, students themselves strive to solve problems, education turns into self-education.
Performing creative tasks contributes to the application of school knowledge in real life situations.
Lesson type: combined lesson
Lesson Objectives:
- cognitive:
- to ensure the conscious assimilation by students of the concept of direct and inverse proportionality in solving problems;
- check the level of knowledge on a given topic through various forms of work.
- Educational:
- to activate the mental activity of students through the participation of each of them in the process of work;
- develop attention, memory, intellectual and creative abilities;
- develop the emotional sphere of students in the learning process;
- develop control and self-control.
- Educational:
- to form a sense of cooperation, mutual assistance;
- to form practical skills;
- generate interest in the subject being studied.
Lesson plan:
- Organizational moment (2 min.)
- Mental account (4 min.)
- Analysis of problems solved by students (5 min.)
- Physical education (2 min.)
- Consolidation of the studied material, group work (16 min.)
- Independent work (13 min.)
- Summing up the lesson (2 min.)
- Homework(1 min.)
DURING THE CLASSES
1. Organizational moment
Mutual greeting, recording the topic of the lesson. Organization of work with self-control cards.
2. Repetition of material
a) The solution by two students on the board of problems for direct and inverse proportionality
b) the rest verbally repeat the basic concepts:
- what are the names of the numbers x and y in the proportion x: a = b: y?
- the equality of two relations is called ...
- What is a direct proportional relationship?
- what kind of relationship is inversely proportional?
- one hundredth of a number is...
Work with self-control cards (maximum number of points - 1).
3. Mental account
1. The game "Silent"
a) Which of the equalities can be called proportions?
If the proportion is correct, then the students raise the green cards, if not, then the red ones.
b) Are the following relationships directly or inversely proportional?
1) the number of readers from the number of books in the library;
2) the path traveled by the car at a constant speed and time of its movement;
3) the age of the person and the size of his shoes;
4) the perimeter of the square and the length of its sides;
5) speed and time during the passage of the same section of the path.
If the statement is true, then the students raise the green cards, if not, then the red ones.
Work with self-control cards (maximum score for oral score 2).
2. Analysis of problems solved by students on the board.
a) A swallow flew some distance in 0.5 hours at a speed of 50 km/h. In how many minutes will a swift fly the same distance if its speed is 100 km / h?
Decision:
Let x hours be the flight time of the swift.
| 50 km/h - 0.5 h 100 km/h - X h |
0.25 h = 25/100 = 1/4 h = 15 min.
Answer: 15 minutes.
b) Beets were brought to the sugar factory, from which 12% of sugar is obtained. How much sugar will be obtained from 30 tons of beets of this variety?
Decision:
Let x tons of sugar come out.
Answer: 3.6 tons
4. Physical education
5. Group work
You have cards on the tables. They have 4 tasks. Groups 1, 3, 5 decide starting with #1. Groups 2, 4, 6 decide starting with #4 (in reverse order).
1) 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such a potato.
Decision:
Let x% of starch be found in potatoes.
17.5% is starch.
Answer: 17, 5 %
2) You can swim from one village to another along the river in 1.5 hours. How long will it take for a motor boat to make this journey if the speed of the boat is 3 km/h and the speed of the boat is 13.5 km/h?
Decision:
Let x hours be the time of the boat
| 3 km/h 13.5 km/h |
– 1.5 h – X h |
Answer: 20 minutes
3) When cleaning sunflower seeds, 28% is the husk. How much pure grain will be obtained from 150 tons of sunflower seeds?
Decision:
Let x t grains turn out.
150 - 42 = 108 (t)
108 tons of grain.
Answer: 108 tons
4) It took 48 cars with a carrying capacity of 7.5 tons to transport cargo. How many cars with a carrying capacity of 4.5 tons are needed to transport the same cargo?
Decision:
Let x cars be taken with a carrying capacity of 4.5 tons.
Answer: 80 cars.
Checking the solution of problems on the board.
Work with self-control cards (maximum number of points - 8; each task 2 points)
5. Individual independent work 4 options.
I option
1) Dad paid 48 rubles for 4 identical boxes of pencils. How much do 7 of these boxes of pencils cost?
2) Three students weeded the garden in 4 hours. How many hours will it take 2 students to complete the same task?
II option
1) When cooking meat, 65% of the mass remains. How much boiled meat will be obtained from 2 kg of raw meat?
2) Four masons can complete the job in 15 days. In how many days can three masons complete this work?
III option
1) Linden blossom loses 74% of its weight. How much dry lime blossom can be obtained from 300 kg of fresh?
2) A motorcyclist traveled 3 hours at a speed of 60 km/h. How many hours will it take him to travel the same distance at a speed of 45 km/h?
IV option
1) Cuban farmers offer us sugar cane to produce sugar. Sugarcane, when processed into sugar, loses 91% of its original mass. How much sugar cane does it take to get 900 kg of sugar?
2) On a hot day, 6 mowers drank a barrel of kvass in 1.5 hours. How many mowers will drink the same barrel in 3 hours?
7. Summing up the lesson
What types of problems did we solve in class?
Students sum up the lesson in self-control cards and give grades
16-17 points - "5"
13-15 points - "4"
9-12 points - "3"
– The objectives of the lesson were achieved, and most importantly, the work was carried out in a creative atmosphere.
8. Homework
Repeat steps 13-18.
Textbook task: No. 817, No. 812, differentiated No. 818.
Literature
- Mathematics textbook grade 6 educational institutions, authors: N. Ya. Vilenkin, V. I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd, Moscow. "Mnemosyne", 2011.
- Collection test items for thematic and final control Mathematics 6th grade Moscow, "Intellect Center" 2009.
- A. I. Ershova, V.V. Goloborodko. Mathematics 6. Independent and test papers.– M: Ileksa, 2011.
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Slides captions:
"Direct and inverse proportional dependencies" Grade 6 Mathematics teacher of the MAOU "Kurovskaya secondary school No. 6" Chugreeva T.D.
Mathematics is the basis and queen of all sciences, And I advise you to make friends with it, my friend. If you follow her wise laws, You will increase your knowledge, You will begin to apply them. Can you swim in the sea, Can you fly in space. You can build a house for people: It will stand for a hundred years. Do not be lazy, work, try, Knowing the salt of sciences Try to prove everything, But tirelessly.
Finish the phrase: 1. A direct proportional relationship is such a dependence of quantities at which ... 2. An inverse proportional relationship is such a dependence of quantities at which ... 3. To find the unknown extreme member of the proportion ... 4. The middle member of the proportion is ... 5. The proportion is correct, if ... C) ... when one value increases several times, the other decreases by the same amount. X) ... the product of the extreme terms is equal to the product of the middle terms of the proportion. A) ... when one value is increased several times, the other increases by the same amount. P) ... you need to divide the product of the middle terms of the proportion by the known extreme term. Y) ... when one value is increased several times, the other increases by the same amount. E) ... the ratio of the product of the extreme terms to the known mean.
The growth of the child and his age are directly proportional. 2. With a constant width of a rectangle, its length and area are directly proportional. 3. If the area of a rectangle is a constant value, then its length and width are inversely proportional. 4. The speed of the car and the time of its movement are inversely proportional.
5. The speed of the car and its distance traveled are inversely proportional. 6. The income of the cinema box office is directly proportional to the number of tickets sold, sold at the same price. 7. Carrying capacity of machines and their number are inversely proportional. 8. The perimeter of a square and the length of its side are directly proportional. 9. At a constant price, the cost of a commodity and its mass are inversely proportional.
Come on, pencils aside! No papers, no pens, no chalk! Verbal counting! We do this business Only by the power of the mind and soul! VERBAL COUNTING
Find the unknown term of the proportion? ? ? ? ? ? ?
"DIRECT PROPORTIONAL DEPENDENCE" LESSON TOPIC AND INVERSE
a) A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed? b) 8 identical pipes fill the pool in 25 minutes. How many minutes will it take 10 such pipes to fill the pool? c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days, working at the same productivity? d) From 5.6 kg of tomatoes, 2 liters of tomato sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes? Make proportions for solving problems:
Answers: a) 3:x=75:125 b) 8:10= X:2 5 c) 8: x=10: 15 d) 5.6:54=2: X
To heat the school building, coal was harvested for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this reserve last if it is spent daily at 0.5 tons? Solve the problem
Short record: Mass (t) for 1 day Number of days At the rate of 0.6 180 0.5 x Let's make a proportion: ; ; Answer: 216 days. Decision.
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? #793 Solve the problem
Number of parts Mass Iron 7 73.5 Impurities 3 x; Answer: 31.5 kg of impurities. Decision. ; №793
An unknown number is denoted by the letter x. The condition is written in the form of a table. The type of dependence between quantities is established. Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows. The proportion is recorded. An unknown member is located. Algorithm for solving problems for direct and inverse proportionality:
Solve the equation:
No. 1. On the way from one village to another at a speed of 12.5 km / h, the cyclist spent 0.7 hours. At what speed did he have to go to cover this path in 0.5 hours? No. 2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes will be obtained from 17.5 kg of fresh plums? No. 3. The car drove 500 km, having spent 35 liters of gasoline. How many liters of gasoline do you need to travel 420 km? No. 4. 12 crucians were caught in 2 hours. How many carp will be caught in 3 hours? #5 Six painters can do some work in 18 days. How many more painters need to be invited to complete the job in 12 days? Independent work Solve problems by making proportions.
Solving problems from independent work Solution: No. 1 Short entry: Speed (km / h) Time (h) 12.5 0.7 x 0.5 Answer: 17.5 km / h Solution: No. 2 Short entry: Plums (kg ) Prunes (kg) 5 1.5 17.5 x; ; kg Answer: 5.25 kg; ; ;
Solving problems from independent work Solution: No. 3 Solution: No. 5 Brief record: Brief record: Distance (km) Gasoline (l) 500 35 420 x; Answer: 29.4 liters. Number of babies Time (days) 6 18 x 12; ; painters will complete the work in 12 days. 1) 9 -6 = 3 painters still need to be invited. Answer: 3 painters.
Additional task: #6. A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles. for one. How many such cars can the enterprise buy if the price for one car becomes 15 thousand rubles? Decision: No. 1 Brief entry: Number of cars (pcs) Price (thousand rubles) 5 12 x 15; cars. ; Answer: 4 cars.
Home rear No. 812 No. 816 No. 818
Thank you for the lesson!
Preview:
Chugreeva Tatyana Dmitrievna 206818644
Math lesson in 6th grade
on the topic "Direct and inverse proportional relationships"
Developed
mathematic teacher
MAOU "Kurovskaya secondary school No. 6"
Chugreeva Tatyana Dmitrievna
Lesson Objectives:
educational- update the concept of "dependence" between quantities;
Educational - through solving problems, setting additional questions and tasks to develop the creative and mental activity of students;
Independence;
self-esteem skills;
Educational- to cultivate interest in mathematics as a part of human culture.
Equipment: TCO necessary for the presentation: a computer and a projector, sheets for recording answers, cards for the reflection stage (three each), a pointer.
Lesson type: a lesson in the application of knowledge.
Lesson organization forms:frontal, collective, individual work.
During the classes
- Organizing time.
The teacher reads: (slide number 2)
Mathematics is the basis and queen of all sciences,
And I advise you to make friends with her, my friend.
Her wise laws, if you follow,
Increase your knowledge
You will be using them.
Can you swim in the sea
You can fly in space.
You can build a house for people:
It will stand for a hundred years.
Don't be lazy, work hard
Knowing the salt of sciences.
Try to prove everything
But don't give up.
2. Checking the studied material.
- Finish the sentence:(slide 3). (Children first complete the task on their own, writing down only the letters corresponding to the correct answer on the sheets. Then they raise their hand. After that, the teacher reads the question aloud, and the students answer).
- A direct proportional relationship is such a dependence of quantities in which ...
- An inverse proportional relationship is such a dependence of quantities at which ...
- To find the unknown extreme term of the proportion...
- The middle term of the proportion is...
- The proportion is correct if...
C) ... when one value increases several times, the other decreases by the same amount.
X) ... the product of the extreme terms is equal to the product of the middle terms of the proportion.
A) ... when one value is increased several times, the other increases by the same amount.
P) ... you need to divide the product of the middle terms of the proportion by the known extreme term.
Y) ... when one value is increased several times, the other increases by the same amount.
E) ... the ratio of the product of the extreme terms to the known mean.
Answer: SUCCESS. (slide 6)
- Oral counting: (slides 6-7)
Come on, pencils aside!
No papers, no pens, no chalk!
Verbal counting! We're doing this thing
Only by the power of the mind and soul!
Exercise: Find the unknown term of the proportion:
Answers: 1) 39; 24; 3; 24; 21.
2)10; 3; 13.
- The topic of the lesson. slide number 8 (Provides motivation for students to learn.)
- The theme of our lesson is "Direct and inverse proportional relationships."
- In previous lessons, we considered direct and inverse proportional dependence of quantities. Today in the lesson we will solve different problems using proportions, establishing the type of relationship between data. Let's repeat the main property of proportions. And the next lesson, concluding on this topic, i.e. lesson - control work.
- The stage of generalization and systematization of knowledge.
1) Task1.
Make proportions for solving problems:(work in notebooks)
a) A cyclist travels 75 km in 3 hours. How long will it take a cyclist to travel 125 km at the same speed?
b) 8 identical pipes fill the pool in 25 minutes. How many minutes will it take 10 such pipes to fill the pool?
c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days, working at the same productivity?
d) From 5.6 kg of tomatoes, 2 liters of tomato sauce are obtained. How many liters of sauce can be obtained from 54 kg of tomatoes?
Check answers. (Slide number 10) (self-assessment: put + or - in pencil innotebooks; analyze errors)
Answers: a) 3:x=75:125 c) 8:x=10:15
b) 8:10= X:2 5 d) 5.6:54=2: X
Solve the problem
№788 (p. 130, Vilenkin's textbook)(after parsing by yourself)
In the spring, during the greening of the city, lindens were planted on the street. 95% of the milestones of planted lindens were accepted. How many lindens were planted if 57 lindens were taken?
- Read the task.
- What two quantities are mentioned in the problem?(about the number of limes and their percentages)
- What is the relationship between these quantities?(directly proportional)
- Make a short note, proportion and solve the problem.
Decision:
Lindens (pcs.) | Percentage % |
|
planted | ||
Accepted |
; ; x=60.
Answer: 60 lindens were planted.
Solve the problem: (slide No. 11-12) (after parsing, decide on your own; mutual check, then the solution is displayed on the screen slide No. 23)
To heat the school building, coal was harvested for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this reserve last if it is spent daily at 0.5 tons?
Decision:
Brief entry:
Weight (t) for 1 day | Quantity days |
|
According to the norm | ||
Let's make a proportion:
; ; days
Answer: 216 days.
No. 793 (p. 131) (field parsing by yourself; self-control.
(Slide number 13)
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?
Solution: (slide number 14)
Quantity parts | Weight |
|
Iron | 73,5 |
|
impurities |
Answer: 31.5 kg of impurities.
So, let's formulate an algorithm for solving problems using proportions.
Algorithm for solving problems directly
and inversely proportional relationships:
- An unknown number is denoted by the letter x.
- The condition is written in the form of a table.
- The type of dependence between quantities is established.
- Directly proportional dependence is indicated by equally directed arrows, and inversely proportional dependence is indicated by oppositely directed arrows.
- The proportion is recorded.
- An unknown member is located.
Repetition of the studied material.
No. 763 (i) (p. 125) (with commentary at the board)
6. Stage of control and self-control of knowledge and methods of action.
(slide №17-19)
Independent work(10 - 15 min) (Mutual check: on the finished slides, students check each other independent work, while setting + or -. The teacher at the end of the lesson collects notebooks for review).
Solve problems by making proportions.
No. 1. On the way from one village to another at a speed of 12.5 km / h, the cyclist spent 0.7 hours. At what speed did he have to go to cover this path in 0.5 hours?
Decision:
Brief entry:
Speed (km/h) | Time (h) |
|
12,5 | ||
Let's make a proportion:
; ; km/h
Answer: 17.5 km/h
No. 2. From 5 kg of fresh plums, 1.5 kg of prunes are obtained. How many prunes will be obtained from 17.5 kg of fresh plums?
Decision:
Brief entry:
Plums (kg) | Prunes (kg) |
|
17,5 |
Let's make a proportion:
; ; kg
Answer: 5.25 kg
No. 3. The car drove 500 km, having spent 35 liters of gasoline. How many liters of gasoline do you need to travel 420 km?
Decision:
Brief entry:
Distance (km) | Gasoline (l) |
|
Let's make a proportion:
; ; l
Answer: 29.4 liters.
№4 . 12 crucians were caught in 2 hours. How many carp will be caught in 3 hours?
Answer: the answer does not exist. these quantities are neither directly proportional nor inversely proportional.
№5 Six painters can do some work in 18 days. How many more painters need to be invited to complete the job in 12 days?
Decision:
Brief entry:
Number of painters | Time (days) |
|
Let's make a proportion:
; ; painters will complete the work in 12 days.
1) 9 -6=3 painters still need to be invited.
Answer: 3 painters.
Additional (slide number 33)
No. 6. A mining enterprise needs to purchase 5 new machines for a certain amount of money at a price of 12 thousand rubles. for one. How many such cars can the enterprise buy if the price for one car becomes 15 thousand rubles?
Decision:
Brief entry:
Number of machines (pcs.) | Price (thousand rubles) |
|
Let's make a proportion:
; ; cars.
Answer: 4 cars.
- The stage of summing up the lesson
- What did we learn in the lesson?(The concepts of direct and inverse proportional dependence of two quantities)
- Give examples of directly proportional quantities.
- Give examples of inversely proportional quantities.
- Give examples of quantities whose dependence is neither directly nor inversely proportional.
- Homework (slide 21)
№ 812, 816, 818.
Thanks for the lesson slide number 22
Chapter 3 RELATIONS AND PROPORTIONS
Proportions can be used to solve problems.
You know, for example, that the value of a commodity depends on its quantity: the more a commodity is bought, the greater its value will be. Such quantities are called directly proportional.
Remember!
Two quantities are said to be directly proportional if, when one quantity increases (decreases) several times, the other quantity increases (decreases) by the same number of times.
Task 1. For 2 kg of sweets they paid 72 UAH. How much will 4.5 kg of these sweets cost?
Solutions.
Note:
if two quantities are directly proportional, then the proportion is formed by the ratio of the corresponding values of these quantities.
In practice, in addition to the direct proportional dependence of quantities, there is also an inverse proportional dependence. For example, on the way to school, when time is running out, you increase the speed of your movement so as not to be late for class. Therefore, the speed of your movement depends on the hour of movement: the shorter the time of movement, the greater your speed will be. Such quantities are called inversely proportional.
Remember!
Two quantities are called inversely proportional if, when one quantity increases (decreases) several times, the other quantity decreases (increases) by the same number of times.
Task 2. A car, moving at a speed of 90 km/h, traveled the distance from Cherkassy to Kyiv in 2 h 3 what speed did he move in the opposite direction, if he covered the distance from Kyiv to Cherkasy in 2.5 h?
Solutions.
Note:
if two quantities are inversely proportional, then the proportion is formed by the mutually inverse ratios of the corresponding values of these quantities.
Are two quantities always directly proportional or inversely proportional? Let's discuss. For example, during an illness, a child's temperature may rise and fall for several days. And here there is no dependence, which means that there can be no proportionality. But the growth of the child is constantly increasing with increasing age. Consequently, there is a relationship between the quantities, which means that there is reason to analyze proportional to these quantities. It is clear that there is no proportional dependence here, therefore, it is not necessary to find out exactly how these proportional values are directly or vice versa. If two quantities are proportional, then only two options are possible that mutually exclude each other - either direct proportionality or inverse proportionality.
Find out more
The name of the Italian mathematician monk is indirectly connected with the history of the golden section. Leonardo of Pisa (1180-1240 pp.), better known as Fibonacci (son of Bonacci).
He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting boards) was published, in which all the problems known at that time were collected. One of the tasks was: “How many pairs of rabbits will be born from one pair in one year?”. Arguing on this topic, Fibonacci built the following series of numbers:
0, 1, 1,2, 3, 5, 8, 13,21, 34,55, ... .
Now this sequence of numbers is known as the Fibonacci series. The peculiarity of this sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two:
0 + 1 = 1; 1+1 = 2; 1+2 = 3; 2 + 3 = 5;
3 + 5 = 8; 5 + 8=13; 8 + 13 = 21; 13 + 21=34
the like, and the ratio of neighboring numbers of the series approaches the ratio of the golden section. For example:
21: 34 = 0.617, a34: 55 = 0.618.
REMEMBER THE MAIN THINGS
1. What quantities are called directly proportional? Give examples.
2. How do you solve problems for direct proportionality?
3. What quantities are called inversely proportional? Give examples.
4. Do I solve inverse proportionality problems?
5. Are two quantities always proportional?
589". Two values are directly proportional. How will one value change if the other: a) increase by 5 times; b) decrease by 2 times?
Explain the answer.
590". According to the condition of the problem, they made an abbreviated record:
1)3-36, 2) 70-3, 3) 2-100,
4-48; 60-2; 4-50.
Are these quantities directly proportional?
591". Two values are inversely proportional, How will one value change if the other:
a) will increase by 4 times; b) decrease by 6 times?
Explain the answer.
592". According to the condition of the problem, they made an abbreviated record:
1) 80-4, 2)3-18, 3)10-8,
160 - 2; 5 - 30; 4 - 20.
Are these quantities inversely proportional?
593°. Determine whether this dependence of the quantities is directly proportional:
1) the cost of goods purchased at one price and the quantity of goods;
2) the mass of the box of sweets and the number of identical sweets in the box;
3) the path that the car has traveled at a constant speed, and the time of movement;
4) the speed of movement and the time of movement to overcome a certain distance;
5) the person's weight and height;
b) the mass of berries and the mass of sugar for making jam;
7) the perimeter of the rectangle and the length of one of its sides;
8) the length of the side of the square and its perimeter.
594°. From the abbreviated notation of the problem, find x if the quantities are directly proportional.
1) 3 kg of sweets -36 UAH, 2) 15 parts - 3 hours,
6 kg of sweets x; x -2 hours.
595°. How much do 10 kg of sweets cost if 128 UAH were paid for 4 kg of such sweets?
596°. For 3 kg of apples they paid 24 UAH. How much do 7 kg of these apples cost?
597°. The boat traveled 80 km in 4 hours. How far will the boat travel in 2 hours at the same speed?
598°. A tourist walked 20 km in 5 hours. How many hours does it take a tourist to cover a distance of 28 km, moving at the same speed?
599°. When baking bread from 1 kg of rye flour, 1.4 kg of bread is obtained. How much flour is needed to get 42 quintals of bread?
600°. From 3 kg of raw coffee beans, 2.5 kg of roasted beans are obtained. How many kilograms of raw coffee beans do you need to take to get 10 kg of roasted?
601°. The car traveled a distance of 210 km in 3 hours. What distance is easier for a car in 2 hours, moving at the same speed?
602°. A tailless gibbon monkey, jumping from tree to tree, covers a distance of 32 km in 2 hours. How far will a gibbon travel in 3 hours?
603°. Determine whether this dependence of the quantities is inversely proportional:
1) the price of the goods and the purchase price;
2) the mass of the box of sweets and its value;
3) the speed of movement and the time of movement to overcome a certain distance;
4) the speed of the car and the path that it traveled at a constant speed;
5) the amount of work performed and the time of its implementation;
6) labor productivity and time for its implementation of a certain amount of work;
7) the number of cars and the cargo that they will transport in a certain time;
8) the length of the side of the square and its area.
604°. Using the abbreviated notation of the problem, find x if the quantities are inversely proportional.
1) 3 h - 80 km/h, 2) 5 -8 working days,
4 h - x; x -10 days.
605°. 3 carpenters completed an order for the manufacture of furniture in 12 days. In how many days will it take 6 carpenters to complete the order if their labor productivity is the same?
606°, In how many days will 6 workers complete the task if 2 workers can complete this task in 9 days?
607°. The red kangaroo moved for 3 hours at a speed of 55 km/h. What should be the speed of a kangaroo so that it can cover this distance in 2.5 hours?
608°. What should be the speed of the train according to the new schedule in order to cover the distance between two stations in 4 hours, if, according to the old schedule, moving at a speed of 100 km/h, it covered it in 5 hours?
609. For 4 kg of cookies they paid 56 UAH. How much will 3 kg of sweets cost 2 UAH more than the price of cookies?
610. 5 kg of apples cost 40 UAH. Find the cost of 2 kg of pears, the price of which is 4 UAH more than the price of apples.
611. Wall clock pendulum makes 730 swings in 15 minutes. How many oscillations will he make in 1 hour? How long does it take for the pendulum to make 2190 oscillations?
612. Natalia paid UAH 60 for 24 notebooks. How much do 20 of these notebooks cost? How many of these notebooks can be bought for 45 UAH?
613. There are 12 liters of milk in a can. It was poured equally into 6 cans. How many liters of milk are in each jar? How many three-liter jars can be filled with milk from this can?
614. 6 liters of water flows through a water tap in a minute. How much water will run out of the faucet in half an hour? How long will it take for 27 liters of water to flow through the faucet?
615. The distance between the stations is 360 km. How long will it take a train to cover 90 km in one hour? What must be the speed of the train to cover this distance in 4 hours and 30 minutes?
616. The distance between the villages is 18 km. How much easier is the distance for a cyclist whose speed is 12 km/h? At what speed does the pedestrian need to move to cover this distance in 6 hours?
617. Two tractors plowed the field in 6 days. How many days will it take 4 tractors to dig this field if they work with the same labor productivity? How many tractors does it take to plow this field in 2 days?
618. Eight trucks can carry cargo in 3 days. In how many days will 6 such trucks be able to transport the goods? How many trucks will it take to transport this cargo in 2 days?
619. Compose and solve a problem for:
1) direct proportionality, for the solution of which you need to make a proportion
2) inverse proportionality, for the solution of which you need to make up the proportion x: 4 \u003d 120: 160.
620. Make up and solve the problem for: 1) direct proportionality, for the solution of which you need to make a proportion
2) inverse proportionality, for the solution of which it is necessary to make a proportion of 3: x \u003d 90: 60.
621*. Tarasik can walk from the railway station to the village in 20 minutes. How long will it take him to ride a bike from the station to the village, if the speed of his movement on a bicycle is 2 times greater than the speed of movement on foot?
622*. The master, working independently, completes the work in 3 days, and together with the student - in 2 days. In how many days can the student complete this work on their own?
623*. Dima runs 4 laps on the treadmill in the same time as Katya runs 3 laps. Katya ran 12 laps. How many laps did Dima run during this time?
624*. Water can be pumped out of the pool in 1 hour and 15 minutes. How long after the start of work in the pool will there be 0.2 of the amount of water that was at first?
APPLY IN PRACTICE
625. For the printing of the book, it was supposed to place 28 lines on each page, 40 letters in each line. However, it turned out that it was more expedient to place 35 lines on each page. In this case, how many letters will be placed in each line of letters during the printing of this book, if the number of letters per page does not change?
626. To prepare 12 cakes, you need to take the protein of one egg and 3 tablespoons of sugar. How many of these products should be taken to prepare 24 such stacks? How many cakes will you get if you have 3 eggs?
REPETITION TASKS
627. What number should be entered in the last cell of the chain?
628. Solve the equation:
The easiest way to understand a directly proportional relationship is to use the example of a machine that manufactures parts at a constant speed. If in two hours he makes 25 parts, then in 4 hours he will make twice as many parts - 50. How many times longer the time he will work, the same number of times more details he will produce.
Mathematically it looks like this:
4: 2 = 50: 25 or like this: 2:4 = 25:50
Directly proportional quantities here are the operating time of the machine and the number of manufactured parts.
They say: The number of parts is directly proportional to the operating time of the machine.
If two quantities are directly proportional, then the ratios of the corresponding quantities are equal. (In our example, this is the ratio of time 1 to time 2 = the ratio of the number of parts in time 1 to number of parts in time 2)
Inverse proportionality
An inversely proportional relationship is often found in speed problems. Speed and time are inversely proportional. Indeed, the faster an object moves, the less time it will take to travel.
For example: 
If the quantities are inversely proportional, then the ratio of the values of one quantity (speed in our example) is equal to the inverse ratio of the other quantity (time in our example). (In our example, the ratio of the first speed to the second speed is equal to the ratio of the second time to the first time.
Task examples
Task 1:
Decision:
Let's write a brief condition of the problem: 
Task 2:
Decision:
Brief entry:
If a CNC machine produces 28 parts in 2 hours, then in twice the time, i.e., in 4 hours, it will produce twice as many such parts, i.e. 28 2 \u003d 56 parts. How many times longer the machine will work, how many times more parts it will produce. This means that the ratios 4: 2 and 56: 28 are equal. Therefore, the proportion 4: 2 \u003d 56: 28 is correct. Such quantities as the operating time of the machine and the number of manufactured parts are called directly proportional quantities.
If two quantities are directly proportional, then the ratios of the corresponding values of these quantities are equal.
Let a train travel from city A to city B in 12 hours at a speed of 40 km/h. e. 6 hours How many times the speed of movement will increase, the time of movement will decrease by the same amount. In this case, the ratio of 80:40 will not be equal to the ratio of 6:12, but the inverse ratio of 12:6. Therefore, the correct proportion is 80:40 = 12:6. Such quantities as speed and time are called inversely proportional quantities.
If quantities are inversely proportional, then the ratio of the values of one quantity is equal to the inverse ratio of the corresponding values of the other quantity.
Not every two quantities are directly proportional or inversely proportional. For example, the height of a child increases with increasing age, but these values are not proportional, since when the age is doubled, the height of the child does not double.
Proportional problems can be solved using proportions.
Task 1. 115.2 rubles were paid for 3.2 kg of goods. How much should I pay for 1.5kg of this item?
Decision. Let us briefly write down the condition of the problem in the form of a table, denoting by the letter x the cost (in rubles) of 1.5 kg of this product.
The entry will look like this:
The relationship between the quantity of goods and the cost of purchase is directly proportional, since if you buy several times more goods, then the purchase price will increase by the same amount. We conventionally denote such a dependence by equally directed arrows.
Let's write the proportion: .
Answer: 54 p.
Problem 2. Two rectangles have the same area. The length of the first rectangle is 3.6 m and the width is 2.4 m. The length of the second rectangle is 4.8 m. Find the width of the second rectangle.
Decision. Denoting the width (in meters) of the second rectangle with the letter x, we write briefly the condition of the problem:
The relationship between the width and length for the same value of the area of the rectangle is inversely proportional, since if you increase the length of the rectangle several times, then you need to reduce the width by the same amount. We conventionally denote such a dependence by oppositely directed arrows.
Let's write the proportion:
Now let's find the unknown term of the proportion:
Answer: 1.8 m.
Questions for self-examination
- What quantities are directly proportional? What can be said about the ratios of the corresponding values of such quantities?
- Give examples of directly proportional quantities.
- What quantities are called inversely proportional? What can be said about the ratios of the corresponding values of such quantities?
- Give examples of inversely proportional quantities.
- Give examples of quantities whose dependence is neither directly nor inversely proportional.
Do the exercises
782. Determine whether the relationship between the quantities is directly proportional, inversely proportional or not proportional:
- a) the path traveled by the car at a constant speed, and the time of its movement;
- b) the cost of goods purchased at one price, and its quantity;
- c) the area of the square and the length of its side;
- d) the mass of the steel bar and its volume;
- e) the number of workers performing some work with the same labor productivity, and the time it takes to complete this work;
- f) the cost of goods and their quantity, bought for a certain amount of money;
- 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;
- l) a fraction and its numerator, if the denominator does not change.
Solve problems No. 783 - 794 by making a proportion.
783. A steel ball with a volume of b cm 3 has a mass of 46.8 g. What is the mass of a ball made of the same steel if its volume is 2.5 cm 3?
784. From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
785. 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?
786. It took 24 trucks with a carrying capacity of 7.5 tons to transport the cargo. How many trucks with a carrying capacity of 4.5 tons are needed to transport the same cargo?
787. To determine the germination of seeds, peas were sown. Out of 200 peas sown, 170 sprouted. What percentage of peas germinated (germination rate)?
788. In the spring, during the greening of the city, lindens were planted on the street. 95% of all planted lindens were accepted. How many lindens were planted if 57 lindens were taken?
789. There are 80 students in the ski section. Among them, 32 girls. What percentage of the participants in the section are girls and what percentage are boys?
790. The plant was supposed to smelt 980 tons of steel per month according to the plan. But the plan was fulfilled by 115%. How many tons of steel did the plant smelt?
791. 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?
792. In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of all beets, if you work with the same productivity?
793. 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?
794. To prepare borscht, 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?
795. Calculate orally:
796.
Express as the sum of two fractions with a numerator of 1 each of the following fractions: 
.
797. From the numbers 3, 7, 9 and 21, make two correct proportions.
798. The middle members of the proportion are 6 and 10. What can be the extreme members? Give examples.
799. At what value of x is the proportion correct:
800. Find the relation:
- a) 2 min to 10 s;
- b) 0.3 m 2 to 0.1 dm 2;
- c) 0.1 kg to 0.1 g;
- d) 4 hours to 1 day;
- e) 3 dm 3 to 0.6 m 3.
801. Where on the coordinate ray should the number c be located in order for the proportion to be correct (Fig. 34)?
Rice. 34
802. Develop your memory! Close the table with a sheet of paper. Open the first line for a few seconds and then, closing it again, try to repeat or write down the three numbers of this line. If you correctly reproduced all the numbers, go to the second row of the table. If a mistake is made in any line, write yourself several sets of the same number of two-digit numbers as in the line and practice memorizing them. If you can reproduce at least five two-digit numbers without errors, you have a good memory.
803. Solve the equation:
804. Is it possible to make the correct proportion of the following numbers:
805. From the equality of the products 3 24 \u003d 8 9, make three correct proportions.
806. The length of segment AB is 8 dm, and the length of segment CD is 2 cm. Find the ratio of the lengths of segments AB and CD. What part of the length of segment AB is the length of segment CD?
807. There are 460 vacationers in the sanatorium, of which 70% are adults, and the rest are children. How many children rested in the sanatorium?
808. Find the value of the expression:
809. Solve the problem:
- When processing a part from a casting weighing 40 kg, 3.2 kg was wasted. What percentage is the mass of the part of the mass of the casting?
- When sorting grain out of 1750 kg, 105 kg went to waste. What percentage of grain is left?
810. Find the value of the expression:
- 6,0008: 2,6 + 4,23 0,4;
- 2,91 1,2 + 12,6288: 3,6.
811. From 20 kg of apples, 16 kg of applesauce is obtained. How much applesauce will be made from 45 kg of apples?
812. 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 if all the painters work at the same productivity?
813. A concrete slab with a volume of 2.5 m 3 has a mass of 4.75 tons. What is the volume of a slab of the same concrete if its mass is 6.65 tons?
814. 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.
815. The sunflower seeds of the new variety contain 49.5% oil. How many kilograms of such seeds should be taken to contain 29.7 kg of oil?
816. 80 kg of potatoes contain 14 kg of starch. Find the percentage of starch in such potatoes.
817. Flax seeds contain 47% oil. How much oil is in 80 kg of flax seeds?
818. 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?
819. Find the value of the expression:
- a) 203.81: (141 - 136.42) + 38.4: 0.75;
- b) 96: 7.5 + 288.51: (80 - 76.74).