Direct proportional dependence. Direct and inverse proportional relationships Self-test questions
Math lesson in 6th grade
on the topic "Direct and inverse proportional relationships"
Developed
mathematic teacher
Municipal educational institution "Mikhailovskaya secondary school named after
Hero Soviet Union V.F. Nesterov"
Kleymenova D.M.
Lesson Objectives :
1. Didactic :
promote the formation and consolidation of problem solving skills using proportions;
teach how to identify two quantities in problem conditions and establish the type of relationship between them;
write a short note and make a proportion;
consolidate skills and abilities to solve equations that have the form of proportions.
2. Developmental :
develop memory, attention, continue the development of students’ mathematical speech;
promote the development of students' creative activity and interest in the subject of mathematics.
3. Educational :
cultivate accuracy, develop interest in mathematics;
cultivate the ability to listen carefully to the opinions of others, cultivate self-confidence, cultivate a culture of communication.
Equipment: TSO required for the presentation: computer and projector, sheets of paper for writing down answers, cards for conducting the reflection stage (three for each), pointer.
Lesson type: lesson in applying knowledge.
Forms of lesson organization:frontal, collective, individual work.
Lesson structure:
Organizing time, greetings, wishes.
Checking the studied material.
Lesson topic message.
Repetition of learned material.
The stage of control and self-control of knowledge and methods of action.
Stage of summarizing the lesson.
Homework.
Reflection.
During the classes
Organizing time.
(slide 3)
(Greeting, recording absentees, checking students’ preparedness for the educational process, distributing leaflets and cards for reflection, checking the readiness of the classroom for the lesson, organizing the student’s attention).
The teacher reads: (slide No. 3)
Mathematics is the basis and queen of all sciences,
And I advise you to make friends with her, my friend.
If you follow her wise laws,
You will increase your knowledge
Will you start using them?
Can you swim on 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, try,
Understanding the salt of science.
Try to prove everything
But tirelessly.
2. Checking the studied material.
(Identifies problems in students’ knowledge and methods of activity and determines the reasons for their occurrence, eliminates identified gaps during the test.)
Oral survey: (slide No. 4)
What is the ratio of two numbers?
How to find a fraction of a number?
What is proportion?
What quantities are called directly proportional?
What does the ratio of two numbers show?
How to find a number by its fraction?
The main property of proportion.
What quantities are called inversely proportional?
Finish the sentence: (slide 5). (Children first complete the task independently, writing down on pieces of paper only the letters corresponding to the correct answer. Then they raise their hands. After that, the teacher reads the question out loud, and the students answer).
Direct proportional dependence is such a dependence of quantities in which...
An inverse proportional dependence is a dependence of quantities in which...
To find the unknown extreme term of the proportion...
The average term of the proportion is...
The proportion is correct if...
WITH) …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 increases 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.
U) ...as one value increases several times, the other increases by the same amount.
E) ...the ratio of the product of the extreme terms to the known average.
Answer:SUCCESS.(slide 6)
Graphic dictation(slides 7-10).
Don't say "yes" or "no"
And draw an icon.
“Yes” with a “+” sign, no with a “-” sign.
(Students work independently. Answers are written down on pieces of paper. Self-test using slide No. At the end of the lesson, the teacher looks at the pieces of paper)
If the area of a rectangle is constant, then its length and width are inversely proportional.
A child's height and age are directly proportional.
If the width of a rectangle is constant, its length and area are directly proportional.
The speed of a car and the time it moves are inversely proportional.
The speed of a car and its distance traveled are inversely proportional.
The revenue of a cinema box office is directly proportional to the number of tickets sold, sold at the same price.
The carrying capacity of machines and their number are inversely proportional.
The perimeter of a square and the length of its side are directly proportional.
At a constant price, the cost of a product and its mass are inversely proportional.
Answer: + - + + - + + - -(Slide No. 10)
Get an assessment.(slide No. 11)
8 -9 correct answers - “5”
6-7 correct answers - “4”
4-5 correct answers - “3”
Oral counting: (slides 12-13)
Come on, put the pencils aside!
No papers, no pens, no chalk!
Verbal counting! We're doing this thing
Only by the power of mind and soul!
Exercise: Find the unknown term of the proportion:
Answers: 1) 39; 24; 3; 24; 21.
2)10; 3; 13.
Lesson topic message. slide number 14 (Provides motivation for schoolchildren to study.)
The topic of our lesson is “Direct and inverse proportional relationships.”
In previous lessons, we looked at the direct and inverse proportional dependence of quantities. Today in the lesson we will solve various problems using proportions, establishing the type of connection between data. Let us repeat the basic property of proportions. And the next lesson, concluding on this topic, i.e. lesson - test.
Demonstrated slide number 15
The stage of generalization and systematization of knowledge.
1) Task1.
Create proportions to solve 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 a pool in 25 minutes. How many minutes will it take to fill a pool with 10 such pipes?
c) A team of 8 workers completes the task in 15 days. How many workers can complete this task in 10 days while 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 No. 16) (self-assessment: put + or - in pencilnotebooks; analyze errors)
Answers:a) 3:x=75:125c) 8: x=10: 15
b) 8:10= X:2 5 d) 5.6:54=2: X
2) Physical education minute. (slide no. 17-22)
We quickly got up from our desks
And they walked on the spot.
And then we smiled
They stretched higher and higher.
Sat down - stood up, sat down - stood up
In a minute we gained strength.
Straighten your shoulders
Raise, lower,
Turn right, turn left
And sit down at your desk again.
3) Solve the problem (slide number 23)
№788 (p. 130, Vilenkin’s textbook)(after parsing it yourself)
In the spring, during the city's landscaping work, linden trees were planted on the street. 95% of all planted linden trees were accepted. How many linden trees were planted if 57 linden trees were planted?
Read the problem.
What two quantities are discussed in the problem?(about the number of linden trees and their percentages)
What is the relationship between these quantities?(directly proportional)
Make a short note, proportion and solve the problem.
Solution:
| Linden trees (pcs.) | Interest % |
|
| They imprisoned | ||
| Accepted |
; 
; x=60.
Answer: 60 linden trees were planted.
4) Solve the problem: (slide No. 24-25) (after analysis, decide on your own; mutual verification, then the solution is displayed on the screen, slide No. 23)
To heat the school building, coal was stored for 180 days at a consumption rate of 0.6 tons of coal per day. How many days will this supply last if 0.5t is spent daily?
Solution:
Brief entry:
| Weight (t) in 1 day | Quantity days |
|
| According to the norm | ||
Let's make a proportion:
; 
; 
days
Answer: 216 days.
5) No. 793 (p. 131)(parsing field independently; self-control.
(Slide No. 26)
In iron ore, for every 7 parts iron there are 3 parts impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
Solution: (slide No. 27)
| Quantity parts | Weight |
|
| Iron | 73,5 |
|
| Impurities |
; 
; 
Answer: 31.5 kg of impurities.
6) Summing up the results of the stage. (slide No. 28)
So, let’s formulate an algorithm for solving problems using proportions.
Algorithm for solving direct problems
and inversely proportional relationships:
An unknown number is denoted by the letter x.
The condition is written in table form.
The type of relationship between quantities is established.
Directly proportional dependence is indicated by identically directed arrows, and an inversely proportional relationship is indicated by oppositely directed arrows.
The proportion is recorded.
Her unknown member is located.
5. Repetition of the studied material. (slide No. 29)
№763(s)(page 125)(with commenting at the board)
6. Stage of control and self-control of knowledge and methods of action.
(slide No. 30-32)
Independent work (10 - 15 min) (Mutual check: students check each other using ready-made slides independent work, while setting + or -. At the end of the lesson, the teacher collects the notebooks for review).
Solve problems by making proportions.
№1. The cyclist spent 0.7 hours traveling from one village to another at a speed of 12.5 km/h. At what speed did he have to travel to cover this path in 0.5 hours?
Solution:
Brief entry:
| Speed (km/h) | Time (h) |
|
| 12,5 | ||
Let's make a proportion:
; 
; 
km/h
Answer: 17.5 km/h
№2. From 5 kg of fresh plums you get 1.5 kg of prunes. How many prunes will 17.5 kg of fresh plums yield?
Solution:
Brief entry:
| Plums (kg) | Prunes (kg) |
|
| 17,5 |
Let's make a proportion:
; 
; 
kg
Answer: 5.25 kg
№3. The car drove 500 km, using 35 liters of gasoline. How many liters of gasoline will be needed to travel 420 km?
Solution:
Brief entry:
| Distance (km) | Gasoline (l) |
|
2. Proportional system.
The obvious injustice towards the political parties participating in elections, which the majoritarian system often brings, has given rise to a system of proportional representation of parties and movements, abbreviated as the proportional system. Its main idea is that each party receives a number of mandates in parliament or other representative body proportional to the number of votes cast for its candidates in the elections.
Proportional representation systems are most common in countries Latin America And of Eastern Europe, and also make up one third of Africa's electoral systems.
Most proportional systems are characterized by party list voting, which assumes that each party will be ready to propose a list of candidates for voters' consideration. Voters vote for parties, and they receive their share of seats in parliament in proportion to the number of votes received.
This system has its own advantages:
1. Does not lead to abnormal results characteristic of a majoritarian system, and provides a more representative legislative body.
2. Ensures a fair ratio of votes received to seats in parliament, and therefore makes it possible to avoid destabilizing and “unfair” results.
4. Enables small parties to gain representation in parliament. Any political party, even with a few percent of the votes, can be represented in parliament, unless, of course, the entry barrier is too high or the size of the constituency is too small.
5. Encourages parties to include candidates on their lists who represent different social strata.
6. Gives members of cultural and other minorities a better chance of being elected.
7. Give women more chances to be elected to parliament.
8. The system constrains regional division. Because With proportional representation, small parties receive a small number of seats, this virtually eliminates the situation in which one party receives all the mandates from one province or district.
9. Provides a more visible division of power between parties and interest groups. In most new democratic countries it is impossible to avoid the necessity of dividing power among the majority of the people, whose representatives hold in their hands political power, and a small number of those who hold economic power.
Proportional representation systems criticized for two main reasons:
firstly, for their tendency to form coalition governments with all their shortcomings;
secondly, for the failure of some of these systems to provide a strong geographical connection between the MP and his constituents. The most common arguments against proportional representation systems are:
1. The formation of a coalition government leads to legislative stupor and further inability to pursue a consistent policy regarding the most important problems.
2. Destabilizing fragmentation. Polarized pluralism can give small parties the opportunity to win over large ones and enter into negotiations with them to create coalitions. In this aspect, wide representation is cited as a disadvantage.
3. The basis for the activities of extremist parties.
4. Creation of a ruling coalition in which there is not enough understanding about the necessary political course, and which does not enjoy the support of the population.
5. The impossibility of eliminating the party from power.
6. Weakening the connection between voters and deputies.
7. Places too much power in the hands of the party center and senior party leadership. A candidate's place on the party list, and therefore the likelihood with which he can get into parliament, depends on the favor of the party bosses, and relations with voters fade into the background.
8. The system is little known to most countries that have a history of English or French colonial conquest.
Chapter 3 RELATIONS AND PROPORTIONS
Using proportions you can solve problems.
You know, for example, that the cost of a product depends on its quantity: the more quantity of a product is purchased, the greater its value will be. Such quantities are called directly proportional.
Remember!
Two quantities are called directly proportional if, when one quantity increases (decreases) several times, the other quantity increases (decreases) the same number of times.
Problem 1. For 2 kg of sweets we 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 short, you increase your speed 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) the same number of times.
Problem 2. A car, moving at a speed of 90 km/h, covered 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 Cherkassy 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 speculate. For example, during an illness, a child’s temperature may rise and fall over the course of several days. And here there is no dependence, which means there can be no proportionality. But a child’s height constantly increases as his age increases. Consequently, there is a relationship between the quantities, which means there is reason to analyze the proportional data of the quantities. It is clear that there is no proportional dependence here, so there is no need to find out exactly how these proportional quantities are direct or inverse. If two quantities are proportional, then only two options are possible, which are mutually exclusive - 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 ratio 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 “Book of Abacus” (counting boards) was published, which collected all the problems known at that time. 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, ... .
This sequence of numbers is now known as the Fibonacci series. The peculiarity of this sequence of numbers is that each of its members, starting from the third, equal to the sum two previous ones:
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 in a series approaches the ratio of the golden ratio. For example:
21: 34 = 0.617, a34: 55 = 0.618.
REMEMBER THE IMPORTANT
1. What quantities are called directly proportional? Give examples.
2. How do they solve problems involving direct proportionality?
3. What quantities are called inversely proportional? Give examples.
4. Do I solve problems involving inverse proportionality?
5. Are two quantities always proportional?
589". Two quantities are directly proportional. How will one quantity change if the other: a) increases by 5 times; b) decreases by 2 times?
Explain your answer.
590". Based on the conditions of the problem, we made an abbreviated entry:
1)3-36, 2) 70-3, 3) 2-100,
4-48; 60-2; 4-50.
Are these quantities directly proportional?
591". Two quantities are inversely proportional. How will one quantity change if the other:
a) will increase 4 times; b) will decrease by 6 times?
Explain your answer.
592". Based on the conditions of the problem, we made an abbreviated entry:
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 a box of chocolates and the number of identical chocolates in the box;
3) the distance traveled by the car at a constant speed, and the time of movement;
4) speed of movement and time of movement to cover a certain distance;
5) a 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°. Using the abbreviated form 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 you paid 128 UAH for 4 kg of such sweets?
596°. For 3 kg of apples we paid 24 UAH. How much do 7 kg of such apples cost?
597°. In 4 hours the boat traveled 80 km. How far will the boat travel in 2 hours, moving at the same speed?
598°. A tourist walked 20 km in 5 hours. How many hours will it take for 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 make 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 coffee beans?
601°. The car covered a distance of 210 km in 3 hours. What is the distance traveled by a car in 2 hours, moving at the same speed?
602°. The tailless gibbon monkey, jumping from tree to tree, covers a distance of 32 km in 2 hours. How far will the gibbon cover in 3 hours?
603°. Determine whether this dependence of quantities is inversely proportional:
1) the price of the product and the purchase price;
2) the mass of the box of chocolates and its cost;
3) speed of movement and time of movement to cover a certain distance;
4) the speed of the car and the distance it covered at a constant speed;
5) the amount of work performed and the time it took to complete it;
6) labor productivity and the time it takes to complete a certain amount of work;
7) the number of cars and the cargo they will transport in a certain time;
8) the length of the side of the square and its area.
604°. Using the abbreviated form 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 the order for furniture production in 12 days. In how many days will 6 carpenters be able to complete an order if their labor productivity is the same?
606°, How many days will it take 6 workers to 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 must be the speed of the kangaroo so that it can cover this distance in 2.5 hours?
608°. What must be the speed of the train according to the new schedule in order to travel 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 we paid 56 UAH. How much will 3 kg of sweets cost, the price of which is 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. The pendulum of a wall clock makes 730 oscillations in 15 minutes. How many oscillations will it make in 1 hour? How long will it take the pendulum to make 2190 oscillations?
612. Natalya paid 60 UAH for 24 notebooks. How much do 20 of these notebooks cost? How many of these notebooks can you buy 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 can? 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 flow out of the tap in half an hour? How long will it take for 27 liters of water to flow out of a tap?
615. The distance between stations is 360 km. How long will it take a train to cover this distance if it covers 90 km in an hour? What must be the speed of the train so that it can cover this distance in 4 hours 30 minutes?
616. The distance between villages is 18 km. How long does it take a cyclist whose speed is 12 km/h to complete this distance? How fast does a pedestrian need to move to cover this distance in 6 hours?
617. Two tractors plowed a field in 6 days. How many days will it take 4 tractors to clear this field if they work with the same labor productivity? How many tractors are needed to plow this field in 2 days?
618. Eight trucks can transport cargo in 3 days. How many days will it take 6 such trucks to transport cargo? How many trucks will it take to transport this load in 2 days?
619. Compose and solve a problem on:
1) direct proportionality, to solve which you need to create a proportion
2) inverse proportionality, to solve which you need to make up the proportion x: 4 = 120: 160.
620. Compose and solve a problem on: 1) direct proportionality, for the solution of which you need to create a proportion
2) inverse proportionality, to solve which you need to make the proportion 3: x = 90: 60.
621 *. Tarasik can walk from the railway station to the village in 20 minutes. How long will it take him to ride his bike from the station to the village if his speed on a bike is 2 times faster than his speed on foot?
622*. A master, working independently, completes the work in 3 days, and together with a student - in 2 days. In how many days can a student complete this work independently?
623*. Dima runs 4 laps on the treadmill in the same time as Katya runs 3 laps. Katya ran 12 laps. How many laps does Dima run during this time?
624*. Water can be pumped out of a pool in 1 hour 15 minutes. How long after starting work will there be 0.2 of the amount of water left in the pool that was there at first?
PUT IT IN PRACTICE
625. To print the book, it was planned to place 28 lines on each page, with 40 letters in each line. However, it turned out that it makes more sense to place 35 lines on each page. How many letters will be placed on each line during printing of this book, if the number of letters on the page does not change?
626. To prepare 12 cakes, you need to take the white of one egg and 3 tablespoons of sugar. How many of these products do you need to take to prepare 24 of these pieces? How many of these cakes will you get if you have 3 eggs?
REVIEW PROBLEMS
627. What number should be entered in the last cell of the chain?
628. Solve the equation:
Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Shvartsburd for 6th grade in mathematics on the topic:
§ 4. Relations and proportions:
22. Direct and inverse proportional relationships
1 For 3.2 kg of goods they paid 115.2 rubles. How much should you pay for 1.5 kg of this product?
SOLUTION
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 is 4.8 m. Find its width.
SOLUTION
782 Determine whether the relationship between the quantities is direct, inverse, or not proportional: the distance covered by the car at a constant speed and the time of its movement; the cost of the goods purchased at one price and its quantity; the area of the square and the length of its side; the mass of the steel bar and its volume; the number of workers performing some work with the same productivity, and the time of completion; the cost of the product and its quantity purchased for a certain amount of money; the age of the person and the size of his shoes; the volume of the cube and the length of its edge; the perimeter of the square and the length of its side; a fraction and its denominator if the numerator does not change; a fraction and its numerator if the denominator does not change.
SOLUTION
783 A steel ball with a volume of 6 cm3 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 cm3?
SOLUTION
784 From 21 kg of cottonseed, 5.1 kg of oil was obtained. How much oil will be obtained from 7 kg of cottonseed?
SOLUTION
785 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?
SOLUTION
786 To transport the cargo, 24 vehicles with a carrying capacity of 7.5 tons were required. How many vehicles with a carrying capacity of 4.5 tons are needed to transport the same cargo?
SOLUTION
787 To determine the germination of seeds, peas were sown. Of the 200 peas sown, 170 sprouted. What percentage of the peas sprouted (germinated)?
SOLUTION
788 During the city greening Sunday, linden trees were planted on the street. 95% of all planted linden trees were accepted. How many of them were planted if 57 linden trees were planted?
SOLUTION
789 There are 80 students in the ski section. Among them are 32 girls. What percentage of section participants are girls and boys?
SOLUTION
790 According to the plan, the plant was supposed to smelt 980 tons of steel in a month. But the plan was fulfilled by 115%. How many tons of steel did the plant produce?
SOLUTION
791 In 8 months, the worker completed 96% of the annual plan. What percentage of the annual plan will the worker complete in 12 months if he works with the same productivity?
SOLUTION
792 In three days, 16.5% of all beets were harvested. How many days will it take to harvest 60.5% of the beets if you work at the same productivity?
SOLUTION
793 In iron ore, for every 7 parts of iron there are 3 parts of impurities. How many tons of impurities are in the ore that contains 73.5 tons of iron?
SOLUTION
794 To prepare borscht, for every 100 g of meat you need to take 60 g of beets. How many beets should you take for 650 g of meat?
SOLUTION
796 Express each of the following fractions as the sum of two fractions with numerator 1.
SOLUTION
797 From the numbers 3, 7, 9 and 21, form two correct proportions.
SOLUTION
798 The middle terms of the proportion are 6 and 10. What can the extreme terms be? Give examples.
SOLUTION
799 At what value of x is the proportion correct.
SOLUTION
800 Find the ratio of 2 min to 10 sec; 0.3 m2 to 0.1 dm2; 0.1 kg to 0.1 g; 4 hours to 1 day; 3 dm3 to 0.6 m3
SOLUTION
801 Where on the coordinate ray should the number c be located for the proportion to be correct.
SOLUTION
802 Cover the table with a sheet of paper. Open the first line for a few seconds and then, closing it, try to repeat or write down the three numbers of that line. If you have reproduced all the numbers correctly, move on to the second row of the table. If there is an error in any line, write several sets of the same number yourself double digit numbers and practice memorization. If you can reproduce at least five two-digit numbers without errors, you have a good memory.
SOLUTION
804 Is it possible to formulate the correct proportion from the following numbers?
SOLUTION
805 From the equality of the products 3 · 24 = 8 · 9, form three correct proportions.
SOLUTION
806 The length of segment AB is 8 dm, and the length of segment CD is 2 cm. Find the ratio of the lengths AB and CD. What part of AB is the length CD?
SOLUTION
807 A trip to the sanatorium costs 460 rubles. The trade union pays 70% of the cost of the trip. How much will a vacationer pay for a trip?
SOLUTION
808 Find the meaning of the expression.
SOLUTION
809 1) When processing a casting part weighing 40 kg, 3.2 kg was wasted. What percentage is the mass of the part from the casting? 2) When sorting grain from 1750 kg, 105 kg went to waste. What percentage of grain is left?