Relationships in mathematics From a piece of matter 5 m long, 2 m were cut off. What part of the piece of matter was cut off? 5 m 2 m Solution \u003d 0.4 \u003d 40 0 / 0 The quotient of two numbers is called the ratio of these numbers. What does attitude show? The answer can also be written as decimal fraction or in percentage. 2:5=
What does attitude show? The ratio shows how many times the first number is greater than the second 16 kg 8 kg 16: 8 \u003d 2 (p.) Or what part the first number is from the second. 4 m 20 m 4: 20 \u003d 0.2 (parts) If two quantities are measured by the same unit of measurement, then the ratio of their values \u200b\u200bis called the ratio of these quantities. Mass ratio Length ratio To test
PROPORTION AND PROPORTION “Proportion-proportionality. 1) A certain ratio of parts to each other. Proportionality in nature, art, architecture means the observance of certain ratios between the sizes of individual parts of a plant, sculpture, building and is a sine qua non correct and beautiful image of the subject. 2) In mathematics: the equality of two ratios. Ozhegov S.I.
Proportions The ratios 3.6:1.2 and 6.3:2.1 are equal. Therefore, we can write the equation 3.6:1.2=6.3:2.1 or a: b = c:d Average terms of the proportion Extreme terms of the proportion In the correct proportion, the product of the extreme terms is equal to the product of the averages. a * d = b * c How to check if the proportion is correct? To a question
Proportion The main property of proportion: If the product of the extreme terms is equal to the product of the middle terms of the proportion, then the proportion is correct. Check if the proportion is correct? 20:16=5:
Exercises Make, if possible, proportions from the following ratios: a) 20:4 and 60: Make, if possible, proportions from the four given numbers: a) 100; 80; 4; Check if the equality is true in two ways: a) 49:14=14: Make a proportion from the following equalities: a) 40*30=20* Find the unknown term of the proportion: a) x:30=54:40
Test 1. Relationships. 1. Which of these ratios is equal? a) 7:2; b) 4:14; c) 7:17.5; d) 12:17; 7:24:147:17.512:17 2. Find the ratio of 1.2 m to 10 cm. a) 12; b) 12 m; c) 0.12; d) another answer. 1212 m 0.12 another answer 3. How does one third of an hour relate to eighteen minutes? a) 1:54; b) 10:8; c) 1:6; d) another answer. 1:5410:81:6 another answer. 4. The ratio a:b is 5:3. Find the ratio 3a:10c. a) 1:2; b) 2; c) 9:30; d) another answer. 1:229:30 another answer.
Test 2. Proportions. 1. Find the product of the average members of the proportion: a) 9.8; b) 0.98; c) 80; d) another answer. 9,80,9880 another answer. 2. Find the unknown member of the proportion: a) 0.05; b) 20; c) 0.5; d) another answer 0.05200.5 another answer. 3. From these proportions, select the correct one: a) 82:72=64:78; b)15:8=13:6;82:72=64:7815:8=13:6 c)17:2=34:4; d) 22:23=81:82.17:2=34:422:23=81:82
Task 4 The distance on the map from the Earth to the Moon is 38.4 cm. Find the distance between them if the map scale is 1:
in exercises, tasks and tests
Zheleznogorsk
sanatorium-forest school
§ 1 Ordinary fractions (review).
item 1. addition and subtraction of fractions with different denominators.
item 2. Joint actions with decimal and ordinary fractions.
§ 2 Relations and proportions
item 3. relationship.
item 4. Proportions. Basic property of proportion.
item 6. Inverse proportionality quantities.
§ 3 Positive and negative numbers
item 7. Coordinate line.
item 8. positive numbers. The absolute value of a number.
item 9. Comparison of numbers.
§ 10. Addition of rational numbers.
item 11. Addition of numbers with different signs.
§ 12. Laws of addition of rational numbers.
item 13. Subtraction.
item 14. Distance between points.
§ 15. Multiplication of rational numbers.
item 16. Division.
Item 17. Properties of actions with rational numbers.
§ 4 Solutions of equations
p. 18. Opening brackets.
§ 19. Reduction of similar terms.
Section 20. Solution of Equations.
§ 5 Problem solving
item 21. Tasks.
p. 22. Repetition.
ChapterI. Ordinary fractions (review).
§ 1 addition and subtraction of fractions with different denominators.
Abstract
1. Comparison of fractions.
a) if the denominators are the same: that fraction\u003e which has a larger numerator?
https://pandia.ru/text/78/170/images/image003_13.png" align="left" width="43" height="41 src=">
c) if different numerators and denominators: bring to a common denominator.
https://pandia.ru/text/78/170/images/image005_9.png" align="left" width="41" height="41 src=">.png" align="left" width="47 "height="41 src=">.png" align="left" width="40" height="41 src=">.png" align="left" width="47" height="41 src=" >.png" align="left" width="57" height="41 src=">.png" align="left" width="67" height="41 src=">.png" align="left " width="33" height="41 src=">.png" align="left" width="36" height="41 src=">r)
3. Calculate:
https://pandia.ru/text/78/170/images/image021_0.png" align="left" width="79" height="41 src=">.png" align="left" width="65 "height="41 src=">a)
https://pandia.ru/text/78/170/images/image025_0.png" align="left" width="89" height="41 src=">.png" align="left" width="65 " height="41 src=">.png" align="left" width="65" height="41 src=">.png" align="left" width="75" height="41 src=" >.png" align="left" width="73" height="41 src=">
withindependent work
4. Choose the correct answer.
https://pandia.ru/text/78/170/images/image035_0.png" align="left" width="100" height="45 src="> a) b)
https://pandia.ru/text/78/170/images/image037_0.png" align="left" width="92" height="45 src="> c) d)
https://pandia.ru/text/78/170/images/image039.png" align="left" width="15" height="41 src=">.png" align="left" width="16 " height="41 src=">.png" align="left" width="143" height="48 src=">Express your answer in minutes.
https://pandia.ru/text/78/170/images/image045.png" align="left" width="49" height="23 src="> c) Morning milk yield was 81/2 l, evening – 63/10 l, and at lunch the milk yield was 3/5 l less than in the morning What is the milk yield per day?
6. Compare fractions and draw a conclusion.
https://pandia.ru/text/78/170/images/image047.png" align="left" width="52" height="41 src=">.png" align="left" width="39 " height="41 src=">.png" align="left" width="51" height="41 src=">b)
https://pandia.ru/text/78/170/images/image053.png" align="left" width="45" height="41 src=">.png" align="left" width="21 " height="41 src=">.png" align="left" width="21" height="41 src=">
8. Arrange in descending order.
https://pandia.ru/text/78/170/images/image059.png" align="left" width="16" height="41 src=">.png" align="left" width="91 " height="41 src=">.png" align="left" width="73" height="41 src=">a)
https://pandia.ru/text/78/170/images/image065.png" align="left" width="67" height="41 src=">.png" align="left" width="100 "height="45 src=">.png" align="left" width="115" height="41 src="> in d)
https://pandia.ru/text/78/170/images/image071.png" align="left" width="25" height="41 src=">.png" align="left" width="33 " height="41 src=">.png" align="left" width="107" height="41 src=">
Png" align="left" height="17 src=">.png" align="left" width="16" height="41 src=">.png" align="left" height="17 src= ">.png" align="left" height="17 src=">.png" align="left" width="15" height="41 src=">.png" align="left" width=" 21" height="41 src=">.png" align="left" width="15" height="41 src=">a) b)
Png" align="left" width="16" height="41 src=">.png" align="left" height="17 src=">
Png" align="left" height="17 src=">2). The sum of the fractions is:
https://pandia.ru/text/78/170/images/image081.png" align="left" width="68" height="41 src=">
3). The value of the expression is:
https://pandia.ru/text/78/170/images/image082.png" align="left" width="57" height="41 src=">
4). Add fractions:
https://pandia.ru/text/78/170/images/image084.png" align="left" width="57" height="41 src=">
5). Calculate:
https://pandia.ru/text/78/170/images/image085.png" align="left" width="99" height="45 src=">6).
https://pandia.ru/text/78/170/images/image086.png" align="left" width="72" height="41 src=">.png" align="left" width="44 " height="41 src=">.png" align="left" width="85" height="41 src=">.png" align="left" width="140" height="41 src=" >.png" align="left" width="81" height="41 src="> Examples: a) b) c)
(can be reduced by 2) (fraction cannot be reduced) (can be reduced by 25)
https://pandia.ru/text/78/170/images/image096.png" align="left" width="59" height="41 src=">.png" align="left" width="51 "height="41 src="> a)
, since the denominator has a divisor of 5 and, 2: 5 \u003d 0.4
https://pandia.ru/text/78/170/images/image100.png" align="left" width="31" height="41 src="> in)
- cannot be converted to decimal, because the denominator does not have divisors 2 and 5
13. Find fractions that can be written as decimals and decipher the word.
https://pandia.ru/text/78/170/images/image102.png" align="left" width="29" height="69 src=">.png" align="left" width="27 " height="69 src=">.png" align="left" width="28" height="69 src=">.png" align="left" width="28" height="69 src=" >.png" align="left" width="24" height="69 src=">
14. Pay common fraction to decimal.
https://pandia.ru/text/78/170/images/image112.png" align="left" width="32" height="41 src=">.png" align="left" width="20 " height="41 src=">.png" align="left" width="21" height="41 src=">.png" align="left" width="55" height="41 src=" >.png" align="left" width="55" height="41 src=">.png" align="left" width="49" height="41 src=">.png" align="left " width="61" height="41 src=">.png" align="left" width="147" height="45 src=">.png" align="left" width="100" height= "41 src="> a) b) c)
19. Independent work.
one). Calculate:
https://pandia.ru/text/78/170/images/image130.png" align="left" width="55" height="41 src=">.png" align="left" width="55 "height="41 src="> a) b) c) d)
2). Find the value of the expression:
https://pandia.ru/text/78/170/images/image134.png" align="left" width="81" height="41 src="> a) b)
20. Find the value of fractions.
https://pandia.ru/text/78/170/images/image136.png" align="left" width="35" height="44 src=">.png" align="left" width="28 "height="44 src="> a) b) c) d)
21. Independent work (test yourself).
Find the value:
https://pandia.ru/text/78/170/images/image143.png" align="left" width="33" height="44 src=">.png" align="left" width="21 " height="41 src=">.png" align="left" width="83" height="41 src=">.png" align="left" width="21" height="41 src=" >.png" align="left" width="25" height="41 src=">
one). a B C D)
https://pandia.ru/text/78/170/images/image153.png" align="left" width="32" height="41 src=">.png" align="left" width="32 " height="41 src=">.png" align="left" width="31" height="41 src=">.png" align="left" width="21" height="41 src=" >.png" align="left" width="32" height="41 src=">
4). a B C D)
23. Take action.
https://pandia.ru/text/78/170/images/image163.png" align="left" width="199" height="45 src=">
https://pandia.ru/text/78/170/images/image165.png" align="left" width="135" height="83 src=">
hProblems for finding fractions of a number.
To find a fraction of a number, you need to multiply the number by that fraction.
24. Find a fraction of a number.
https://pandia.ru/text/78/170/images/image167.png" align="left" width="83" height="45 src=">.png" align="left" width="73 "height="45 src=">.png" align="left" width="80" height="45 src=">
https://pandia.ru/text/78/170/images/image173.png" align="left" width="97" height="45 src=">
https://pandia.ru/text/78/170/images/image175.png" align="left" width="92" height="45 src=">.png" align="left" width="100 " height="45 src=">.png" align="left" width="181" height="45 src=">.png" align="left" width="177" height="45 src=" >
https://pandia.ru/text/78/170/images/image183.png" align="left" width="185" height="45 src=">.png" align="left" width="193 " height="45 src=">.png" align="left" width="183" height="45 src=">
33. Mowed 3/7 of the meadow. Find the area of the meadow if you mowed 21 hectares.
34. In the first hour, the car covered 5/7 of the entire distance. What is the total distance if the car traveled 70 km in the first hour.
35. 2/7 of all machine tools of the workshop were repaired. How many machines are in the workshop if 28 machines have been repaired?
36. Repaired 5/6 of the road, which is 30 km. What is the length of the entire road?
37. In the first hour, the car covered 27% of the intended path, after which it had another 146 km to go. How many kilometers is the length of the intended route?
38. 32% of vegetables were sold, after which 136 tons more remained to be sold. How many tons of vegetables were in the store?
39. 29% of the entire book was read, after which there were 142 more pages to read. How many pages are in the book?
40. Test № 1.
https://pandia.ru/text/78/170/images/image189.png" align="left" width="209" height="45 src=">
2). Compare
41. A 120 ha field was planted with potatoes, cabbage and carrots. 3/4 of this field was planted with potatoes, 80% of the remainder was planted with cabbage, and the rest of the field with carrots. How many hectares were planted with carrots?
42. Control work number 2.
one). Find the value of the expression:
https://pandia.ru/text/78/170/images/image191.png" align="left" width="39" height="44 src=">.png" align="left" width="205 " height="45 src=">.png" align="left" width="64" height="41 src=">.png" align="left" width="63" height="41 src=" >.png" align="left" width="16" height="41 src=">2).
and if a > b , then shows how many times a > b
Png" align="left" width="16" height="41 src=">
b) if a < b , it shows what part is a from b
https://pandia.ru/text/78/170/images/image202.png" align="left" width="391" height="45 src=">
https://pandia.ru/text/78/170/images/image204.png" align="left" width="191" height="45 src=">.png" align="left" width="193 "height="45 src=">
43. What do relationships show?
https://pandia.ru/text/78/170/images/image208.png" align="left" width="351" height="45 src=">
44. There are 6 white and 12 red roses in the flower bed. What the relationship shows:
a) 6:12; b) 12:6; c) 6:18; d) 18:12?
45. Simplify the relationship (i.e., reduce the fractions).
https://pandia.ru/text/78/170/images/image210.png" align="left" width="60" height="44 src=">.png" align="left" width="29 "height="41 src=">
a) 4: 5; b) c) d) 77: 28; e)
https://pandia.ru/text/78/170/images/image214.png" align="left" width="53" height="41 src=">
47. The garden occupies 5.6 a, and the vegetable garden 3.2 a. How many times the area of the garden more area vegetable garden? What part of the total area is occupied by the garden?
48. Serzha walked 5.6 km and traveled 12.6 km by bus. How many times is the distance traveled on foot less than the journey by bus? What part of the entire journey did Seryozha travel by bus?
49. There are 25 people in the brigade, 20 of them are men. What percentage of all people in the team are men?
50. out of 32 students in the class, 4 students were absent due to illness. What percentage of the students were present?
51. Instead of the planned 75 parts, the worker produced 80 parts. How many percent of the plan was completed?
52. To determine the germination of seeds, 300 seeds were planted. Of these, 273 sprouted. What is the percentage of seed germination?
53. Independent work.
a) We bought vegetables for 2.6 rubles and fruits for 9.1 rubles. How many times more did they pay for fruits than for vegetables? How much of the total purchase was vegetables?
b) The length of the entire road is 360 km. Asphalted 240 km. this road. What part of the road is paved? How many times longer is the entire road than its paved part?
c) Out of 250 seeds, 200 sprouted. Find the percentage of germination.
§ 4 Pproportions. Basic property of proportion.
Abstract
1. Attitude = relation
https://pandia.ru/text/78/170/images/image216.png" align="left" width="27" height="34 src=">.png" align="left" width="46 "height="3 src="> proportion
a : b = c : d or
https://pandia.ru/text/78/170/images/image219.png" align="left" width="139" height="41 src="> 3.
