The percentage (or ratio) of two numbers is the ratio of one number to another multiplied by 100%.

The percentage of two numbers can be written using the following formula:

Percentage example

For example, there are two numbers: 750 and 1100.

The percentage of 750 to 1100 is

The number 750 is 68.18% of 1100.

The percentage of 1100 to 750 is

The number 1100 is 146.67% of 750.

Example task 1

The vehicle manufacturing plant has a production rate of 250 vehicles per month. The plant assembled 315 vehicles in a month. Question: by what percentage did the plant exceed the plan?

Percentage of 315 to 250 = 315: 250 * 100 = 126%.

The plan was fulfilled by 126%. The plan was overfulfilled by 126% - 100% = 26%.

Example task 2

The company's profit for 2011 was $ 126 million, in 2012 the profit was $ 89 million. Question: by what percentage did profit fall in 2012?

The percentage of 89 million to 126 million = 89: 126 * 100 = 70.63%

Profit fell by 100% - 70.63% = 29.37%

The percentage expression of pot odds and the expression in the form of a ratio are two points that need to be seriously thought about and dealt with. This knowledge will be useful to you not only directly to improve your understanding of the pot odds themselves, but also give an idea of ​​the chances of being able to complete your draw, and will also be useful during other mathematical calculations.

Below you will find two tables to help you learn how to convert ratios to percentages and vice versa.

• The first table shows the exact odds you will use based on the number of your improvement outs.
• The second table presents the rounded odds that you can use to quickly calculate pot odds. Those. if you need to call $ 5 to win a $ 20 pot, your odds are 4 to 1 (or 20% as a percentage).

Representing Outs as a Ratio and a Percentage

Number of outs	Improvement on the next card - Attitude	Improvement on the next map -%
1	46.0 to 1	2.1%
2	22.5 to 1	4.3%
3	14.7 to 1	6.4%
4 (gutshot)	10.8 to 1	8.5%
5	8.4 to 1	10.6%
6	6.8 to 1	12.8%
7	5.7 to 1	14.9%
8 (straight draw)	4.9 to 1	17.0%
9 (flush draw)	4.2 to 1	19.1%
10	3.7 to 1	21.3%
11	3.3 to 1	23.4%
12	2.9 to 1	25.5%
13	2.6 to 1	27.7%
14	2.4 to 1	29.8%
15 (straight + flush draw)	2.1 to 1	31.9%
16	1.9 to 1	34.0%
17	1.8 to 1	36.2%
18	1.6 to 1	38.3%
19	1.5 to 1	40.4%
20	1.4 to 1	42.6%
21	1.2 to 1	44.7%
22	1.1 to 1	46.8%

Simple conversion from relationship to interest and vice versa

Attitude	Interest -%
10 to 1	9%
9 to 1	10%
8 to 1	11%
7 to 1	13%
6 to 1	14%
5 to 1	17%
4 to 1	20%
3 to 1	25%
2.5 to 1	29%
2 to 1	33%
1.5 to 1	40%
1 to 1	50%

If you do not want to constantly refer to these tables, you can download yourself the hoRatio odds converter program, which will do all the dirty work for you.

Decoding a row of lines with outs

Gutshot Is a special kind of straight draw that requires only one card to complete. Here's a simple example: you have a board in your hands. You can only complete a straight combination if any comes on the turn or river.

Straight draw- a standard open street (OESD - open-ended straight draw) with a lot of outs for improvement. Example: on your board. You will be able to complete the combination of a straight if any or comes on the turn or river.

Flush draw- a situation when you are holding on the board and the coming out of one more chirv card will complete your draw.

Straight + flush draw- a combination of OESD and flush draw at the same time. For example, when you have a board.

How to use conversion tables

The first table will be useful for comparing the ratio and percentage of probabilities versus the number of outs to improve your hand. Just by looking at the chart, you can see that the flush draw has 9 outs to improve and the odds are 4.2: 1 as a ratio, or 19.1% as a percentage.

The second table will be useful for comparing and converting odds. Therefore, with this table at hand, you can calculate pot odds on the fly. For example, you need to call $ 10 to win a $ 50 pot. Pot odds are 5: 1. We look at the table and see that this corresponds to about 17%.

As we mentioned earlier, you can also use the hoRatio program to quickly convert any percentage expressions to ratios and vice versa. Perhaps it will turn out to be much more convenient and useful.

Converting odds in the mind

How to get a percentage from a fraction

To get a percentage from a fraction, you need to add two numbers from this fraction and divide the resulting number by 100.

For example, if you have a flush draw on the turn, the odds of completing your draw are 4.1: 1 (we'll use an approximate value of 4: 1).

• The odds are 4 to 1, so we add two numbers from the ratio: 4 + 1 = 5.
• 100 / 5 = 20%.

Thus, if you have a 4: 1 chance of improving, then there is a 20% chance you will be able to complete your draw. It's simple.

How to get a fraction from a percentage

To get a fraction from a percentage, you need to divide 100 by the number of percent. Then subtract 1 (one) from the resulting number. As a result, you get the number "x", which can be substituted into the fraction "x: 1".

For example, if you have a flush draw on the turn and you know that the probability of completing your draw is 19.6% (we will assume that it is 20%), then you get the following:

• 100 / 20 = 5.
• 5 - 1 = 4.

Thus, the ratio will be 4 to 1.

Feel free to round up percentages to whole numbers to make it easier for you to divide in your head and make the calculations as easy as possible.

Sinitsina Svetlana Ivanovna - teacher of mathematics

MBOU Secondary School No. 20 named after N.I. Milevsky Kushchevsky district

Lesson number 86 (skill development lesson) Grade 6

(The design is parallel with a separate description of the UUD and the combination of the visual row column with the columns of student and teacher activity.).

Lesson topic (40 minutes).

Lesson objectives:

Educational - Decide tasks, including tasks with practical content, with real data, to find the percentage of two values.

Developing - teach techniques and methods of reasoning, develop logical thinking of students, mathematical speech (oral and written), attention.

Educational - to develop the intellectual and creative abilities of students, cognitive activity, interest in mathematics; encourage students to be independent when looking for ways to solve a problem.

Venue: study room

Equipment: electronic supplement to the textbook “Mathematics. Arithmetic. Geometry. Grade 6 "(E.A. Bunimovich and others), interactive whiteboard (ID), presentation, workbooks, drawing tools

Lesson plan:

Lesson steps	Formable student learning activities
1. Organizational moment (1 min.)	Self-regulation
2. Actualization of knowledge (10 min.)	Compare and analyze, observe and refute wrong decisions. Assessment of available computational skills.
3. Goal setting and motivation (1 min.)	Forecasting, reflection
4. Learning new material (5 min.)	Understand the information presented. Building speech structures, rationalizing, applying an algorithm, proposing and testing hypotheses, the ability to analyze and respond to incoming answers
5. Physical exercise (2 minutes)	Aesthetic perception, health preservation, self-regulation
6. Consolidation of the studied material	Formulate your thoughts orally, be able to interact with a neighbor while completing an educational task; establish and compare different points of view before making a decision and making a choice. Compare your mode of action with the standard. Argue your point of view, argue and defend your position in a way that is not hostile to opponents
8. Summing up the lesson, reflection	Subject reflection, awareness of the relevance of the studied material. Comparison and comparison of personal successes with others.

During the classes

Stages	Teacher activity	Student activities
	Organizing time Greet and check the general readiness and individual students for the lesson.	Greet teachers, control their own readiness (on desks - notebooks, textbooks, pens, pencils, rulers, squares, diaries)
	Updating knowledge, checking D / s Slides 1-3	Pupils have exchanged notebooks and check d / z, then solve frontally 1) 36: 1.6 = 22.5 (mph) 2) Took - 12000 rubles Percentage - 16% per year How much RUB should I contribute monthly? 1) 12000 1.16 = 13920 (rub) - return at the end of the year 2) 13920: 12 = 1160 (rub) per month Answer: 1160 rubles
	Goal setting and motivation Today in the lesson we will continue to solve problems and express the attitude as a percentage. Who will try to formulate the topic of the lesson?	Students write in a notebook: Classwork “Percentage. Expression of the relationship as a percentage. Solving word problems "
	Learning new material read page 117 task 4 (both methods)	Students discuss solutions
	Fizminutka Slides 6-10	Students do eye exercises
	Consolidation of the studied material Problem book number 433, 435-437	Problem book No. 433 a) Total - 40 liters Cast - 8 L Left - ? l What percentage of the canister volume is the remaining volume? 1) 40-8 = 32 (l) - left 2)32/40=8/10 =0,8 = 80 % b) Total - 20 games Won - 12 games Lost -? Games How many percent of all games did the team lose? 1) 20-12 = 8 (games) - lost 8/20= 4/10= 0,4 = 40 % Answer: 40% № 435 1) 4: 5 = 0.8 = 80% per winner 2) 1: 5 = 0.2 = 20% for the loser Answer: for the winner -80%, for the loser -20% № 436 Boys - 65 Girls - 55 What percentage of all sixth graders are boys? Girls? 1) 65 + 55 = 120 (academic) - sixth graders 2) 65/120 = 13/24 = 0.54 = 54% boys 3) 100-54 = 46% girls Answer: boys - 54%, girls - 46% In which city are voters more active? 1) 13/21 = 0, 62 = 62% in city A 2) 11/19 = 0.58 = 58% in city B Answer: in the city of A.
	Summing up the lesson, reflection Summarizes the lesson, evaluates student work, reports homework T.T. No. 159-160 Depict, depending on your self-esteem, in your notebooks one of the options for the "smiley". Slide 13 Thank you for the lesson. Slide 14	Write down homework in diaries.

Literature:

1. Lesson thematic planning. 6th grade. / [L.V. Kuznetsova, S. S. Minaeva, L. O. Roslov, S. B. Suvorov]. - M .: Education, 2011 .-- 45 p.

2. Mathematics. Work programs. The subject line of the textbooks "Spheres". 5-6 grades: a guide for teachers of educational institutions / [LV Kuznetsova, SS Minaeva, LO Roslova, SB Suvorova]. - M .: Education, 2011 .-- 80 p.

3. Mathematics. Arithmetic. Geometry. Grade 6: textbook. for general education institutions / E.A. Bunimovich, L.V. Kuznetsova, S.S. Minaeva and others; publishing house "Education". - M .: Education, 2013 .-- 240 p .: ill. - (Academic school textbook) (Spheres)

4. Mathematics. Arithmetic. Geometry. Exercise notebook. 6th grade. E.A. Bunimovich, L.V. Kuznetsova, S.S. Minaeva and other publishing house "Education". - M .: Education, 2012 .-- 160s.

5. Internet resources.

Approximate lesson planning of teaching material

Textbook item	Number of lessons	Workbook	Didactic materials	Characteristics of the main activities of students
6.1. What is attitude		79-80 (p. 32)	O-31, P-22	Explain what the ratio of two numbers shows, use and understand standard phrases with the word “ratio”. Create relationships, explain the meaningful meaning of the relationship. Explain how to find the ratio of the same and unlike quantities, find the ratio of quantities. Model the relationship of quantities using pictures and drawings. Recognize and solve problems that require relational use, including real-life problems. Analyze the relationship between the relationships of the sides of the squares, their perimeters and areas. Explain what the scale shows (map, plan, drawing, model). Apply knowledge of scale to solve practical problems. Build "copies" of a figure at a given scale
6.2. Division in this respect		-	O-32, P-23	Solve problems of dividing numbers and quantities in this respect, including tasks of a practical nature. Analyze how, with a constant perimeter, the area of ​​a rectangle changes depending on the ratio of its sides
6.3. "Main" task for interest		75, 77 (p. 30)	O-33, P-24	Express percentages as a decimal. Characterize the proportions of a quantity in various equivalent ways - using a decimal or an ordinary fraction, percent. Solve problems to find a few percent of the value, to increase (decrease) the value by several percent, to find the value by its percentage. Apply the concept of percentage to solve problems of practical content, problems with real data. Carry out self-control when finding the percentage of the value, using the techniques of estimation
6.4. Expression of a relationship as a percentage		76, 78 (p. 30-31)	O-34, O-35, "Check yourself", P-25	Move from decimal to percentage. Express the ratio of two values ​​as a percentage. Solve problems of finding the percentage of two values, including a problem with a practical context, with real data. Analyze the text of the problem, simulate a condition using diagrams and pictures, explain the result
Review and control

Basic goals: introduce the concept of relationship, continue to study interest, develop estimation and estimation skills.

Chapter overview. The concept of relationship is introduced in the course of considering some life situations. As a result of studying the material, students should learn how to find the ratio of two quantities, as well as solve problems for dividing the quantity in this respect.

The development of students' ideas about percentages continues. Percentages are now considered in relation to decimal fractions. Students should learn to express percentage as a decimal, move from decimal to percent, solve problems to calculate the percentage of a certain value, and also express the ratio of two values ​​as a percentage.

A large place among the tasks of the textbook continues to be occupied by tasks for estimating, for developing a "feeling" of percentage as a certain fraction of a value, for applying knowledge in practical situations.

Control materials.

Examination manual. Credit 4. Relationships and interest.

Manual "Thematic tests". Test 9. Relationships and percentages.

What is attitude

Methodical commentary

The introduction of the term "relation" is preceded by a discussion of an important practical question about different ways of comparing numbers and quantities. Example 1 (textbook, p. 122) illustrates the comparison of quantities by finding their ratios. In the course of the exercises, students move from the term "particular" to the term "relation", learn to compose relationships, to explain the meaning of each of the compiled relationships. Note that this section deals with the relations of both the same and opposite quantities. Analyzing the material, it should be emphasized that in actions with quantities of the same name, the data are first expressed in the same units and then the ratio is found (number, exercises 469-471 ); in actions with opposite values ​​receive a new value (exercise 472 ). The concept of "scale" is directly related to the concept of "relation". Exercise 475 , 476 , 481-484 included in this paragraph will contribute to the formation of the necessary practical skills used in related disciplines.

Exercise commentary

462. a) Additional question: "What does each of the relationships show?" For example, the ratio shows how many times the length AB is greater than the length of the AC, and the ratio shows how much of the length AC is from length AB.

466. b) Since the ratio is less than 1, then AC is less than BC, and therefore point C should be marked closer to point A.

474. a) Let's compose the relationship and compare them:, therefore, Boris's result is better.

478 , 479. Performed orally. The student must explain the meaning of each of the relationships.

480. Equal are the ratios of the sides and perimeters of the squares. It is useful to draw and illustrate once again the fact that the ratio of the areas of the squares is not equal to the ratio of their sides. You can offer students a few more similar problems by changing the numeric data.

Division in this respect

Methodical commentary

The ability to solve problems in division in this respect is based on the ability to solve problems in parts. Therefore, in a weak class, before considering an example (textbook, p. 128), you can offer a preparatory exercise:

1) Take a segment AB and divide it into 5 equal parts and mark a point on it WITH(fig. 5). In what respect is the point WITH divides the segment AB?

2) It is clear that AS : SV= 2: 3. If the length AB equals 15 cm, then you can find the lengths of the formed parts: AS= 15: 5 × 2 = 6 (cm), SV =
= 15: 5 × 3 = 9 (cm).

The division of values ​​in this respect is conveniently illustrated with the help of figures. We advise you to “draw” the problem more often at the first stage. For example, to the task 489 "A" you can make a schematic drawing (Fig. 6). Students got used to such schemes already in the 5th grade, solving problems in parts.

Exercise commentary

490. b) You can consider different methods of calculation, for example, the following: (h) = 40 (min); (h) = 50 (min). You can express 1.5 hours in minutes and then perform calculations.

491. a) We express the mass in one unit of measurement:

2 kg 550 g = 2550 g, or 2 kg 550 g = 2.55 kg.

Pay attention to the students that in the answer to this problem we indicate only one value:

1 kg 200 g (1.2 kg).

494. It is advisable to solve the problem on the board, dividing it into 4 parts.
In a notebook, the solution can be presented visually, depicting rectangles to scale, taking as the length of the perimeter, for example, 36 cells.

495. First, we find how many parts are in the segment SV: 5 - 2 = 3 (parts). From here we get: a) AS : SV= 2: 3; b) SV : AB= 3: 5; v) AB : AS =
= 5: 2; G) AB : SV = 5: 3.

496. If the ratio of the number of boys to the number of girls is 5: 4, then the number of boys is 5 parts, girls - 4 the same parts, and the number of all students in the school - 9 parts of the same. Therefore, boys from the number of all students in the school make up, and girls -.

497. First, you need to find the relationship in which the owner divided the food:
9 kg to 3 kg is 9: 3, ie 3: 1. Answer:.

498. This preparatory task for solving group problems B... It is necessary to be able to determine which of the two values ​​given in the relation is given in the condition, to be able to express the difference between the two given values ​​"in parts". It is advisable to consistently solve all the problems under this number in the classroom.

501. The whole number of pencils must be expressed in parts. The number of pencils in the small box is 5 pieces and the large box is 9 pieces. Three small boxes have 15 pieces and two large boxes have 18 pieces. We have: there are 66 pencils for 15 + 18 = 33 (parts), therefore, for 1 part there are 2 pencils. In a small box 2 × 5 = 10 (pencils), in a large box 2 × 9 = 18 (pencils).

503. The task is difficult, therefore, for a better understanding on the board, it is advisable to complete the drawing (Fig. 7). Now it becomes clear that the number of siskins is 5 parts, snakes - 4 parts, hedgehogs - 2 parts, and in total
11 parts. After that, another reasoning can be shown: multiplying both terms of the second ratio by 2 (so that its first term becomes equal to 4), we get 2: 1 = 4: 2. We get the same distribution of parts. Answer: 50 siskins,
40 snakes and 20 hedgehogs.

Main »task for interest

Methodical commentary

The study of the topic is a continuation of work begun at the beginning of the school year, when the concept of "percentage" was introduced and students became familiar with a wide range of problems in which it was encountered. Let's remind that the tasks were solved mainly meaningfully, on the basis of understanding the meaning of interest. The next step in mastering the concept of percentage is to familiarize students with the ability to associate percentages with decimal fractions and find the percentage of a number by multiplying by a fraction. Note, however, that when solving problems in which it is required to find the percentage of a number, the student can choose the solution himself.

Knowing by heart some facts (20% - this, 25% - this, etc.) is used in solving problems, and, in particular, it is very useful for solving problems for estimation (exercise 520 ).

Tasks involving an increase (decrease) in the value by several percent, during frontal work, it is desirable to solve in two ways, as shown in example 3 (p. 132 of the textbook), but the student should be given the right to limit himself to the first method or prefer the second.

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