Solving problems for percentages in VPR in Mathematics

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Regional scientific conference

“Discovery”

Solving problems for percentages in VPR in Mathematics

Performed by student of 8t hform

Varlamova Sofya

Supervisors :

Nakorenok Dina Alexandrovna

Artemova  Natalia Ivanovna

Saratov, 2024

Content

Introduction                                                                                        3

1.The main kinds of percent Math problems                                        4

2. Modern percent Math problems’ examples                                         9

3.My searches                                                                                15

Conclusion                                                                                        17

References                                                                                        18

Introduction

Percent (lat. Percent «one hundredth») ; is denoted with the sign «%»; is used for meaning of part at the ratio to the whole, integral.

 At first, I looked through the History, Geography and Biology students’ books. The searching shows the information in percent can be seen in every school student’s book. Then I asked 38 students two questions:

1. Is it important to study “Percent”?

2. Where can “Percent” knowledge be used?

My survey results are next: 1. Important - 34 students (89 %). Unimportant -3 students (8 %). I don’t know - 1 student (3 %).

I decided to prove the importance of “Percent” in my work and show a wide variety of using percent in a daily life.

My search goal is to consider solutions to tasks studied in the 6th, 7th and 8th forms and study the application area of percent. More than that I would like to prove, that we can not avoid percent Math problems in our life.

My search scopes are:

-to study the main kinds of percent Math problems;

-to show the way of using “percent” by solving real Math problems from different human life fields;

-to pursue research in statistics;

-to make a conclusion.

My search object is percent Math problems from our daily life.

My search subject is the way of solving real percent problems and using percent calculation in different fields of human being.  

Methods: searching method by using study materials and necessary information through the Internet; practical technique by percent calculating from daily life; analysis of results.

1. The main kinds of percent Math problems

Marks for Math VPR are equivalent to marks for control works or tests. As a rule, the teachers give out good marks in the class register. But for the objectivity of the assessment of schools, the state uses clear criteria. On Math VPR in 6th form, in order to get excellent, you need to score from 14 to 16 points, for '' good '', 10 to 13 points are enough, '' satisfactory '' will be given by typing from 6 to 9 points. It is important to understand that Math VPR (not only Math VPR in mathematics, but also in any other subject) is a standardized work. All tasks are typical, each of them tests a specific skill or topic. All these skills are developed during six years of studying at school.

The work lasts for one hour and it contains 13 tasks. Tasks are divided into three levels of difficulty: basic, advanced and the high standard of knowledge. The maximum score for test is 16 points. For tasks 9, 11, 13, you can get 2 points! Let's take a closer look at what are the specifics of Math VPR tasks in mathematics for grade 6.

Task 1 - the ability to operate with integers.

Task 2 - the ability to operate with ordinary fractions.

Task 3 - the ability to find a part of a number (or vice versa, a whole by its part)

Task 4 - the ability to operate with decimal fractions.

Task 5 - the ability to assess the real size of the objects around us.In order to solve the first 5 tasks, you should be focused and attentive as it’s possible. They check how much the student is counting correctly.

Task 6 - Ability to work with tables and diagrams.

Task 7 - the ability to operate with the number module.

Task 8 - the ability to compare rational fractions.

Task 9 - it is required to find the value of the expression (if there is a mistake in one of the actions, 1 point is given).

Task 10 - Ability to work with logical problems (foundations of set theory)

Task 11 - the ability to solve word problems on percentages and relationships (with one mistake or insufficient justification, 1 point is given).

Task 12 - the ability to apply geometry in the simplest problems (the answer must be presented in the form of a picture).

Task 13 - the ability to think logically and conduct mathematical reasoning (with logical gaps in the solution, 1 point is given).

Analysis of Math VPR in the 6 form in 2021.

And now we consider the criteria for evaluating work in the 7th form.

 Tasks 1 and 2 will check the knowledge of the concepts of negative number, common fraction, decimal and numerical digits.

Task 3 tests the ability to extract information presented in tables or graphs.

In task 4, the mastery of the basic units of measurement of length, area, volume, mass, time, speed is checked.

Task 5 tests the ability to solve word tasks by percentage.

Task 6 is aimed at testing the ability to solve simple logical tasks, as well as find an intersection, union, subset in the simplest situations.

Task 7 tests the ability to find out the information presented in the diagrams and make estimates.

Task 8 checks the knowledge of the concepts '' function '', '' function graph '', '' ways of defining a function ''.

 In task 9, the ability to solve the tests.

Task 10 check the ability to extract the necessary information from the task; make estimates for practical calculations.

 In task 11, the ability to perform transformations of literal expressions using abbreviated multiplication formulas.

 Task 12 tests the ability to compare fractions, decimals and mixed numbers.

 Tasks 13 and 14 test the ability to operate with the properties of geometric shapes, apply geometric facts to solve problems.

 Task 15 tests the ability to present data in the form of tables, diagrams, graphs.

 Task 16 tests the ability to solve word problems for productivity, shopping, movement.

In tasks 1-9, 11 and 13, you only need to write down the answer.

 In task 12, you need to mark points on the number line.

 In task 15, you need schematically plot a function graph. In tasks 10, 14, 16, it is required to write down the solution and the answer.

The correct solution to each of the tasks 1-11, 13, 15 is1 point.

The task is correctly if the student gave the correct answer: wrote down the correct number, the correct value; depicted the correct drawing.

Completion of tasks 12, 14, 16 is estimated from 0 to 2 points.

Converting points to grade

"2": 0-6;  "3": 7-11;  "4": 12-15;  "5": 16-19.

Analysis of Math VPR in the 8 ’’b’’ form (7th form) in September 2020.

We see that most of the class did not cope with the assignments. In the 6th form they did not cope with the tasks 6 and 11, and in the 7th form they did not cope with the tasks of 3,5,7,10,15.

We know from the task execution script that these are interest tasks. In my opinion, the most difficult topic istasks for percentages.

Therearethreemain kinds of percent Math problems:

Problem1. Find the specified percent by a given number. The given number can be multiplied of the specified percent, and then the intersection is divided by one hundred.

500 pumpswereproducedbyafactory in aquarter. 60 % from these pumps were of the highest quality. Howmanypumpsofthehighestqualitywere produced by the factory?

Solving:

Find60 % of 500 (the whole pumps’ quantity).60 % = 0,6500 • 0,6 = 300 pumps of the highest quality.

Answer: 300 pumps of the highest quality.

 Problem 2.Find the specified percent by a given number and its coefficient in percent of an unknown number. The given number is divided in its percent and multiplied of 100. 

A student read 138 pages, what makes 23% of the whole number of pages in the book. How many pages are there in the book?

Solving:

So, we don’t know how many pages are there in the book. But we know that the part, what was read by the student is 23% of the whole quantity of pages in the book. As138 pages are only a part of the whole, the number of pages in the whole book is higher than 138.

Problem3. Find percent of one number at the other. To find out percent of one number at another, they must the part, which is asked divide into the total number and multiply of 100 %.

16 watermelons from 200 ones turned out to be unripe. What percent of all watermelons make unripe ones?

Solving:

What is asked about? It is asked about unripe watermelons. We divide 16 into the whole quantity of watermelons and multiply of 100 %.

Answer: 8 % - make unripe watermelons of all ones.

2. Modern percent Math problems’ examples

1. «Seasonsale»

Problem №1.The shop “September” reduced prices for rubber shoes by 15% during the season sale. How many rubles can be saved by buying rubber shoes, if we know that they cost 1100 rubles before price reducing?  

Solving:

1. 1100:100 * 15 =165 (rubles) - sale
2. 1100-165=935 (rubles) – the shoes price during the season sale.

Answer: 935 rubles.

Problem №2. A customer can get a present, which costs 20 % of spent money during the New Year sale action in the shop. Can the customer hope to get a picture, which costs 365 rubles, if he or she has already spent 1850 rubles? Solving:

1. 1850 :100 *20=370 (rubles)

Answer: The customer can hope to get a present as 20 % of spent sum is more than the picture price.

2.«Credit»

Problem №1.A mother took out a loan 30 000 rubles for one year at interest 12%for buying a computer . She has to pay back a loan monthly with equal sum of money to return the whole sum with percent in a year. How much money has she to pay back every month?

Solving:

Let’s find how much money has the mother to pay back – to sum 30 000 we add 12% of 30000 rubles: 30000+(30000:100*12) = 33600 (rubles) – the whole debt.

33600 : 12 =2800 (rubles) – monthly debt sum.

Answer: 2800 rubles.

3.«Bankdeposit»

A bank assess 13% interest of damages and the sum, which is paid in comes to 100 000 rubles. What sum will be assessed depositor’s bill in three years? Solving:

 1)100000+100000*0,12=112000 (rubles) – in one year. 

1. 112000+112000*0,12=125440 (rubles) – in two years.
2. 125440+125440*0,12=140492,8 (rubles) – in three years.

Answer: 140492, 8 rubles.

4.“Social payments”

The pension of my grandmother payment was 14 440 rubles in March. In April she got 15 760 rubles. How many percent did my grandmother's pension payment become higher?

We can solve this problem through the ratio:

14 440 rubles – 100 %

15 760 rubles  -х %

Solving:

Х=15 760 *100 : 14440 =109 (%) – became the pension payment.

109 – 100 = 9 %

Answer: the pension payment became higher 9 %.

5. “Using of household appliances”

Allowable voltage for washing-machine (according to the pass)  is 220 V ± 5 %. What range of electrical voltage can the machine be used in?

Solving:

220:100 *5= 11 (V).

220-11=209(V) 220+11=231 (V)

Answer: the range of electrical voltage can vary from 209 till 231 V.

6. “Salary”

A father got 36 700 rubles including award. In work contract it is denoted that accident free work is paid 15 % of base pay. What is the father’s salary without award?

Solving: 36 700 rubles – 115 %

Хrubles - 100 %

Х =36 700 *100: 115 =31 913 (rubles).

Answer: 31 913 (rubles) is the father’s salary without award.

7. “Travelling”

A railway adult’s ticket costs 820 rubles, student’s one costs  40% of adult’s ticket price. A family consists of two school-aged children and two adults. How much money does the family spend on tickets?

Solving:

1) 820 :100*40=328 (rubles).

2) 820 *2+328*2=2296 (rubles).

Answer: 2296 (rubles).

8. “Cooking”

Problem № 11.A refectory bought 50 kg of fish for cooking fried fish. Find out how many kg of scaled fish is needed for the dish, if the by-product norm makes 27 %?

Solving:

50 : 100 *27=13,5 (kg) –by-products

50-13,5=36,5 (kg) fish.

For example:

1.The price of electric kettle was made 21% higher and made 3025 rubles. How much does the kettle cost before the price was made higher?

Pay attention that 3025 rubles is the price after making it 21% higher. Let’s make a proportion: 3025 is 121%, 100% - the price before rising.

3025 rubles - 121%

х rubles     -  100%

Answer: 2500

2. A T-shirt costs 1200 rubles. After the price reducing it cost 972 rubles. How many percent was the T-shirt cheaper after that?

We find out what percent from 1200 makes the reduced price.

1200 rubles - 100%,

972 rubles - х %.

Make a proportion:

1200 rubles   -  100%

972 rubles      -   х %

It means that 972 is 81% from 1200 rubles.

The price was 100-81=19% cheaper.

Answer: 19

3. In N town live 250000 citizens. 15 % of children and teenagers live with. 35% of citizens do not work (pensioners, students, housewives). How many citizens work?

Why do students seldom solve this kind of problems correctly? The fact is that “15 percent” or “35 percent” are relative numbers. Every time different numbers can be considered as 100%. As in the rule: in different occasions we take up everything we are comparing for 100%.So, children and teenagers make15% of 250000 of all citizens. It means that their number is 15% from 250000,

Answer: There are 37500 of children and teenagers in the town.

Consequently there are 250000-37500=212500 adults in the town.35%adults do not work. Now we take up number of adults for 100%. It turns out that the number of citizens who do not work is35% from  212500.

Make a proportion:

212500      -  100%

 х              -   35%

So, the number of working adults is 212500-74375=138125 men.

Be attentive bysolving such problems. Read statement of problems attentively, do not forget what you have to find out.

Answer: 138125

4.Railway adult’s ticket costs 720 rubles. Railway children’s ticket costs 50% from the adult’sticket price. A group consists of 15 students and 2 adults. How much do the tickets for the whole group cost?

1) 720 :100*50=360 (rubles).

2) 720 *2+360*15=6840 (rubles).

Answer: 6840

4. Mysearches

At the beginning of the study of the topic, the class and I solved 10 problems for percentages, but I was very upset the results.

               Work statistics

Then I offered the class the problems I had analyzed for percentages. We have calculated and analyzed each of them. Then they wrote a similar first test work.

Work statistics

The result surprised and delighted, my work was useful to my classmates. The quality of the work has increased.

Girls and boys percentage in the 8 form

Conclusion: The girls’number are 30 % more in our class then boys.

Quality and progress in the second term

Conclusion: The progress in our form is 40 % higher than knowledge quality.

Conclusion

The work is finished. It was interesting for me. I have learnt the history of percent better and found out that percent’s are very important in our everyday life.  To sum up everything I have searched I make a conclusion that the theme is actual.

Many life branches require knowledge of percent counting: taking on a loan, deposits, buying goods on credit, tax countingand advertisement’s actions: rebate counting in percentage and so on.

Nowadays everybody have to understand and to use percent.

I would like to conclude that percent knowledge and the counting are important for every modern person not only in one’s occupation,and in everyday life.

References

1. Виленкин, Н. Л., Жохов, В. И., Чесноков, А. С., Шварцбурд, С. И. Математика 6. – М.: Дрофа, 2006. – 288с.

2. Виленкин, Н. Л. За страницами учебника математики. – М.: Просвещение, 1989. –73с.

3. Захарова А.Е. Несколько задач «про цены».// Журнал «Математика в школе». – 2002– №8

4. Зубарева И. И. Еще раз о процентах.// Журнал «Математика в школе». – 2006– №10

5. http://mathist.narod.ru/razmerz.htm математический кружок в 5-6

6. https://math6-vpr.sdamgia.ru

7. https://oge.sdamgia.ru

8. https://math7-vpr.sdamgia.ru

9. https://math8-vpr.sdamgia.ru