Presentation on the topic “Interest”
Slide 0
Presentation on the topic “Interest”
Slide 1
Content. 1) Determination of percentage. 2) From the history of interest. 3) Three main types of problems involving percentages. 4) Examples of solving three main types of problems involving percentages. 5)Tasks for exercises. 6)Tasks for independent work. 7) Answers and solutions.
Slide 2
Percent (from the Latin pro cento - from a hundred) The hundredth part of any value or number is called. Indicated by: %
Slide 3
It is known that a percentage is one hundredth of a number, i.e. fraction. An interesting system of fractions was in ancient Rome. It was based on dividing a unit of weight into 12 parts, which was called ass. The twelfth part of an ace was called an ounce. And the path, time and other quantities were compared with a visual thing - weight. Due to the fact that in the duodecimal system there are no fractions with denominators of 10 or 100, the Romans had difficulty dividing by 10, 100, etc. from the history of interest
Slide 4
When dividing 1001 asses by 100, one Roman mathematician first received 10 asses, then divided the asses into ounces, etc. But he didn't get rid of the rest. To avoid having to deal with such calculations, the Romans began to use percentages. They took interest from the debtor (i.e., money in excess of what was lent). At the same time they said: not “the interest will be 16 hundredths of the amount of the debt,” but “for every 100 sesterces of the debt you will pay 16 sesterces of the interest.” And the same thing was said and there was no need to use fractions.
Slide 5
The % symbol was due to a typo. In manuscripts the word "prosentum" was often replaced by "cento". And in 1685 A book was published in Paris - a manual on commercial arithmetic, in which the typesetter mistakenly typed % instead of Cto. This is how this symbol appeared. Origin of the % symbol
Slide 6
The word "percent" comes from the Latin procentum, which literally means "per hundred." Already in the first codification of Roman law that has come down to us, Justinian’s Digest, dating back to the 5th century, one can find a completely modern use of percentages. The “Fisk” (imperial treasury) does not pay interest on contracts concluded by it, but receives interest itself: for example, from renters of public restrooms if these renters pay money too late; also in case of late payment of taxes. When the fiscus is the successor of a private person, it usually pays interest. Origin of the word "percentage"
Slide 7
The use of the word percentage as a norm in the Russian language begins at the end of the 18th century. This is evidenced by examples of problems on deposits: Problem of E. Voityakhovsky A merchant traded the 100 rubles put into the auction at a loss, so that the remaining amount after the first year without 4/25 of the total capital is equal to the remaining amount after two years. The question is: since he received a loss of 100 rubles. every year? Problem of T.P. Osipovsky Let us assume, for example, that capital consisting of 10,000 rubles at 5% is given to a pawnshop and another 800 rubles are deposited annually. The question is: after 12 years, how big will this capital be?
Slide 8
Ancient people tried to use percentages to solve problems, although they had no idea what they were. Your work is much easier: you just need to understand, imagine the significance of percentages and learn to work with them. And to begin with, let the following quatrain accompany you: At school, the teacher gives grades in the journal for our deeds. We call a hundredth of any number a percentage.
Slide 9
Three main types of percentage problems. 1) Find the indicated percentage of a given number. 2) Find the number based on the given value of its indicated percentage. 3) Find the expression of one number as a percentage of another.
Slide 10
Example 1. Problem 1. Of the 1800 hectares of the collective farm field, 558 hectares are planted with potatoes. What percentage of the field is planted with potatoes? Solution. 1800 hectares - 100% of the field 558 hectares - X% of the field Let's make a proportion. X=558*100/1800=31 31% - fields are planted with potatoes. Answer: 31%.
Slide 11
Example 2 Problem 2. A garment factory produced 1,200 suits. Of these, 32% are suits of a new style. How many new style suits did the factory produce? Solution. 1200 suits – 100% of the production of X suits – 32% of the new style Let’s make a proportion. X=1200*32/100=384 384 suits of a new style were produced by the factory. Answer: 384 suits.
Slide 12
Example 3. Problem 3. For a test in mathematics, 12 students received an o, which is 30% of all students. How many students are there in the class? Solution. 12 students is 30% of the class. X students is 100% of the class. Let's make the proportion X=12*100/30=40 40 students in the class. Answer: 40 students.
Slide 13
Task 4. To determine the germination of seeds, peas were sown. Of the 200 sown peas, 170 sprouted. What percentage of the peas sprouted?
Slide 14
Problem 5. In 8 months, the worker completed 96% of the annual plan. How much percent of the annual plan will the worker complete in 12 months if he works with the same productivity?
Slide 15
Problem 6. Sugar beets contain 18.5% sugar. How much sugar is contained in 38.5 tons of sugar beets? Round your answer to the nearest tenth of a ton.
Slide 16
Problem 1: Winnie the Pooh loved honey very much and began to breed bees, in the first year the bees gave 10 kg of honey, but Winnie the Pooh this was not enough, in the second year the bees increased honey production by 10%, but this was not enough Winnie the Pooh, he calculated that he needs about 13 kg of honey. The question is how many years must Winnie the Pooh wait to satisfy his needs, provided that the bees increase honey production by 10% every year?
Slide 17
Problem 7: When Tom Sawyer decided to give part of the money to his aunt and keep part of it for himself, so that by putting it in the bank at 5% per annum every year he would receive this interest for personal expenses, he even calculated that he needed approximately 300 per year dollars. How much should he put in the bank?
Slide 18
Task 8. The library has books in English and English German. English books make up 36% of all books, French books make up 75% of English books, and the remaining 185 books are German. How many books are there in the library?
Slide 19
Problem 6. The deposit placed in the savings bank two years ago reached an amount equal to 1312.5 rubles. What was the initial contribution at 25% per annum? Solution: To solve this problem, you need to understand that the result 1312.5 is the amount for the first year and plus 25% or 125% or 100% = 1050 rubles. We do the same with the amount of 1050, since the contribution was for two years 125% = 1050 rubles or 100% = 840 rubles. You can solve it in the second way, using the formula for compound interest 1312.5 = X · (1+ 0.25)2 X = 840 rubles. Answer: 840 rubles.
Slide 20
1 Task 1. Determine the percentage of components in each of these vitamin preparations (code positives 1–3).
Slide 21
Code positive 2
Slide 22
Code positive 3
Slide 23
Task 2. Determine the percentage of each type of flower in the bouquet, if there are 100 flowers in each bouquet. Code positive 4
Slide 24
Task 4 (Code Positive 6). In the 17th century, rhubarb was imported to Russia from China. Calculate the percentage of the number, use the answer key and name the Siberian historian and cartographer who indicated where rhubarb grows in Siberia. Each person completes the task individually. Code positive 5
Slide 25
Correct answer: Remezov.
Slide 26
Task 5. Determine the mass of each component in the recipe. Code positive 7
Slide 27
Answer to task 5. Determine the mass of each component in the recipe. Code positive 10
Slide 28
Task 6. Determine the percentage of each component in the recipe. Code positive 9
Slide 29
Answer to task 6. Determine the mass of each component in the recipe. Code positive 6
Slide 30
Task 7. Determine the mass of each component in the recipe. Code positive 8
Slide 31
Answer to task 7. Determine the mass of each component in the recipe. Code positive 11
Slide 32
Task 8. Perform the calculations and you will find out by what percentage the number of microbes in the room is reduced by volatile phytoncides of indoor plants.
Slide 33
Task 8. Perform the calculations and you will find out by what percentage the number of microbes in the room is reduced by volatile phytoncides of indoor plants.
Slide 34
Solution. Sowed 200 g - 100% Germinated 170 g - X% Let's make the proportion 200/170=100/X 200X=17000 X=17000/200=85 Germination percentage 85% Answer: 85%
Slide 35
Solution Completed 8 months - 96% Completed 12 months - X% 8:12=96:X X=96*12:8=144% 144% - the worker will complete the annual plan in 12 months. Answer: 144%
Slide 36
Solution. Sugar beets 38.5 t - 100% Sugar X t - 18.5% Let's make the proportion: 38.5:X=100:18.5 X=38.5*18.5:100=7.1 t 7.1 tons of sugar in 38.5 tons of sugar beets. Answer: 7.1 t.
Slide 37
Solution. 5%-300 dollars 100%-X dollars Let's make a proportion: X=300*100:5=6000 dollars. Tom must deposit $6,000 in the bank. Answer: $6000.
Slide 38
Solution. 75% = 3:4 means 36% · 3:4 = 27% French, books of the total number. 36% + 27% = 63% are English and French books combined. 100% – 63% = 37% of all German books. 185: 37% = 5 books is 1%. Total books in libraries 100% · 5 = 500 books. Answer: 500 books.
Slide 39
Solution: In order to find out how long Winnie the Pooh needs to wait, you need to find out how much he will have in a year, which will be 11 kg, in two years 12.1 kg, and only in the third year will he satisfy his needs. Answer: 3 years.
Presentation for the lesson “Percentages”
Percentage problems
Mathematic teacher
Samokhina Olga Vasilievna
Didactic part
- Preliminary preparation for the lesson:
U
Students should know what is called a percentage, be able to convert a decimal into a percentage, and convert percentages into a decimal.
- Lesson objectives:
1) Educational
– consolidation of students’ knowledge and skills on the topic “Percentages”, development of thinking skills when solving problems with percentages.
2) Developmental
– testing the ability to independently apply knowledge in standard, as well as in modified non-standard conditions, development of logical thinking, outlook, attention.
3) Educational
– fostering mutual assistance, mathematical culture, and diligence.
- Lesson type:
combined. - Teaching methods:
verbal, visual, control and self-control. - Forms of lesson organization:
individual, frontal. - Equipment:
screen, media projector, personal computer.
Lesson plan:
- Organizing time.
- Independent work.
- Learning new material.
- Consolidation. Problem solving.
- Summarizing. Homework.
Independent work
Checking independent work
4
Learn to solve problems using percentages.
The simplest problems involving percentages can be divided into 3 types.
1 type
.
The problem requires finding the percentage ratio of two numbers. Problem:
A master turned 40 parts in 1 hour.
Using a cutter, he began turning 10 more parts per hour. By what percentage did labor productivity increase? Solution:
1) 10: 40 = 0.25 – the part that 10 makes up from 40.
2) 0,25 = 25 %
The labor productivity of the master increased by 25%.
Answer:
25%.
5
Conclusion:
To find what percentage one number is of another, you need to divide the first number by the second and write the resulting fraction as a percentage
5
Type 2 The problem requires you to find the percentage of a given number.
Task:
The master turned 40 parts in 1 hour.
By using a cutter, he increased labor productivity by 25%. How many more parts per hour did the master begin to turn? Solution:
1) 25 % = 0,25
2) 40 × 0.25 = the master began to grind 10 (children) more per hour.
Answer:
10 details.
5
5
Conclusion:
To find the specified percentage of a given number, you need to replace the percentage with a decimal fraction and multiply this number by the resulting decimal fraction.
5
Type
3 The problem requires finding a number if it is known what percentage of the whole the given number is. Task:
A master turned a certain number of parts in 1 hour.
Using a cutter, he began turning 10 more parts per hour, which was 25% of the previous number of parts. How many parts per hour did the master turn earlier? Solution:
1) 25% = 0,25
Let
x the required number of parts. According to the problem, 25% of the number x is 10 parts.
Let’s create and solve the equation: 0.25
x = 10
x
= 10: 0.25
x
= 40
The master had previously turned 40 parts per hour.
This can be solved more briefly: 2) 10: 0.25 = 40 (details) per hour the master turned earlier.
Answer:
40 parts.
5
Conclusion:
To find a number by its percentage, you need to replace the percentage with a decimal fraction and divide the number by the resulting decimal fraction.
- To find a number by its percentage, you need to replace the percentage with a decimal fraction and divide the number by the resulting decimal fraction.
- To find a number by its percentage, you need to replace the percentage with a decimal fraction and divide the number by the resulting decimal fraction.
5
Let's write three rules in one table
5
Consolidation. Problem solving.
- №1.
The school has 700 students. Among them are 357 boys. What percentage of the students at this school are boys?
- Read the problem.
- How many students are there in the school?
- How many of them are boys?
- Tell us how you will solve this problem.
Solution:
1)
357 : 700 = 0.51
2)
0.51 = 51% of school students are boys.
Answer:
51%.
5
- №2
Cotton harvesters were working on the field, whose area is 620 hectares. In one day they removed 15% of the entire field. How many hectares of cotton were harvested in one day?
- Read the problem.
- What is this problem about?
- What is the area of the field?
- How much was removed per day?
- Tell us how you will solve this problem.
Solution:
1)
15% = 0.15
2)
620 × 0.15 = 93 (ha) cotton was harvested per day.
Answer: 93 hectares.
- №3
The student read 138 pages, which is 23% of all pages in the book. How many pages are in the book?
- Read the problem.
- How many pages did the student read?
- Did he read the whole book?
- Tell us how you will solve this problem.
Solution:
1)
23% = 0.23
2)
138 : 0.23 = 600 (pages) in the book.
Answer: 600 pages.
Summarizing. Homework.
Summarizing:
- What is a percentage?
- How to find the percentage of two numbers?
- How to find the percentage of a number?
- How to find a number by its percentage?
- What percentage problems have you learned?
Homework
Presentation “Compound Interest in Real Life”
#Secondary vocational education #Project activity #Presentation
Compound interest in real life Maxim Denisovich Isakov, 2nd year student. State Autonomous Professional Educational Institution of the Samara Region "Tolyatti College of Service Technologies and Entrepreneurship" Scientific supervisor: Dashkina Mariyam Nikolaevna
Goal: To understand complex financial mechanisms; choose for yourself the optimal strategy for managing your own funds. Objectives: Consider the concept of compound interest; Learn to solve problems involving compound interest; Conduct a study using the example of three banks in the Russian Federation and determine where it is profitable to invest money. Relevance: A huge number of people invest their funds in banks at certain interest rates. This is the relevance of my work, in which I want to show the use of compound interest in life.
Compound Interest in Real Life Albert Einstein said that compound interest is the most powerful force on earth and the most remarkable discovery of mankind. There are many examples in history that prove the magical power of compound interest. For example, we can recall the remarkable act of Benjamin Franklin. Franklin, who died in 1790, bequeathed $5,000 each to two of his favorite cities, Boston and Philadelphia. According to the terms of the will, the cities could receive this money in two installments, 100 and 200 years after the will came into force. After 100 years, each city could take $500,000 to finance public works, and after another 100 years, all the money from the account. 200 years later, in 1990, the cities received approximately $20,000,000 each. Franklin very clearly demonstrated the benefits of compound interest.
Compound interest in real life Compound interest is a form of income calculation based on adding accrued but unpaid interest to the amount of debt, calculating interest on interest, calculating interest for two or more periods, carried out in such a way that interest is calculated not only on the original amount , but also on the interest accrued in the previous period.
Compound interest formula Meaning of symbols: I – annual interest rate; j – The number of calendar days in the period following which the bank capitalizes accrued interest; k– Number of days in a calendar year (365 or 366); P – Initial amount of funds attracted to the deposit; n is the number of operations to capitalize accrued interest during the total period of raising funds; S - the amount of funds due to be returned to the depositor at the end of the deposit period. It consists of the amount of the deposit plus interest.
Simple interest. You invested RUB 100,000. for 15 years at 20%. There are no additional fees. You withdraw all profits. SUM = 15 * (100,000 / 100 * 20) = 300,000 rubles Profit amounted to 200,000 rubles.
Compound interest You invested 100,000 rubles. for 15 years at 20%. There are no additional fees. Each year, interest profits are added to the principal amount. rubles The profit will be 1,700,000 rubles, which is 1,500,000 rubles more than for simple interest. Investments using compound interest are an order of magnitude more profitable than with simple interest.
Investments using compound interest are much more profitable than investments using simple interest. Simple interest Compound interest Amount Profit for the year Amount Profit for the year After 1 year 60,000 rub. 10,000 rub. 60,000 rub. 10,000 rub. After 2 years 70,000 rub. 10,000 rub. 72,000 rub. 12,000 rub. After 3 years 80,000 rub. 10,000 rub. 86,400 rub. 14,400 rub. After 4 years 90,000 rub. 10,000 rub. RUB 103,680 RUR 17,280 After 5 years, 100,000 rubles. 10,000 rub. RUB 124,416 RUR 20,736 After 6 years 110,000 rub. 10,000 rub. RUB 149,299 RUR 24,883 After 7 years 120,000 rub. 10,000 rub. RUB 179,159 RUB 29,860 After 8 years 130,000 rub. 10,000 rub. RUR 214,991 RUB 35,832 After 9 years 140,000 rub. 10,000 rub. RUB 257,989 RUR 42,998 After 10 years, 150,000 rubles. 10,000 rub. RUB 309,587 RUB 51,598 After 11 years, 160,000 rubles. 10,000 rub. RUB 371,504 RUB 61,917 After 12 years, 170,000 rubles. 10,000 rub. RUB 445,805 RUB 74,301 After 13 years, 180,000 rubles. 10,000 rub. RUB 534,966 RUB 89,161 After 14 years, 190,000 rubles. 10,000 rub. RUB 641,959 RUB 106,993 After 15 years, 200,000 rubles. 10,000 rub. RUB 770,351 RUB 128,392 Total profit: 150,000 rub. RUB 720,351
In the case of simple interest, the graph of capital increase is linear, since you withdraw profits and do not let them work and bring in new profits. In the case of compound interest, the graph turns out to be exponential; over time, the curve for increasing capital becomes steeper and tends more and more upward. This happens because, year after year, profits accumulate and create new profits.
Compound interest in real life Sberbank Deposit “Save” A deposit for reliably preserving your savings and receiving a guaranteed stable income. P = 100,000 rubles n = 12 I = 7.70% j = 30 days k = 365 days Thus, = 107,864 (rubles) This means that if you invest a deposit of 100,000 rubles in the “Save” deposit at the end of the period of 1 year, the profit will be 7,864 rubles.
Compound interest in real life Deposit “Manage” (replenishable, with partial withdrawal) Deposit for secure storage of your savings with the ability to withdraw part of the funds before the expiration of the deposit without loss of interest. P = 100,000 rubles n = 12 I = 6.15% j = 30 days k = 365 days Thus, = 106,237 (rubles) This means that when investing a deposit of 100,000 rubles into the “Save” deposit (without replenishment) at the end of the period in 1 year profit will be 6,237 rubles.
Compound interest in real life VTB 24 “Profitable” deposit Do you need a deposit with increased profitability? Choose the “Profitable” deposit! Trust us to take care of preserving and increasing your finances. P = 100,000 rubles n = 12 I = 8.05% j = 30 days k = 365 days Thus, = 108,535 (rubles) This means that if you invest a deposit of 100,000 rubles in the “Profitable” deposit, at the end of the period of 1 year the profit will be 8,535 rubles.
Compound interest in real life “Accumulative” deposit (with the possibility of replenishment) Do you prefer to approach the issue of saving money wisely? Top up your deposit without restrictions and get additional benefits from your savings! P = 100,000 rubles n = 12 I = 7.90% j = 30 days k = 365 days Thus, = 108,076 (rubles) This means that when a deposit of 100,000 rubles is invested in the “Accumulative” deposit (without additional contributions) at the end of the term in 1 year the profit will be 8,076 rubles.
Compound interest in real life Alfa Bank Deposit “Victory” P = 100,000 rubles n = 12 I = 8.84% j = 30 days k = 365 days Thus, = 109,075 (rubles) This means when investing a deposit of 100,000 rubles into the deposit “Victory” at the end of the 1-year period, the profit will be 9,075 rubles.
Compound interest in real life From this study we see that of the banks we reviewed, it is more profitable to invest money in Alfa-Bank. Sberbank is inferior to other banks in terms of profitability due to the low interest rate. But, despite this, about 50% of Russian citizens choose Sberbank. This is due to the fact that this bank is considered the most reliable, and also has the most developed branch network. From this we can conclude that the placement of one’s temporarily free funds depends not only on profitability, but also on the image of the bank and on the development of its branch network.
Compound interest in real life Conclusion In the process of work, I researched compound interest, analyzed the deposits of three banks in the Russian Federation and learned to solve problems on compound interest and compile them myself. Thus, the goal of the work has been achieved. This work carries significant practical significance, which in the future will help to place your funds more wisely. We see further prospects for our work as considering a larger number of banks, perhaps not only in Russia, and also solving more difficult problems on the topic under consideration.
