The rule for finding a number by its fraction:
To find a number by given value its fractions, you need to divide this value by the fraction.
Let's look at how to find a number by its fraction, using specific examples.
Examples.
1) Find a number whose 3/4 are equal to 12.
To find a number by its fraction, divide the number by that fraction. To do this, you need to multiply this number by the inverse of the fraction (that is, by an inverted fraction). To do this, you need to multiply the numerator by this number and leave the denominator unchanged. 12 and 3 by 3. Since we got one in the denominator, the answer is an integer.
2) Find a number if 9/10 of it equals 3/5.
To find a number given the value of its fraction, divide this value by this fraction. To divide a fraction by a fraction, multiply the first fraction by the inverse of the second (inverted). To multiply a fraction by a fraction, multiply the numerator by the numerator, and the denominator by the denominator. We reduce 10 and 5 by 5, 3 and 9 by 3. As a result, we get the correct irreducible fraction, which means this is the final result.
3) Find a number whose 9/7 are equal
To find a number by the value of its fraction, divide that value by that fraction. Mixed number and multiply it by the inverse of the second number (inverted fraction). We reduce 99 and 9 by 9, 7 and 14 by 7. Since we got improper fraction, it is necessary to select an entire part from it.
1. The distance between two villages is 24 km. During the first week, the team paved this distance. How many kilometers remain to be paved?
2. 12 birds were sitting on a branch; more of them flew away. How many birds are left sitting on the branch?
3. There are 32 students in the class, all students went skiing. How many students have not skied?
4. The cyclists covered 48 km in two days. On the first day they drove all the way. How many kilometers did they travel on the second day?
5. Dad, having 3,500 rubles, spent his money. How much money does he have left?
6. The notebook has 24 pages. The entries cover the numbers of all pages of the notebook. How many blank pages are there in the notebook?
7. Autotourists drove 360 km in three days. On the first day they traveled, and on the second day - the entire distance. How many kilometers did the motor tourists travel on the third day?
8. There are several boys and 24 girls in the drama club. The number of boys equals the number of girls. How many students are in the drama club?
9. What is the amount of money if 12 rubles constitute the available amount?
10. During the first week, the brigade paved 15 km, which was the distance between the two villages. What is the distance between villages?
11. Determine the length of the segment whose length is 15 cm.
12. My son is 10 years old. His age is the age of his father. How old is father?
13. Daughter is 12 years old. Her age is the age of her mother. How old is the mother?
14. In 1 hour the bus covers the entire distance. How many hours will it take him to cover the entire distance?
15. The boy read the entire book in 10 minutes. How long can he read the entire book?
16. There are 18 boys and 16 girls in the class. boys and girls study in a literary circle. How many students are in the literature club?
17. The typist has 120 sheets of paper. She used all the sheets first, and then the remaining ones. How many sheets of paper did the typist use?
18. When all the apples have been cut for the compote, there are still 4 apples left. How many apples were there in total?
19. The boy had 240 rubles. He spent this amount and the rest. How much money did he spend?
20. It was 1000 rubles. We spent this amount on the first purchase, and the rest on the second. How many rubles are left?
21. When you have read 35 pages, there are books left to read. How many pages are in the book?
22. On the first day they read, and on the second - the numbers of all the pages of the book. After that, there are 80 pages left to read. How many pages are there in the book?
23. Tourists walked 48 km in three days. On the first day they walked the entire distance, and on the second day they walked the rest. How many kilometers did they walk on the third day?
24. Half of the books in the school library are textbooks, and a sixth of all textbooks are mathematics textbooks. What proportion of all books are mathematics textbooks?
25. Mom spent half the money and the rest. She has 6,000 rubles left. How much money was there initially?
26. 4 friends came to Vasya’s birthday. The first received the pie, the second - the remainder, the third - the new remainder. Vasya divided the rest of the pie equally with his fourth friend. Who got the most?
27. Reduce 90 rubles. on this amount.
28. Increase 80 rubles by this amount.
29. The son is 8 years old, his age is the age of his father. The father's age is the grandfather's age. How old is grandpa?
| № | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |
| otv | 9 | 4 | 8 | 16 | 1000 | 9 | 81 | 33 | 16 | 24 | 25 | 35 | 30 | 6 | |
| № | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 25 | 26 | 27 | 28 |
| otv | |||||||||||||||
| 50 | 8 | 60 | 12 | 150 | 200 | 49 | 300 | 16 | 1/12 | 18000 | 81 | Vasya and 4 friends | 81 | 116 |
|
BASIC TYPES OF SOLVING PERCENTAGE PROBLEMS
I. FINDING A PART OF THE WHOLE
To find a part (%) of a whole, you need to multiply the number by the part (percent converted to a decimal fraction).
EXAMPLE: There are 32 students in the class. During test work 12.5% of students were absent. Find how many students were absent?
SOLUTION 1: The integer in this problem is the total number of students (32).
12,5% = 0,125
32 · 0.125 = 4
SOLUTION 2: Let x students be absent, which is 12.5%. If 32 students –
total number of students (100%), then
32 students – 100%
x students – 12.5%
ANSWER: There were 4 students missing from the class.
II. FINDING THE WHOLE BY ITS PART
To find a whole from its part (%), you need to divide the number by the part (percents converted to a decimal fraction).
EXAMPLE: Kolya spent 120 crowns in the amusement park, which amounted to 75% of all his pocket money. How much pocket money did Kolya have before coming to the amusement park?
SOLUTION 1: In this problem you need to find the whole if the given part and value are known
this part.
75% = 0,75
120: 0,75 = 160
SOLUTION 2: Let Kolya have x crowns, which is a whole, i.e. 100%. If he spent 120 crowns, which was 75%, then
120 CZK – 75%
x CZK – 100%
ANSWER: Kolya had 160 crowns.
III. EXPRESSION AS A PERCENTAGE OF THE RATIO OF TWO NUMBERS
SAMPLE QUESTION:
WHAT % IS ONE VALUE FROM ANOTHER?
EXAMPLE: The width of the rectangle is 20m and the length is 32m. What % is the width of the length? (Length is the basis for comparison)
SOLUTION 1:
SOLUTION 2: In this problem, the length of a rectangle of 32m is 100%, then the width of 20m is x%. Let's compose and solve the proportion:
20 meters – x%
32 meters – 100%
ANSWER: The width is 62.5% of the length.
NB! Notice how the solution changes as the question changes.
EXAMPLE: The width of the rectangle is 20m and the length is 32m. What % is the length of the width? (Width is the basis for comparison)
SOLUTION 1:
SOLUTION 2: In this problem, the width of a rectangle of 20m is 100%, then the length of 32m is x%. Let's compose and solve the proportion:
20 meters – 100%
32 meters – x%
ANSWER: The length is 160% of the width.
IV. EXPRESSION AS PERCENTAGE OF CHANGE IN QUALITY
SAMPLE QUESTION:
BY HOW MUCH % DID THE INITIAL VALUE CHANGE (INCREASED, DECREASED)?
To find the change in value in % you need to:
1) find how much the value has changed (without %)
2) divide the resulting value from step 1) by the value that is the basis for comparison
3) convert the result to % (by multiplying by 100%)
EXAMPLE: The price of the dress has decreased from 1250 CZK to 1000 CZK. Find by what percentage the price of the dress has decreased?
SOLUTION 1:
2) The basis for comparison here is 1250 CZK (i.e. what it was originally)
3)
ANSWER: The price of the dress has decreased by 20%.
NB! Notice how the solution changes as the question changes.
EXAMPLE: The price of the dress increased from 1000 CZK to 1250 CZK. Find by what percentage the price of the dress has increased?
SOLUTION 1:
1) 1250 –1000= 250 (kr) how much the price has changed
2) The basis for comparison here is 1000 CZK (i.e. what it was originally)
3)
Solving a problem in one step:
SOLUTION 2:
1250 –1000= 250 (cr) how much the price has changed
In this problem, the initial price of 1000 kroner is 100%, then the change in price of 250 kroner is x%. Let's compose and solve the proportion:
1000 CZK – 100%
250 CZK – x%
x =
ANSWER: The price of the dress has increased by 25%.
V. CONSEQUENTIAL CHANGE OF QUANTITY (NUMBER)
EXAMPLE: The number was reduced by 15% and then increased by 20%. Find by what percentage the number has changed?
The most common mistake: the number increased by 5%.
SOLUTION 1:
1) Although the original number is not given, for ease of solution it can be taken as 100 (i.e. one integer or 1)
2) If the number is decreased by 15%, then the resulting number will be 85%, or from 100 it would be 85.
3) Now the result obtained must be increased by 20%, i.e.
85 – 100%
and the new number x is 120% (since it has increased by 20%)
x =
4) Thus, as a result of the changes, the number 100 (original) changed and became 102, which means that the original number increased by 2%
SOLUTION 2:
1) Let the initial number X
2) If the number decreased by 15%, then the resulting number will be 85% of X, i.e. 0.85X.
3) Now the resulting number must be increased by 20%, i.e.
0.85Х – 100%
what about the new number? – 120% (since increased by 20%)
? =
4) Thus, as a result of changes, the number X (initial) is the basis for comparison, and the number 1.02X (obtained), (see IV type of problem solving), then
ANSWER: The number increased by 2%.
§ 1 Rules for finding a part from a whole and a whole from its part
In this lesson, we will formulate the rules for finding a part from a whole and a whole from its part, and also consider solving problems using these rules.
Let's consider two problems:
How many kilometers did the tourists walk on the first day, if the entire tourist route is 20 km?
Find the length of the entire tourist path.
Let's compare these problems - in both, the entire path is taken as a whole. In the first problem the whole is known - 20 km, and in the second it is unknown. In the first task you need to find a part of a whole, and in the second - a whole from its part. The quantity known in the first problem, 20 km, is unknown in the second problem, and vice versa, what is known in the second problem, 8 km, must be found in the first. Such problems are called mutually inverse, since in them the known and sought quantities are swapped.
Let's consider the first problem:
The denominator 5 shows how many parts the whole was divided into, i.e. If the whole 20 is divided by 5, we find out how many kilometers one part is, 20: 5 = 4 km. Numerator 2 shows that the tourists walked 2 parts of the path, which means 4 must be multiplied by 2, the result is 8 km. On the first day, tourists walked 8 km.
The result is expression 20: 5 ∙ 2 = 8.
Let's move on to the second problem.
Therefore, one part will be equal to the quotient of 8 and 2, the result is 4, the denominator is 5, which means there are 5 parts in total.
4 multiplied by 5, you get 20. The answer is 20 km, the length of the entire path.
Let's write the expression: 8: 2 ∙ 5 = 20
Using the meaning of multiplying and dividing a number by a fraction, the rules for finding a part of a whole and a whole from its part can be formulated as follows:
To find a part of a whole, you need to multiply the number corresponding to the whole by the fraction corresponding to this part;
To find a whole from its part, you need to divide the number corresponding to this part by the fraction corresponding to the part.
Accordingly, the solution to the problems can now be written differently:
for the first problem 20 ∙ 2/5 = 8 (km),
for the second problem 8: 2/5 = 20 (km).
To avoid any difficulties, we write the solution to such problems as follows:
Whole: all the way, known - 20 km.
Answer: 8 km.
Whole: the whole path is unknown.
Answer: 20 km.
§ 2 Algorithm for solving problems of finding a whole from its part and part of the whole
Let's create an algorithm for solving such problems.
First, let's analyze the condition and question of the problem: let's find out what the whole is, whether it is known or not, then we'll find out how a part of the whole is represented and what needs to be found.
If you need to find a part of a whole, then multiply the whole by the fraction corresponding to this part; if you need to find a whole by its part, then divide the number corresponding to the part by the fraction corresponding to this part. As a result, we get the expression. Next, we will find the meaning of the expression and write down the answer, having first read the question of the problem again.
So, before solving such problems, it is necessary to answer the following questions:
What quantity is accepted as a whole?
Is this quantity known?
What do you need to find: a part of the whole or a whole from its part?
Let's summarize: in this lesson you learned about the rules for finding a part of a whole and a whole from its part, and also learned how to solve problems using these rules.
List of used literature:
- Mathematics. 6th grade: lesson plans to the textbook I.I. Zubareva, A.G. Mordkovich //author-compiler L.A. Topilina. Mnemosyne, 2009.
- Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
- Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others / edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education, M.: Prosveshcheniye, 2010.
- Mathematics. 6th grade: educational. for general education institutions /N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – M.: Mnemosyne, 2013.
- Mathematics. 6th grade: textbook / G.K. Muravin, O.V. Muravina. – M.: Bustard, 2014.
Open lesson on mathematics in grade 5b.
Teacher: Bambutova M.I.
Topic: How to find a part of a whole and a whole from its part.
Goal: learn to solve problems of finding a part from a whole and a whole from its part.
Educational: derive a rule for finding a part from a whole and a whole from its part,
solve problems of finding a part from a whole and a whole from its part.
Educational: develop memory and mathematical speech
Educational: develop communication skills.
Lesson plan:
1).Introductory and motivational stage.
1. Org. Moment
2. Updating basic knowledge
Answer the questions (slide)
1) What does a fraction mean?
2) What does a fraction mean? 
?
3) 
Formulation of the problem:
1 task:
2 tasks per slide
1) draw a rectangle with sides 2 cm and 5 cm. What is its area?
Solve the problem
1) The area of the rectangle is 10 cm 2. Parts of the rectangle's area are shaded. What is the area of the shaded part of the rectangle?
2) The shaded part of the rectangle is equal to 4 cm 2, which is part of the entire rectangle. What is the area of the rectangle?
Answer the questions: ( )
part of the whole , and in which the whole according to its part ?
What do we find in task 1 (the whole by its part), what do we find in task 2 (part of the whole)
Task 2: Read the tasks and answer the questions:
1) Field area – 50 hectares. During the day, a team of tractor drivers plowed the fields. How many hectares did the team plow in a day?
2) During the day, the team plowed 20 hectares, which was the area of the entire field. What is the area of the field?
Answer the questions: ( distribute tasks in the form of cards)
What quantity is taken as an integer in each problem?
In which of the problems is this quantity known and in which is it not?
Which problem requires finding part of the whole , and in which the whole according to its part ?
What are these tasks? (reciprocal)
What do these tasks have in common? What were we looking for in these problems?
-Part of the whole And the whole according to its part.
So what is our topic today? ?
Topic: How to find a part of a whole and a whole from its part .(slide)
Correct solution For the last two problems, see the textbook on page 95.
Now we have solved 4 problems, generalize all the problems and derive a rule for finding a part from a whole and a whole from its part.
Pupils try, to help them scattered word combinations, they need to be collected logically correct sentence, which will be the rule.
which expresses this part.
corresponding to the whole,
To find a part of the whole,
divide by the denominator
and multiply the result by the numerator of the fraction
I need a number
To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.
and multiply the result by the denominator of the fraction,
I need a number
divide by the numerator
which expresses this part.
To find the whole from its part,
corresponding to this part,
To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.
Collect this rule on the board.
Students recite this rule to each other.
3. Primary consolidation. Game “Sorting tasks”.
Problem solving workshop. Option 1 solves problems of finding a part of a whole, option 2 solves problems of finding a whole from its part.
1. There are 80 students in the choir, ¼ of them are boys. How many boys are there in the choir?
2. There are 20 boys in the choir, which is ¼ of all students in the choir. How many students are there in the choir?
3. A small deciduous forest purifies the air from 70 tons of dust per year. And coniferous forest is ½ of this amount. How much dust does a coniferous forest filter out per year?
4. 7/12 of the kerosene that was there was poured out of the barrel. How many liters of kerosene were in the barrel if 84 liters were poured out of it?
5. The girl skied 300 m, which was 3/8 of the entire distance. What is the distance?
6. Cleared snow from 2/5 of the skating rink, which is 200 sq.m. Find the area of the entire skating rink?
7. The girl read ¾ of the book, which is 120 pages. How many pages are in the book?
8. The squirrel prepared 600 nuts in total. In the first week she collected 20% of all nuts. How much did the squirrel collect in the first week?
9. Find the number X, 1/8 of which is equal to 1/24.
10. The girl collected 40 plums, which was 1/3 of all plums. How many plums were collected in total?
11. Mom bought 6 kg of sweets. Vitya immediately ate 2/3 of all the candies and felt sick. After how many sweets did Vitya have a stomach ache?
12. The boy collected 80 nuts, which is 2/3 of all the collected nuts. How many nuts were collected?
13. There were 40 chickens in the chicken coop. In a week, the fox carried away 3/8 of all the chickens. How many chickens did the fox take?
14. Alice fell into a fairy well and flew 90 m in 1 minute. What is the depth of the well if Alice flew ¾ of the entire distance in 1 minute?
15. Before the ball, the stepmother gave Cinderella a lot of work. It took Cinderella 6 hours to complete 3/5 of this work. How long will it take Cinderella to complete all the work?
4. Reflection. The rule is to speak it out.
5. Homework: learn the rule, make a card with tasks for finding a part of a whole and a whole from its part (3 tasks for each rule).
