Finding the whole. Examples of solving typical problems for percentages

§ 20. Finding a part of the whole and the whole but its parts - Mathematics Textbook Grade 5 (Zubareva, Mordkovich)

Short description:

It happens that we need to find some part of a number, for example, peel only a third of a potato from a certain number. Or vice versa, when we are told that only a quarter of the class came on an excursion, we need to find out what is the total number of students in the class. Knowing the whole, you can find some given part of it, in the same way, knowing the part, you can determine what the whole was. You will learn about this today from this paragraph of the textbook.
The definition of a part of a whole, and vice versa, is directly related to simple fractions, which you have already studied. Actions in this case do not occur with two numbers, which are denoted by a fraction, but with one fraction and one integer. For example, finding 1/2 of 16 would mean multiplying 16 by 1/2, in which case the denominator of 16 = 1 and the expression can be written as: 1/2 16/1 = 16/2 = 8.
To find an integer by its part, we use the reverse method, and multiply the known number by the inverted fraction (that is, divide by it). In another way, this can be explained as follows: in order to find a whole from its part, you need the known number that corresponds to its part, divide by the numerator and multiply by the denominator of the fraction that denotes this part (which is the action of dividing the fraction, or multiplying to an inverted fraction - you can remember the most convenient way for you to solve such problems). Thus, to find an integer, 3/4 of which are equal to 12, you need 12: 3/4 = 12 4/3 = 48/3 = 16. Or method number 2, which removes unnecessary mathematical operations - the number x, 2/5 from which are 20: x = 20: 2 5 = 50.
Test yourself with the tasks from the textbook and do not forget to review the material to better master and remember it!

Open lesson in mathematics in grade 5b.

Teacher: Bambutova M.I.

Topic: How to find a part of a whole and a whole by its part.

Purpose: to learn to solve problems for finding a part of a whole and a whole by its part.

Educational: derive a rule for finding a part from a whole and a whole from its part,

solve problems to find a part of a whole and a whole by its part.

Developing: develop memory and mathematical speech

Educational: educate communication skills.

Lesson plan:

1). Introductory-motivational stage.

1. Org. Moment

2. Actualization of basic knowledge

Answer the questions (slide)

1) What does the fraction mean?

2) What does the fraction mean ?

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 area of the rectangle are shaded. What is the area of the shaded area of the rectangle?

2) The shaded part of the rectangle is 4 cm 2, which made up parts of the entire rectangle. What is the area of the rectangle?

Answer the questions: ( )

part of the whole , and in what the whole according to its part ?

What we find in task 1 (the whole in its part), what 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 brigade plow in a day?

2) During the day, the brigade plowed 20 hectares, which amounted to the area of the entire field. What is the area of \u200b\u200bthe field?

Answer the questions: ( distribute tasks in the form of a card)

What value is taken as an integer in each problem?

In which of the problems is this value known, and in which is it not?

In which of the tasks you want to find part of the whole , and in what the whole according to its part ?

What are these tasks? (reciprocal)

What do these tasks have in common? What are we looking for in these tasks?

-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 by its part .(slide)

The correct solution of the last two problems is found in the textbook on page 95.

Here we have solved 4 problems, we will generalize all the problems and derive a rule for finding a part from the whole and the whole from its part.

Students try to help them randomly phrases, they need to be assembled into a logically correct sentence, which will be the rule.

which expresses this part.

corresponding to the whole

To find a part of a whole

divide by denominator

and multiply the result by the numerator of the fraction

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,

need a number

divide by numerator

which expresses this part.

To find the whole by its part,

corresponding to this part,

To find a whole by 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.

Put this rule on the board.

Pupils say this rule to each other.

3. Primary fastening. The game "Sort tasks".

Workshop on problem solving. Option 1 solves the problem of finding a part of the whole, Option 2 solves the problem of finding the whole by its part.

1. There are 80 students in the choir, ¼ of them are boys. How many boys are in the choir?

2. There are 20 boys in the choir, which is ¼ of all students in the choir. How many students are in the choir?

3. A small deciduous forest purifies the air from 70 tons of dust per year. A coniferous forest ½ of this amount. How much dust does a coniferous forest filter out in a year?

4. 7/12 of the kerosene 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 length of the distance?

6. We cleared 2/5 of the ice rink from snow, which is 200 sq.m. Find the area of the entire ice rink?

7. The girl read ¾ of the book, which is 120 pages. How many pages are in the book?

8. Squirrel harvested 600 nuts in total. In the first week, she collected 20% of all nuts. How much protein did you collect in the first week?

9. Find a 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 sweets and he became ill. After how many sweets did Vitya get a stomach ache?

12. The boy collected 80 nuts, which is 2/3 of all collected nuts. How many nuts were collected?

13. There were 40 chickens in the chicken coop. For a week, the fox dragged 3/8 of all the chickens. How many chickens did the fox steal?

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 does it take Cinderella to complete all the work?

4. Reflection. Speak rule.

5. Homework: learn the rule, make a card with tasks for finding a part of a whole and a whole for its part (3 tasks for each rule).

Topic: Finding a part from a whole and a whole from its part

Target: To systematize, expand, generalize and consolidate the acquired knowledge on the topic “Finding a part from a whole and a whole by its part. Informatics among us»
Tasks:
Activate students' knowledge about the concepts of fractions, solving problems on fractions.
To teach students to solve problems on a topic, to be able to distinguish between ways to solve problems.
Application of the obtained theoretical knowledge in solving practical problems.
Expand students' horizons in the field of computer science.
Stages of the lesson.

Goal setting - 2 min.
Updating of basic knowledge - 8 min.
Consolidation and generalization of the material. – 23 min.
Summing up the lesson and setting homework. - 5 minutes.

Expected results: students must learn to apply the necessary methods of solving a particular problem, must be able to solve problems, be able to calculate fractions.

During the classes:

Organizing time. - 2 minutes.
Greetings students.
Goal setting - 2 min.
Guess the rebus.

What word is encoded here? That's right, the internet.
What topic are we studying now? (correct, "Finding a part from a whole and a whole from its part")
How will the Internet be related to this topic? (we will solve problems on this topic on knowledge of the Internet0
Who can formulate the topic of today's lesson? (Internet among us)
Do you know what the Internet is? (Give their version)
Internet - (from Latin inter - between and net - network), a global computer network that connects both users of computer networks and users of individual (including home) computers.
Updating of basic knowledge– 8 min.
Perform orally:
a) Find the part of the number:
3/4 of 16;
2/5 of 80;
7/10 from 120;
3/5 of 150;
6/11 from 121;
5/6 from 108

b) Find the number if:
3/8 of it are equal to 15;
2/5 of it is equal to 30;
5/8 of it are equal to 45;
4/9 of it are equal to 36;
7/10 of it are equal to 42;
2/11 of it is equal to 99.

Consolidation and generalization of the material. – 23 min.
Where and when do you think the Internet appeared? (express opinions)
In 1957, after the launch of the first artificial Earth satellite by the Soviet Union, the US Department of Defense decided that in case of war, the US needed a reliable information transmission system. The US Defense Advanced Research Projects Agency has offered to develop a computer network for this.

Now we will solve several problems.

Alena has 140 photos uploaded on her personal page on the Odnoklassniki website. 2/7 of the total number of photos uploaded to the Personal Photos album, 1/4 to the Hobby album, 3/35 to the Rest album, 5/28 to the Family album, and the rest to Na photo of friends. How many photos does Alena have in each album?
140: 7 * 2 = 40 (f) "Personal photos"
140: 4 * 1 = 35 (f) "Hobby"
140: 35 * 3 \u003d 12 (f) "Rest"
140: 28 * 5 = 25 (f) "Family"
140 - 40 - 35 - 12 - 25 \u003d 28 (f) "On the photo of friends"

Misha has 276 emails, which is 3/5 of the number of emails in Kolya's email. How many more letters does Kolya have than Misha?
276: 3 * 5 = 460
460 – 276 = 184.

On a flash card, designed for 4G bytes (1G bytes = 1024 M bytes), there are various files. Photos take up 3/16th of the total memory, movies - 1/8th (of the total memory) more than photos, text documents - 5/64th (of the total memory) more than photos. How many M bytes are in each file?
4 * 1024 = 4096
4096: 16 *3 =768(M bytes) in the photo
4096: 8 * 1=512
768 + 512 = 1280 (M bytes) for movies
4096: 64 *5 = 320
320 +768 = 1088 (M bytes) for text documents.

Guys, what do you need the Internet for?
Communication;
Information;
Games.
What social networks do you know? (express their opinion)
Let's name the "pros" and "cons" of social networks:
"Pros":
Communication;
Information.
"Minuses":
Negative impact on health;
Internet - addiction;
Immersion in the virtual world;
Danger from strangers.

Let's solve the following problem.

Among the 5th grade students of one of the schools, a survey was conducted on the topic “Social networks and children”. To the question "How much time per day do you spend on the Internet", 3/10 of all surveyed schoolchildren answered "5 - 6 hours". How many schoolchildren spend this time on the Internet every day if 150 children participated in the survey?
150: 10 * 3 = 45 (children).
45 children! This is a very large number! After all, every day they spend so much time wasted sitting at the computer.
Guys, what do you think, what harm to health can cause a long pastime on the Internet?
Possible student responses:
visual impairment;
Decreased motor activity;
Psychological overstrain;
The person loses the ability to communicate;
Rachiocampsis;
Headache;
Sleep disturbance.

You see how many negative things you can earn by sitting for several hours on the Internet!

5. Summing up the lesson and setting homework. - 5 minutes.
What new did you learn at the lesson today?
What do you think is the best time to spend on the Internet every day?
What do you mainly use the internet for?
Do you think that 5-6 hours on the Internet every day is the norm?
Homework: prepare a report on the topic "The History of the Internet"
Announcement of grades.
Thank you for the lesson!

§ 20. Finding a part of the whole and the whole but its parts - Mathematics Textbook Grade 5 (Zubareva, Mordkovich)

Short description:

The rule for finding a number by its fraction:

To find a number given the value of its fraction, you need to divide this value by a fraction.

Consider how to find a number by its fraction, using specific examples.

Examples.

1) Find a number whose 3/4 equals 12.

To find a number by its fraction, this number is divided by this fraction. To, you need to multiply this number by the reciprocal of the fraction (that is, by the inverted fraction). To , 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, this value is divided by this fraction. To divide a fraction by a fraction, multiply the first fraction by the reciprocal 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 got 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, this value is divided by this fraction. Mixed number and multiply it by the reciprocal of the second (inverted fraction). We reduce 99 and 9 by 9, 7 and 14 - by 7. Since we got an improper fraction, it is necessary to select an integer part from it.

Portal for the student. Self-training

Topic: Finding a part from a whole and a whole from its part

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