Finding a number from its fraction. Finding a number by its fraction Explanation of the topic finding a number by its fraction

Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Schwarzburd for grade 6 in mathematics on the topic:

Chapter I. Ordinary fractions.
§ 3. Multiplication and division of ordinary fractions:
18. Finding a number by its fraction

1 We cleared 2/5 of the ice rink from snow, which is 800 m2. Find the area of the entire rink.
SOLUTION

2 2400 ha sown with wheat. which is 0.8 of the entire field. Find its area.
SOLUTION

3 Having increased labor productivity by 7%, the worker made 98 more parts over the same period than planned according to the plan. How many parts did the worker have to do according to the plan?
SOLUTION

647 The girl skied 300 m, which was 3/8 of the entire distance. What is the length of the distance?
SOLUTION

648 The pile rises above the water by 1.5 m, which is 3/16 of the length of the entire pile. What is its length
SOLUTION

649 211.2 tons of grain were sent to the elevator, which is 0.88 of the grain threshed per day. How much grain was threshed in a day?
SOLUTION

650 After replacing the engine, the average speed of the aircraft increased by 18%, which is 68.4 km / h. What was the average speed of the aircraft with the same engine.
SOLUTION

651 The mass of dried fish is 55% of the mass of fresh fish. How much fresh should you take to get 231 kg of dried?
SOLUTION

652 The weight of the grapes in the first box is 7/9 of the weight of the grapes in the second. How many kilograms of grapes were in two boxes if the first one contained 21 kg of grapes?
SOLUTION

653 Sold 3/8 of the skis received by the store, after which 120 pairs of skis remained. How many pairs were received by the store?
SOLUTION

654 When dried, potatoes lose 85.7% of their mass. How many raw potatoes do you need to take to get 71.5 tons of dried?
SOLUTION

655 The bank bought several shares of the plant and sold them a year later for 576.8 million rubles, receiving a 3% profit. How much did the bank spend on the purchase of shares?
SOLUTION

656 On the first day, the tourists covered 5/24 of the intended route, and on the second day, 0.8 of what they traveled on the first day. How long is the planned path, if on the second day the tourists walked 24 km?
SOLUTION

657 The student first read 75 pages and then a few more pages. Their number was 40% of what was read for the first time. How many pages are there in a book if 3/4 of the book is read?
SOLUTION

658 The cyclist first traveled 12 1/4 km, and then a few more kilometers, which was 3/7 of the first leg of the journey. After that, he had to drive 2/3 of the whole way. What is its length
SOLUTION

659 3/5 of the number 12 is 1/4 of the unknown number. Find this number.
SOLUTION

660 35% of 128.1 is 49% of the unknown number. Find it
SOLUTION

661 In the kiosk on the first day 40% of all notebooks were sold, on the second 53%, and on the third the remaining 847 notebooks. How many notebooks did the kiosk sell in three days?
SOLUTION

662 On the first day, the vegetable base released 40% of the total available potatoes, on the second day 60% of the rest, and on the third day the remaining 72 tons. How many tons of potatoes were at the base?
SOLUTION

663 Three workers made a number of parts. The first worker made 0.3 of all parts, the second 0.6 of the remainder, and the third the remaining 84 parts. How many parts did the workers make in total?
SOLUTION

664 On the first day, the tractor brigade plowed 3/8 of the plot, on the second 2/5 of the remainder, and on the third the remaining 216 hectares. Determine the area of the plot.
SOLUTION

665 The car traveled 4/9 of the entire distance in the first hour, 3/5 of the remaining distance in the second hour, and the rest of the journey in the third. It is known that in the third hour it traveled 40 km less than in the second. How many kilometers did the car travel in these 3 hours?
SOLUTION

666 Do the calculations. Use a microcalculator to find a number whose 12.7% equals 4.5212; a number 8.52% of which equals 3.0246.
SOLUTION

668 Without dividing, compare.
SOLUTION

669 How many times less than its reciprocal: 1/5; 2/3; 1/6; 0.3?
SOLUTION

670 Think of a number that is 4 times less than its reciprocal; 9 times.
SOLUTION

671 Orally divide the central number into circled numbers.
SOLUTION

672 How many square tiles with a side of 20 cm will be needed to lay the floor in a room that is 5.6 m long and 4.4 m wide. Solve the problem in two ways.
SOLUTION

673 Find the rule for placing numbers in semicircles and fill in the missing numbers
SOLUTION

675 In 3/5 hours a cyclist traveled 7 1/2 km. How many kilometers will a cyclist travel in 2 1/2 hours if he travels at the same speed
SOLUTION

676 In 1/3 hour a pedestrian walked 1 1/2 km. How many kilometers will a pedestrian walk in 2 1/2 hours if he walks at the same speed?
SOLUTION

678 Find the value of the expression
SOLUTION

679 Do steps 10.1 + 9.9 107.1: 3.5: 6.8 - 4.85; 12.3 + 7.7 187.2: 4.5: 6.4 - 3.4
SOLUTION

680 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
SOLUTION

681 Volodya read 234 pages, which is 36% of the entire book. How many pages are in this book?
SOLUTION

682 Using a new tractor to plow a field resulted in a time saving of 70% and took 42 hours. How long would it take to do this job with an old tractor?
SOLUTION

683 A pillar dug into the ground at 2/13 of its length rises 5 1/2 meters above the ground. Find the length of the pillar.
SOLUTION

684 The turner, having turned 145 parts on the machine, exceeded the plan by 16%. How many details did you need to carve according to the plan?
SOLUTION

685 Point C divides segment AB into two segments AC and CB. The length of AC is 0.65 of the length of the segment CB. Find CB and AB if AC = 3.9 cm.
SOLUTION

686 The skiing distance is divided into three sections. The length of the first section is 0.48 of the length of the entire distance, the second - 5/12 of the length of the first section. What is the length of the entire distance if the length of the second section is 5 km? What is the length of the third?
SOLUTION

687 From a full barrel they took 14.4 kg of sauerkraut and then another 5/12 of this amount. After that, 5/8 of the sauerkraut that was previously there remained in the barrel. How many kilograms of cabbage were in a full barrel?
SOLUTION

688 When Kostya has gone 0.3 of the whole way from home to school, he still has 150 m to go to the middle of the way. How long is the way from home to school?
SOLUTION

689 Three groups of schoolchildren planted trees along the road. The first group planted 35% of all available trees, the second group planted 60% of the remaining trees, and the third group planted the remaining 104. How many trees were planted in total?
SOLUTION

690 The shop had turning, milling and grinding machines. Lathes accounted for 5/11 of all these machines. The number of grinding machines is 2/5 of the number of lathes. How many machines of these types were there in the workshop, if there are 8 fewer milling machines than turning ones?
SOLUTION

691 Follow steps (1.704: 0.8 - 1.73) 7.16 - 2.64; 227.36: (865.6 - 20.8 40.5) 8.38 + 1.12; (0.9464: (3.5 0.13) + 3.92) 0.18; 275.4: (22.74 + 9.66) (937.7 - 30.6 30.5).

In this lesson, we will consider the types of tasks for shares and percentages. Let's learn how to solve these problems and find out which of them we can face in real life. We learn the general algorithm for solving such problems.

We do not know what the number was originally, but we know how much it turned out when a certain fraction was taken from it. We need to find the original.

That is, we do not know , but we know and .

Example 4

Grandfather spent his life in the village, which amounted to 63 years. How old is grandpa?

We do not know the original number - age. But we know the share and how many years this share is from age. We create equality. It has the form of an equation with an unknown . We express and find it.

Answer: 84 years old.

Not a very realistic task. It is unlikely that grandfather will give out such information about his years of life.

But the following situation is very common.

Example 5

Discount in the store with a card 5%. The buyer received a discount of 30 rubles. What was the purchase price before the discount?

We do not know the original number - the cost of the purchase. But we know the fraction (the percentages that are written on the card) and how much the discount was.

We compose our standard line. We express the unknown value and find it.

Answer: 600 rubles.

Example 6

More often than not, we are faced with this problem. We see not the size of the discount, but what the cost is after applying the discount. And the question is the same: how much would we pay without a discount?

Let us again have a 5% discount card. We showed the card at the checkout and paid 1140 rubles. What is the price without discount?

To solve the problem in one step, we slightly reformulate it. Since we have a 5% discount, how much do we pay on the full price? 95%.

That is, we do not know the initial cost, but we know that 95% of it is 1140 rubles.

We apply the algorithm. We get the initial value.

3. Website "Mathematics Online" ()

Homework

1. Mathematics. Grade 6 / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemosyne, 2011. Pp. 104-105. item 18. No. 680; No. 683; No. 783 (a, b)

2. Mathematics. Grade 6 / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemozina, 2011. No. 656.

3. The program of school sports competitions included long jumps, high jumps and running. All participants of the competition took part in the running competitions, 30% of all participants in the long jump, and the remaining 34 students in the high jump competitions. Find the number of competitors.

Class: 6

Presentations for the lesson

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Epigraph to the lesson:

“The one who learns on his own succeeds seven times more than the one to whom everything is explained” (Arthur Gitermann, German poet)

Type of lesson: lesson learning new material.

Methods: partial search.

Forms: individual, collective, group, individual.

(Place - 1 lesson on the topic)

Type of lesson: explanatory and illustrative

The purpose of the lesson: to come up with a new way to solve problems in fractions, to consolidate the skills and abilities of solving problems.

to systematize the solution of problems into parts, to derive a new method of solving problems for finding a number by its part.

to help develop students' interest not only in the content, but also in the process of mastering knowledge, to expand the mental horizons of students. Development of students' thinking, mathematical speech, motivational sphere of personality, research skills.

to instill in students a sense of satisfaction from the opportunity to show their knowledge in the lesson. Create a positive motivation for students to perform mental and practical actions. Education of responsibility, organization, perseverance in solving tasks.

Equipment: illustrative material, presentation for the lesson. Sheets with a task for reflection, a textbook on mathematics Mathematics. Grade 6 / N. Ya. Vilenkin, V. I. Zhokhov, A.S. Chesnokov, S. I. Shvartsburd. Moscow: Mnemosyne, 2011.

Lesson plan:

Organizing time.

Updating of basic knowledge and their correction.

Learning new knowledge.

Fizkultminutka.

Primary fastening.

Primary test of understanding of the studied.

Summing up the lesson. Reflection.

Homework.

Estimates.

During the classes

1. Organizational moment.

(didactic task. psychological state of the students

Hello, have a seat. We report the topic, the objectives of the lesson and the practical significance of the topic.

The purpose of our lesson is to come up with a new way to solve fraction problems.

2. Actualization of basic knowledge and their correction

(The didactic task is to prepare students for work in the classroom. Providing motivation and acceptance by students of the goal, educational and cognitive activity, updating basic knowledge and skills).

15; ; 3 6; ; (2; ; 19; c)

Questions for the class:

How do you multiply a fraction by a natural number?

How to find the product of fractions?

How to find the product of a mixed number and a number? (using the distributive property of multiplication or converting a mixed number to an improper fraction)

How to multiply mixed numbers?

2) :2; v:; :; :; (; ; ; X)

Questions for the class:

How do you divide a fraction by a natural number?

How do you divide one fraction into another?

How do you divide a mixed number by a mixed number?

Tables on the slide and supports on the desks of the weak group:

Repeat the algorithms for solving problems for finding a number by its part.

1) We cleared the ice rink from snow, which is 800 m 2. Find the area of the entire rink.

(800:2 5 \u003d 2000 m 2)

2) Winnie the Pooh collected x kg of honey from the hives, which is 30% of the amount that he dreamed of. How much honey did you dream about, Winnie the Pooh? (x:30 100)

3) The boa constrictor gave the monkey “v” bananas, which is from the amount that he always gave. How much did he always give? (a)

Question for the class:

What rule should be remembered here?

(To find a number by its fractional part, you can divide this part by the numerator and multiply by the denominator)
3. Learning new material. “Discovery” of new knowledge by children.

(The didactic task is to organize and direct the cognitive activity of students towards the goal)

Today in the lesson we will try to find an easier way to solve problems of finding a number from its fraction. The learned rules for multiplying and dividing fractions will help us with this.

– Write down the rule in your notebook (а = в: m n).

- Replace the division sign with a fraction bar and try to write it in the form of one action with the number “a” and a fraction.

N = = in = in:

- Translate the resulting rule into mathematical language.

(To find a number by its part, you can divide this part by a fraction) Discovery. Repeat this rule to yourself.

Now work in pairs:

Option 1 tells the rule to option 2, and option 2 to the first.

Why is this rule better than the previous one? (The problem is solved by one action instead of

two)
4. Physical education.

(The task is to relieve stress)

Find all the colors of the rainbow (every hunter wants to know where the pheasant sits). Colored squares are hung in different places around the classroom. You have to rotate to find the right color. Then exercises for the eyes.

Annex 1.

5. Primary fastening.

(The didactic task is to achieve from students the reproduction, awareness, primary generalization and systematization of new knowledge. Consolidation of the method of the student's forthcoming answer during the next survey)

Primary consolidation takes place in the form of frontal work and work in pairs.

(with commentary in loud speech)

1) Find the number if it is 10.

2) Find the number if 1% is 4.

in writing

(with commenting and writing on the board and in notebooks)

1) Masha skied 500 m, which was the entire distance. What is the length of the distance? (500:=800m)

2) The mass of dried fish is 55% of the mass of fresh fish. How much fresh fish to take. To get 231 kg of jerky? (231:=420kg)

3) The mass of strawberries in the first box is equal to the mass of strawberries in the second box. How many kg of strawberries were in two boxes if the first box contained 24 kg of strawberries?

Work in pairs

(collaboration) Make an expression for the tasks.

1) On a beautiful summer morning, a kitten named Woof ate x sausages, which made up his daily diet. How many sausages does Woof the kitten eat per day? (x:= sausages)

2) Dunno read 117 pages, which was 9% of the magic book. How many pages are in the magic book? (117:=1300str)

6. Primary check of understanding of the studied
(in the form of independent work with a check in the class).

(Didactic task– control of knowledge and elimination of gaps on this topic)

One person from each option to call, they will silently work on the wings of the board. Then we check the solution.

1 option

1) find the number if it is 21. (49)

2) find the number if 15% of it is x. ()

3) find the number if 0.88 it is 211.2. (240)

Option 2

1) find the number if it is 24. (64)

2) find the number if 20% of it is x. (5x)

3) find the number if 0.25 it is 6.25. (25)

Evaluate yourself: not a single mistake - “5”; 1 error - "4"; who has more mistakes - to do work on the mistakes.

7. Summing up the lesson.

(Didactic task- to analyze and evaluate the success of achieving the goal and outline the prospects for further work). You made a discovery in class today

came up with a new way to solve problems in fractions, which means they succeeded seven times more than if I had told you everything myself (look again at the epigraph to our lesson)

Reflection.
(Didactic task - mobilization of students to reflect on their behavior, motivation, methods of activity, communication).

And now the guys continue the sentence: Today in the lesson I learned ... Today in the lesson I liked it ... Today in the lesson I repeated ... Today in the lesson I consolidated ... Today in the lesson I gave myself an assessment ... What types of work caused difficulties and require repetition ... In what knowledge I'm sure... Did the lesson help to advance in knowledge, skills, skills in the subject... To whom, over, what else should be worked on...

How effective was the lesson today ... a smiling little man, if you liked the lesson and everything worked out, and a sad little man, if something else doesn’t work (everyone has pictures with little men on their desks).

6
. Homework

(Comment, it is differentiated) (Didactic task - providing an understanding of the purpose, content and methods of doing homework).

Page 104-105. item 18. No. 680; No. 683; №783(а, b)

Additional task No. 656. (for strong students).

For the creative group - come up with tasks on a new topic.

7. Grades for the lesson.

Everyone worked well, absorbing knowledge with appetite. Children! Thank you for the lesson.

We do not know what the number was originally, but we know how much it turned out when a certain fraction was taken from it. We need to find the original.

That is, we do not know , but we know and .

Example 4

Grandfather spent his life in the village, which amounted to 63 years. How old is grandpa?

We do not know the original number - age. But we know the share and how many years this share is from age. We create equality. It has the form of an equation with an unknown . We express and find it.

Answer: 84 years old.

Not a very realistic task. It is unlikely that grandfather will give out such information about his years of life.

But the following situation is very common.

Example 5

Discount in the store with a card 5%. The buyer received a discount of 30 rubles. What was the purchase price before the discount?

We do not know the original number - the cost of the purchase. But we know the fraction (the percentages that are written on the card) and how much the discount was.

We compose our standard line. We express the unknown value and find it.

Answer: 600 rubles.

Example 6

Let us again have a 5% discount card. We showed the card at the checkout and paid 1140 rubles. What is the price without discount?

To solve the problem in one step, we slightly reformulate it. Since we have a 5% discount, how much do we pay on the full price? 95%.

That is, we do not know the initial cost, but we know that 95% of it is 1140 rubles.

We apply the algorithm. We get the initial value.

3. Website "Mathematics Online" ()

Homework

1. Mathematics. Grade 6 / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemosyne, 2011. Pp. 104-105. item 18. No. 680; No. 683; No. 783 (a, b)

2. Mathematics. Grade 6 / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemozina, 2011. No. 656.

The whole rink.

Solution. Let's denote the area of the rink through x m 2. According to the condition of this area, they are equal to 800 m 2, i.e. x \u003d 800.
So x = 800:= 800 = 2000. The area of the rink is 2000 m2.

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

Task 2. 2400 hectares were sown with wheat, which is 0.8 of the entire field. Find the area of the entire field.

Solution. Since 2400:0.8 = 24000:8 = 3000, the area of the entire field is 3000 ha.

Task 3. Having increased labor productivity by 7%, the worker made 98 more parts over the same period than planned according to the plan. How many parts did the worker have to do according to the plan?

Solution. Since 7% \u003d 0.07, and 98: 0.07 \u003d 1400, the worker, according to the plan, had to make 1400 parts.

? Formulate a rule for finding a number given its value fractions. Tell us how to find a number given the value of its percent.

TO 631. The girl skied 300 m, which was the entire distance. What is the length of the distance?

632. The pile rises above the water by 1.5 m, which is the length of the entire pile. What is the length of the entire pile?

633. 211.2 tons of grain were sent to the elevator, which is 0.88 grains threshed per day. How much grain was threshed in a day?

634. For the rationalization proposal, the engineer received 68.4 rubles in excess of the monthly salary, which is 18% of this salary. What is the monthly salary of an engineer?

635. The mass of dried fish is 55% of the mass of fresh fish. How much fresh fish do you need to take to get 231 kg of dried fish?

636. The mass of grapes in the first box is the mass of grapes in the second box. How many kilograms of grapes were in two boxes if the first box contained 21 kg of grapes?

637. Sold the skis received by the store, after which 120 pairs of skis remained. How many pairs of skis did the shop receive?

638. When drying, potatoes lose 85.7% of their mass. How many raw potatoes do you need to take to get 71.5 tons of dried?

639. A Sberbank depositor made a certain amount for a term deposit, and a year later he had 576 rubles on his savings book. 80 k. What was the amount of the deposit if Sberbank pays 3% per annum on term deposits?

640. On the first day, the tourists traveled the intended route, and on the second day, 0.8 of what they traveled on the first day. How long is the planned path, if on the second day the tourists walked 24 km?

641. The student first read 75 pages, and then a few more pages. Their number was 40% of what was read for the first time. How many pages are in the book if the total number of books read?

642. The cyclist first traveled 12 km, and then several more kilometers, which amounted to the first segment of the journey. After that, he had to drive all the way. What is the length of the entire path?

643. from the number 12 is an unknown number. Find this number.

644. 35% of 128D is 49% of an unknown number. Find this number.

645. On the first day, 40% of all notebooks were sold at the kiosk, 53% of all notebooks on the second day, and the remaining 847 notebooks on the third day. How many notebooks did the kiosk sell in three days?

646. The vegetable base released 40% of the total available potatoes on the first day, 60% of the remainder on the second day, and the remaining 72 tons on the third day. How many tons of potatoes were at the base?

647. Three workers made a number of parts. The first worker made 0.3 of all parts, the second 0.6 of the remainder, and the third - the remaining 84 parts. How many parts did the workers make in total?

648. On the first day, the tractor brigade plowed the plot, on the second day the remainder, and on the third day the remaining 216 hectares. Determine the area of the plot.
649. The car passed in the first hour of the entire journey, in the second hour of the remaining journey, and in the third hour the rest of the journey. It is known that in the third hour it traveled 40 km less than in the second hour. How many kilometers did the car travel in these 3 hours?

650. You can find a number by a given value of its percentage using a microcalculator. For example, to find a number whose 2.4% is 7.68, you can use the following program :Do the calculations. Find with a calculator:
a) a number 12.7% of which equals 4.5212;
b) a number, 8.52% of which are equal to 3.0246.

P 651. Calculate orally:

652. Without dividing, compare:

653. How many times less than its reciprocal:

654. Think of a number that is 4 times less than its inverse; 9 times.

655. Orally divide the central number by the number in circles:

656. How many square tiles with a side of 20 cm will be needed to lay the floor in a room that is 5.6 m long and 4.4 m wide. Solve the problem in two ways.

M 657. Find the rule for placing numbers in semicircles and insert the missing numbers (Fig. 29).

658. Perform division:

659. A cyclist traveled 7 km in one hour. How many kilometers will a cyclist travel in 2 hours if he travels at the same speed?

660. In 4~ hours a pedestrian walked 1 km. How many kilometers will a pedestrian walk in 2 hours if he walks at the same speed?

661. Reduce the fraction:

663. Do the following:

1) 10,14-9,9 107,1:3,5:6,8-4,8;
2) 12,34-7,7 187,2:4,5:6,4-3,4.

D 664. Kerosene that was there was poured out of a barrel. How many liters of kerosene were in the barrel if 84 liters were poured out of it?

665. When buying a color TV on credit, 234 rubles were paid in cash, which is 36% of the cost of the TV. How much does a TV cost?

666. A worker received a ticket to a sanatorium at a 70% discount and paid 42 rubles for it. How much does a ticket to the resort cost?

667. A pillar, dug into the ground at its length, rises above the ground by 5 m. Find the entire length of the pillar.

668. The turner, having turned 145 parts on the machine, exceeded the plan by 16%. How many details did you need to carve according to the plan?

669. Point C divides segment AB into two segments AC and CB. The length of segment AC is 0.65 of the length of segment CB. Find the lengths of the segments CB and AB if AC = 3.9 cm.

670. The skiing distance is divided into three sections. The length of the first section is 0.48 of the length of the entire distance, the length of the second section is the length of the Left section. What is the length of the entire distance if the length of the second section is 5 km? What is the length of the third section?

671. From a full barrel they took 14.4 kg of sauerkraut and then another of this amount. After that, the sauerkraut that was previously there remained in the barrel. How many kilograms of sauerkraut were in a full barrel?

672. When Kostya walked 0.3 of the whole way from home to school, he still had to go to the middle of the way 150 m. How long is the path from Kostya's house to school?

673. Three groups of schoolchildren planted trees along the road. The first group planted 35% of all available trees, the second group planted 60% of the remaining trees, and the third group planted the remaining 104 trees. How many trees were planted?

674. The workshop had turning, milling and grinding machines. Lathes made up all of these machine tools. The number of grinding machines was the number of lathes. How many machines of these types were there in the workshop if there were 8 fewer milling machines than turning machines?

675. Do the following:

a) (1.704:0.8 -1.73) 7.16 -2.64;
b) 227.36: (865.6 - 20.8 40.5) 8.38 + 1.12;
c) (0.9464:(3.5 0.13) + 3.92) 0.18;
d) 275.4: (22.74 + 9.66) (937.7 - 30.6 30.5).

N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school

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