Decimal examples with 8 actions. Examples and tasks for all actions with decimal fractions. Dividing a smaller number by a larger one. Advanced level

CHAPTER III.

DECIMAL FRACTIONS.

§ 31. Tasks and examples for all actions with decimal fractions.

Perform the indicated actions:

767. Find the quotient of division:

Perform actions:

772. Calculate:

To find x , if a:

776. The unknown number was multiplied by the difference between the numbers 1 and 0.57 and the product got 3.44. Find an unknown number.

777. The sum of the unknown number and 0.9 was multiplied by the difference between 1 and 0.4 and the product got 2.412. Find an unknown number.

778. According to the diagram on the smelting of pig iron in the RSFSR (Fig. 36), draw up a problem, for the solution of which it is necessary to apply the actions of addition, subtraction and division.

779. 1) The Suez Canal is 165.8 km long, the Panama Canal is 84.7 km shorter than the Suez Canal, and the White Sea-Baltic Canal is 145.9 km longer than the Panama Canal. How long is the White Sea-Baltic Canal?

2) The Moscow metro (by 1959) was built in 5 phases. The length of the first line of the metro is 11.6 km, the second -14.9 km, the length of the third is 1.1 km less than the length of the second line, the length of the fourth line is 9.6 km longer than the third line, and the length of the fifth line is 11.5 km less fourth. What is the length of the Moscow metro by the beginning of 1959?

780. 1) The greatest depth of the Atlantic Ocean is 8.5 km, the greatest depth of the Pacific Ocean is 2.3 km greater than the depth of the Atlantic Ocean, and the greatest depth of the Arctic Ocean is 2 times less than the greatest depth of the Pacific Ocean. What is the greatest depth of the Arctic Ocean?

2) The Moskvich car consumes 9 liters of gasoline per 100 kilometers, the Pobeda car is 4.5 liters more than the Moskvich consumes, and the Volga is 1.1 times more than Pobeda. How much gasoline does the Volga car consume per 1 km of track? (Round off the answer to the nearest 0.01 l.)

781. 1) The student went to his grandfather during the holidays. He traveled by rail for 8.5 hours, and from the station on horseback for 1.5 hours. In total, he traveled 440 km. At what speed did the student ride the railroad if he was riding horses at a speed of 10 kilometers per hour?

2) The collective farmer had to be at a point located at a distance of 134.7 km from his house. For 2.4 hours he traveled by bus at an average speed of 55 km per hour, and the rest of the way he walked on foot at a speed of 4.5 km per hour. How long did he walk?

782. 1) Over the summer, one gopher consumes about 0.12 quintals of bread. In the spring, the pioneers killed 1,250 gophers on 37.5 hectares. How much bread did the schoolchildren save for the collective farm? How much grain is saved per hectare?

2) The collective farm calculated that by destroying ground squirrels on an area of \u200b\u200b15 hectares of arable land, the schoolchildren saved 3.6 tons of grain. How many gophers are destroyed on average per 1 hectare of land if one gopher annually destroys 0.012 tons of grain?

783. 1) When grinding wheat into flour, 0.1 of its weight is lost, and when baking, a bake is obtained equal to 0.4 weight of flour. How much baked bread will be obtained from 2.5 tons of wheat?

2) The collective farm collected 560 tons of sunflower seeds. How much sunflower oil will be made from harvested grain if the grain weight is 0.7 times the weight of sunflower seeds and the weight of the resulting oil is 0.25 times the weight of grain?

784. 1) The cream yield from milk is 0.16 weight of milk and the yield of butter from cream is 0.25 weight of cream. How much milk (by weight) is required to obtain 1 centner of butter?

2) How many kilograms of porcini mushrooms must be collected to obtain 1 kg of dried, if 0.5 weight remains in preparation for drying, and 0.1 weight of the processed mushroom remains during drying?

785. 1) The land allotted to the collective farm is used as follows: 55% of it is occupied by arable land, 35% by meadow, and the rest of the land in the amount of 330.2 hectares is allotted for the collective farm garden and for the farmsteads of collective farmers. How much land is there on the collective farm?

2) The collective farm sowed 75% of the entire sown area with grain crops, 20% with vegetables, and the rest with fodder grasses. How much sown area did the collective farm have if it sowed 60 hectares with fodder grasses?

786. 1) How many centners of seeds will be required to sow a field in the shape of a rectangle 875 m long and 640 m wide, if 1.5 centners of seeds are sown per hectare?

2) How many centners of seeds will it take to sow a rectangular field if its perimeter is 1.6 km? The width of the field is 300 m. Sowing 1 hectare requires 1.5 centners of seeds.

787. How many square plates with a side of 0.2 in. Will fit in a rectangle measuring 0.4 in. X 10 in.?

788. The reading room has dimensions of 9.6 mx 5m x 4.5 m. How many places is the reading room designed for, if 3 cubic meters are needed for each person. m of air?

789. 1) What area of \u200b\u200bthe meadow will a tractor with a trailer of four mowers mow in 8 hours, if the width of each mower is 1.56 m and the tractor speed is 4.5 km per hour? (Time for stops is not taken into account.) (Round off the answer to the nearest 0.1 ha.)

2) The working width of the tractor vegetable seeder is 2.8 m. What area can be sown with this seeder in 8 hours. work at a speed of 5 km per hour?

790. 1) Find the production of a three-body tractor plow in 10 hours. work, if the speed of the tractor is 5 km per hour, the capture of one body is 35 cm, and the non-productive waste of time was 0.1 of the total time spent. (Round off the answer to the nearest 0.1 ha.)

2) Find the output of a five-body tractor plow in 6 hours. work, if the speed of the tractor is 4.5 km per hour, the capture of one body is 30 cm, and the waste of time was 0.1 of the total time spent. (Round off the answer to the nearest 0.1 ha.)

791. The water consumption per 5 km of run for a steam locomotive of a passenger train is 0.75 tons. The water tank of the tender holds 16.5 tons of water. How many kilometers will the train have enough water if the tank was 0.9 of its capacity?

792. Only 120 freight cars can fit on the siding with an average car length of 7.6 m. How many four-axle passenger cars each 19.2 m long will fit on this track if 24 more freight cars are placed on this track?

793. For the strength of the railway embankment, it is recommended to strengthen the slopes by sowing field grasses. For each square meter of the embankment, 2.8 g of seeds are required, worth 0.25 rubles. for 1 kg. How much will it cost to sow 1.02 hectares of slopes if the cost of work is 0.4 of the cost of seeds? (Round off the answer to the nearest 1 rub.)

794. The brick factory delivered bricks to the railway station. 25 horses and 10 trucks worked on the transportation of bricks. Each horse carried 0.7 tonnes per ride and made 4 trips per day. Each car transported 2.5 tons per trip and made 15 trips per day. The transportation lasted 4 days. How many pieces of bricks were delivered to the station if the average weight of one brick is 3.75 kg? (Round off the answer to the nearest thousand.)

795. The flour supply was distributed among three bakeries: the first received 0.4 of the total supply, the second received 0.4 of the remainder, and the third bakery received 1.6 tons less flour than the first. How much flour was distributed in total?

796. In the second year of the institute there are 176 students, in the third year it is 0.875 of this number, and in the first year it is one and a half times more than in the third year. The number of students in the first, second and third years was 0.75 of the total number of students of this institute. How many students were there at the institute?

797. Find the arithmetic mean:

1) two numbers: 56.8 and 53.4; 705.3 & 707.5;

2) three numbers: 46.5; 37.8 & 36; 0.84; 0.69 & 0.81;

3) four numbers: 5.48; 1.36; 3.24 and 2.04.

798. 1) In the morning the temperature was 13.6 °, at noon 25.5 °, and in the evening 15.2 °. Calculate the average temperature for that day.

2) What is the average temperature for the week, if during the week the thermometer showed: 21 °; 20.3 °; 22.2 °; 23.5 °; 21.1 °; 22.1 °; 20.8 °?

799. 1) The school team weeded 4.2 hectares of beets on the first day, 3.9 hectares on the second day, and 4.5 hectares on the third. Determine the average production of the brigade per day.

2) To establish the time standard for the manufacture of a new part, 3 turners were supplied. The first made the part in 3.2 minutes, the second in 3.8 minutes, and the third in 4.1 minutes. Calculate the time rate that was set for the manufacture of the part.

800. 1) The arithmetic mean of two numbers is 36.4. One of these numbers is 36.8. Find another.

2) The air temperature was measured three times a day: in the morning, at noon and in the evening. Find the air temperature in the morning, if at noon it was 28.4 °, in the evening it was 18.2 ° C, and the average day temperature was 20.4 °.

801. 1) The car drove 98.5 km in the first two hours, and 138 km in the next three hours. How many kilometers did a car drive per hour on average?

2) A trial catch and weighing of one year old carps showed that out of 10 carps 4 had a weight of 0.6 kg, 3 of 0.65 kg, 2 of 0.7 kg and 1 of 0.8 kg. What is the average weight of a yearling carp?

802. 1) To 2 liters of syrup worth 1.05 rubles. for 1 liter added 8 liters of water. How much is 1 liter of the obtained water with syrup?

2) The hostess bought a 0.5 liter can of canned borscht for 36 kopecks. and boiled with 1.5 liters of water. How much did a plate of borscht cost if its volume is 0.5 liters?

803. Laboratory work "Measuring the distance between two points",

1st reception. Measurement with a tape measure (measuring tape). The class is divided into links of three people each. Accessories: 5-6 milestones and 8-10 tags.

Work progress: 1) points A and B are marked and a straight line is fixed between them (see problem 178); 2) lay the tape along the fixed line and each time mark the end of the tape with a tag. 2nd reception. Measurement, in steps. The class is divided into links of three people each. Each student walks the distance from A to B by counting their steps. Multiplying the average length of your stride by the resulting number of steps, find the distance from A to B.

3rd reception. Measurement "by eye". Each student extends his left hand with a raised thumb (Fig. 37) and directs the thumb to the milestone at point B (in the figure - a tree) so that the left eye (point A), thumb and point B are on the same straight line. Without changing position, close the left eye and look at the thumb with the right. Measure the resulting displacement by eye and increase it 10 times. This is the distance from A to B.

804. 1) According to the 1959 census, the population of the USSR was 208.8 million, and the rural population was 9.2 million more than the urban population. How many urban and how many rural population were there in the USSR in 1959?

2) According to the 1913 census, the population of Russia was 159.2 million people, and the urban population was 103.0 million less than the rural population. How many urban and rural population were in Russia in 1913?

805. 1) The length of the wire is 24.5 m. This wire was cut into two parts so that the first part was 6.8 m longer than the second. How many meters is each part long?

2) The sum of two numbers is 100.05. One number is 97.06 larger than the other. Find these numbers.

806. 1) At three coal warehouses there are 8656.2 tons of coal, at the second warehouse there is 247.3 tons of coal more than at the first, and at the third by 50.8 tons more than at the second. How many tons of coal are there in each warehouse?

2) The sum of three numbers is 446.73. The first number is 73.17 less than the second and 32.22 more than the third. Find these numbers.

807. 1) The boat went along the river at a speed of 14.5 km per hour, and against the current at a speed of 9.5 km per hour. What is the speed of the boat in still water and what is the speed of the river flow?

2) The steamer passed in 4 hours along the course of the river 85.6 km, and against the stream in 3 hours 46.2 km. What is the speed of a steamer in still water and what is the speed of the river?

808. 1) Two steamers delivered 3,500 tons of cargo, and one steamer delivered 1.5 times more cargo than the other. How much cargo did each ship deliver?

2) The area of \u200b\u200btwo rooms is 37.2 sq. m. The area of \u200b\u200bone room is 2 times larger than the other. What is the area of \u200b\u200beach room?

809. 1) From two settlements, the distance between which is 32.4 km, a motorcyclist and a cyclist simultaneously drove towards each other. How many kilometers will each of them travel before the meeting if the speed of the motorcyclist is 4 times the speed of the cyclist?

2) Find two numbers, the sum of which is 26.35, and the quotient of dividing one number by another is 7.5.

810. 1) The plant dispatched three types of cargo with a total weight of 19.2 tons. The weight of the first type of cargo was three times the weight of the second type, and the weight of the third type was half the weight of the first and second types together. What is the weight of each type of cargo?

2) In three months, a team of miners produced 52.5 thousand tons of iron ore. In March it was mined 1.3 times, in February 1.2 times more than in January. How much ore did the team mine monthly?

811. 1) The Saratov-Moscow gas pipeline is 672 km longer than the Moscow Canal. Find the length of both structures if the length of the gas pipeline is 6.25 times the length of the Moscow Canal.

2) The length of the Don River is 3.934 times the length of the Moscow River. Find the length of each river, if the length of the Don is 1,467 km longer than the Moscow.

812. 1) The difference of two numbers is 5.2, and the quotient of dividing one number by another 5. Find these numbers.

2) The difference of two numbers is 0.96, and their quotient is 1.2. Find these numbers.

813. 1) One number is 0.3 less than the other and is 0.75 of it. Find these numbers.

2) One number is 3.9 more than another number. If the smaller number is doubled, then it will be 0.5 of the larger. Find these numbers.

814. 1) The collective farm sowed 2600 hectares of land with wheat and rye. How many hectares of land were sown with wheat and how many rye, if 0.8 of the area sown with wheat equals 0.5 of the area sown with rye?

2) The collection of the two boys together makes 660 stamps. How many stamps does each boy's collection consist of if 0.5 of the number of stamps of the first boy is equal to 0.6 of the number of stamps in the collection of the second boy?

815. The two students together had 5.4 rubles. After the first one has spent 0.75 of his money, and the second 0.8 of his money, they still have equal amounts of money. How much money did each student have?

816. 1) Two steamships left towards each other from two ports, the distance between which is 501.9 km. How long will they meet if the speed of the first steamer is 25.5 km per hour, and the speed of the second is 22.3 km per hour?

2) Two trains left towards each other from two points, the distance between which is 382.2 km. How long will it take for them to meet if the average speed of the first train was 52.8 km per hour, and the second was 56.4 km per hour?

817. 1) From two cities, the distance between which is 462 km, two cars left at the same time and met in 3.5 hours. Find the speed of each car if the speed of the first car was 12 km per hour more than the speed of the second car.

2) From two settlements, the distance between which is 63 km, a motorcyclist and a cyclist left towards each other at the same time and met in 1.2 hours. Find the speed of the motorcyclist if the cyclist was traveling at a speed of 27.5 km / h less than the speed of the motorcyclist.

818. The student noticed that a train consisting of a steam locomotive and 40 carriages passed by him for 35 seconds. Determine the speed of the train per hour, if the length of the locomotive is 18.5 m and the length of the carriage is 6.2 m. (Give the answer with an accuracy of 1 km per hour.)

819. 1) A cyclist left A to B with an average speed of 12.4 km per hour. After 3 hours 15 minutes. another cyclist left B towards him at an average speed of 10.8 km per hour. In how many hours and at what distance from A will they meet if 0.32 distances between A and B are 76 km?

2) From cities A and B, the distance between which is 164.7 km, a truck from city A and a passenger car from city B drove towards each other. The speed of a truck is 36 km, and that of a passenger car is 1.25 times higher. The passenger car left the truck 1.2 hours later. How long will it take and at what distance from city B will a passenger car meet a truck?

820. Two steamers left the same port at the same time and go in the same direction. The first steamer runs 37.5 km every 1.5 hours, and the second runs 45 km every 2 hours. How long will it take for the first steamer to be 10 km from the second?

821. A pedestrian first left one point, and 1.5 hours after his exit, a cyclist left in the same direction. At what distance from the point did the cyclist catch up with the pedestrian if the pedestrian was walking at a speed of 4.25 km per hour, and the cyclist was traveling at a speed of 17 km per hour?

822. The train left Moscow for Leningrad at 6 o'clock. 10 min. morning and walked at an average speed of 50 km n hour. Later, a passenger plane took off from Moscow to Leningrad and flew to Leningrad simultaneously with the arrival of the train. average speed the aircraft was 325 km per hour, and the distance between Moscow and Leningrad was 650 km. When did the plane take off from Moscow?

823. The steamer went along the river for 5 hours, and against the current for 3 hours and covered only 165 km. How many kilometers did he go with the current and how many against the current, if the river flow rate is 2.5 km per hour?

824. The train left A and must arrive at B at a specific time; after passing half the way and making 0.8 km in 1 minute, the train was stopped for 0.25 hours; further increasing the speed by 100 m in 1 million, the train arrived at B on time. Find the distance between A and B.

825. From the collective farm to the city 23 km. From the city to the collective farm, a postman rode a bicycle at a speed of 12.5 km per hour. 0.4 hours after this, the collective farm IW went to the city on horseback, the collective farmer at a speed that was early 0.6 that of a postman. How long after his departure will the collective farmer meet the postman?

826. From city A to city B, 234 km away from A, a car drove out at a speed of 32 km per hour. 1.75 hours after that, the second car drove out of town B towards the first, the speed of which is 1.225 times higher than the speed of the first. How many hours after leaving the second car will meet the first?

827. 1) One typist can retype a manuscript in 1.6 hours, and another in 2.5 hours. How long will it take for both typists to retype this manuscript working together? (Round off the answer to the nearest 0.1 hour.)

2) The pool is filled with two pumps of different capacities. The first pump, working alone, can fill the pool in 3.2 hours, and the second in 4 hours. How long will it take to fill the pool when these pumps are running simultaneously? (Round off the answer to the nearest 0.1.)

828. 1) One team can complete some order in 8 days. Another takes 0.5 of the first time to complete this order. The third team can fulfill this order in 5 days. How many days will the entire order be completed if three teams work together? (Round off the answer to the nearest 0.1 days.)

2) The first worker can complete the order in 4 hours, the second 1.25 times faster, and the third in 5 hours. How many hours will an order be completed when three workers work together? (Round off the answer to the nearest 0.1 hour.)

829. Two cars are working on street cleaning. The first of them can clean the entire street in 40 minutes, the second takes 75% of the time of the first. Both machines started working at the same time. After working together for 0.25 hours, the second machine stopped working. When did the first machine finish cleaning the street after that?

830. 1) One side of the triangle is 2.25 cm, the second is 3.5 cm larger than the first, and the third is 1.25 cm smaller than the second. Find the perimeter of the triangle.

2) One of the sides of the triangle is 4.5 cm, the second is 1.4 cm smaller than the first, and the third side is equal to half the sum of the first two sides. What is the perimeter of a triangle?

831 ... 1) The base of the triangle is 4.5 cm and its height is 1.5 cm less. Find the area of \u200b\u200ba triangle.

2) The height of the triangle is 4.25 cm, and its base is 3 times larger. Find the area of \u200b\u200ba triangle. (Round off the answer to the nearest 0.1.)

832. Find the areas of the shaded figures (Fig. 38).

833. Which area is larger: a rectangle with sides of 5 cm and 4 cm, a square with a side of 4.5 cm, or a triangle whose base and height are 6 cm each?

834. The room is 8.5 m long, 5.6 m wide and 2.75 m high. The area of \u200b\u200bwindows, doors and stoves is 0.1 of the total area of \u200b\u200bthe walls of the room. How many pieces of wallpaper are needed to cover this room if the piece of wallpaper is 7 m long and 0.75 m wide? (Round off the answer to the nearest 1 chunk.)

835. It is necessary to plaster and whitewash a one-story house outside, the dimensions of which are: length 12 m, width 8 m and height 4.5 m. The house has 7 windows measuring 0.75 mx 1.2 m each and 2 doors each 0.75 mx 2.5 m. How much will the entire work cost if whitewashing and plastering 1 sq. m is 24 kopecks? (Round off the answer to the nearest 1 ruble.)

836. Calculate the surface and volume of your room. Find the dimensions of the room by measuring.

837. The garden has the shape of a rectangle, the length of which is 32 m, the width is 10 m. 0.05 of the entire area of \u200b\u200bthe garden is sown with carrots, and the rest of the garden is planted with potatoes and onions, and the area is 7 times larger than onions with potatoes. How much land is individually planted with potatoes, onions and carrots?

838. The vegetable garden has the shape of a rectangle, the length of which is 30 m and the width is 12 m. 0.65 of the entire area of \u200b\u200bthe garden is planted with potatoes, and the rest - with carrots and beets, with beets planted on 84 sq. m more than carrots. How much soil is separately under potatoes, beets and carrots?

839. 1) The cube-shaped box was sheathed with plywood on all sides. How much plywood is consumed if the edge of the cube is 8.2 dm? (Round off the answer to the nearest 0.1 sq. Dm.)

2) How much paint is required to paint a cube with an edge of 28 cm, if 1 sq. cm will be used up 0.4 g of paint? (Answer, round to the nearest 0.1 kg.)

840. The length of the cast-iron billet, which has the shape of a rectangular parallelepiped, is 24.5 cm, the width is 4.2 cm and the height is 3.8 cm. How much do 200 cast-iron billets weigh, if 1 cubic meter. dm of cast iron weighs 7.8 kg? (Round off the answer to the nearest 1 kg.)

841. 1) The length of the box (with a lid), which has the shape of a rectangular parallelepiped, is 62.4 cm, width 40.5 cm, height 30 cm.How many square meters of boards were used to make the box, if the waste of boards is 0.2 of the surface that should to be planked? (Round off the answer to the nearest 0.1 sq. M.)

2) The bottom and side walls of the pit, which has the shape of a rectangular parallelepiped, must be sheathed with boards. The pit is 72.5 m long, 4.6 m wide and 2.2 m high. How many square meters of planks were used for planking if the waste of planks is 0.2 of the surface to be planked? (Round off the answer to the nearest 1 sq. M.)

842. 1) The length of the basement, which has the shape of a rectangular parallelepiped, is 20.5 m, the width is 0.6 of its length, and the height is 3.2 m. The basement was filled with potatoes by 0.8 of its volume. How many tons of potatoes fit in the basement if 1 cubic meter of potatoes weighs 1.5 tons? (Round off the answer to the nearest 1 m.)

2) The length of the tank, which has the shape of a rectangular parallelepiped, is 2.5 m, the width is 0.4 of its length, and the height is 1.4 m. The tank is filled with kerosene by 0.6 of its volume. How many tons of kerosene is poured into the tank, if the weight of kerosene in a volume of 1 cubic meter m is 0.9 t? (Round off the answer to the nearest 0.1 m.)

843. 1) How long can the air be renewed in a room that is 8.5 m long, 6 m wide and 3.2 m high, if through a window in 1 sec. passes 0.1 cubic meters. m of air?

2) Calculate the time it takes to refresh the air in your room.

844. The dimensions of the concrete block for the construction of the walls are as follows: 2.7 mx 1.4 mx 0.5 m. The void is 30% of the volume of the block. How many cubic meters of concrete will it take to make 100 of these blocks?

845. Grader-elevator (ditch digging machine) in 8 hours. work makes a ditch 30 cm wide, 34 cm deep and 15 km long. How many excavators are replaced by such a machine if one excavator can take out 0.8 cu. m per hour? (Round off the result.)

846. The bins in the form of a rectangular parallelepiped are 12 m long and 8 m wide. In this bin, grain is poured up to a height of 1.5 m. In order to find out how much the whole grain weighs, they took a box 0.5 m long, 0.5 m wide and 0.4 m high, filled it with grain and weighed it. How much did the grain in the bin weigh if the grain in the box weighed 80 kg?

848. 1) Using the diagram "Steel smelting in the RSFSR" (Fig. 39). answer the following questions:

a) How many million tons of steel production increased in 1959 in comparison with 1945?

b) How many times was steel smelting in 1959 greater than smelting in 1913? (Accurate to 0.1.)

2) Using the chart "Sown area in the RSFSR" (Fig. 40), answer the following questions:

a) How many million hectares did the cultivated area increase in 1959 in comparison with 1945?

b) How many times was the sown area in 1959 greater than the sown area in 1913?

849. Construct a linear diagram of the growth of the urban population in the USSR, if in 1913 the urban population was 28.1 million people, in 1926 - 24.7 million, in 1939 - 56.1 million and in 1959 - 99, 8 million people.

850. 1) Make an estimate for the renovation of your classroom, if you need to whitewash the walls and ceiling, as well as paint the floor. The data for the preparation of the estimate (the size of the class, the cost of whitewashing 1 sq. M, the cost of painting the floor of 1 sq. M) should be found out from the school manager

2) For planting in the garden, the school bought seedlings: 30 apple trees at 0.65 rubles. apiece, 50 cherries at 0.4 rubles. apiece, 40 gooseberry bushes for 0.2 rubles. and 100 raspberry bushes for 0.03 rubles. per bush. Write an invoice for this purchase as follows:

Math simulator on the topic

"Joint actions with decimal fractions"

Compiled by the math teacher

Tolmacheva Nadezhda Alekseevna

MBOU SOSH №69, Nizhny Tagil

Explanatory note

The math simulator is designed for students 5kl-6kl, it can be used in work with any teaching materials in mathematics, as well as in preparing 9th grade students for passing the OGE.

The simulator is designed for both classroom work and independent work at home.

The simulator provides the ability to develop a conscious application of all the rules of action with decimal fractions.

The simulator can be used as a primary control of knowledge, as well as in correctional work. The tasks of the simulator allow you to invite the student to perform more calculations in a short time. Thus, not only computational skills are honed, but also attention is trained, the student's working memory develops.

Simulator tasks can be offered for both individual and group work in the classroom.

Math Trainer

Option 1
	15,3 * 5,4 - 4,2* (5,12 – 4,912) + 16,0036
	9,84 - 16,32 * (8 – 7,45) + 2,186
	(2,12 + 1,07) * (2,12 – 1,07)
	86,4 * (17,01: 4,2) : 6,4
	42,26 – 34,68: (33,32: 9,8)
	40 – (7,12 + 11,043: 2,7)
	12,6: (2,04 + 4,26) – 0,564
	7,371: (5 – 3,18) + 2,05 *(17,82 – 7)
	(5,2: 26 + 26: 5,2) *6,1 + 5,25: 5
	27,5967: (8 – 1,186) + 3,02
	(20 – 13,7) * 7,4 + 18: 0,6
	(4,694 - 3,998) : 4,35 + (4,5 * 5,4 – 0,06)
	(4,6 * 3,5 + 15,32) : 31,42 + (7,26 – 5,78) : 0,148
	(101,96 – 6,8 * 7,2) : 4,24 – 3,4 * (10 – 6,35)
	7,72 * 2,25 – 4,06: (0,824 + 1,176) – 12,423
	51,328: 6, 4 + 3,2 * (10 – 4,7) * 2,05
	(42,12 * 0,12 + 112,016* 0,1) : 1,6 – 9,424
	((4,2 *0,81 – 6,8*0,05) : 0,5)) : 200
	2,6* (4,4312 + 15,5688) – 6,66: (8,2 – 6,72)
	(0,624: 4,16 + 6,867: 2,18) *2,08 – 4,664
	4260 + 42,6: (62,06 + 37,94) – 42,6: (52,44 - 52,43)
	5: 0,25 + 0,6 *(9,275 – 4,275) : 0,1
	3,1: 100 + (6 – 0,3: 100) *10
	0,415 +(2,85: 0,6*3,2 – 2,72: 8) + 5,134: 0,17
	0,1: 0,002 – 0,5*(7,91: 0,565 – 11,1:1,48)
	0,2: 0,004 + (7,91: 0,565 – 44,4: 5,92) *0,5
	4,735: 0,5 + 14,95: 1,3 + 2,121: 0,7
	(0,1955 + 0,187) : 0,085
	(86,9 + 667,6) : (37,1 + 13,2)
	(0,008 + 0,992) * (5 *0,6 – 1,4)

Math Trainer

Decimal actions

Option 2
	(130,2 – 30,8) : 2,8 - 21,84
	3,712: (7 – 3,8) + 1,3* (2,74 + 0,66)
	(3,4: 1,7 + 0,57: 1,9)* 4,9 + 0,0825: 2,75
	10,79: 8,3*0,7 - 0,46 * 3,15: 6,9
	(21,2544: 0,9 + 1,02 * 3,2) : 5,6
	4,36: (3,15 + 2,3) + (0,792 – 0,78) * 350
	(3,91: 2,3 * 5,4 – 4,03) * 2,4
	6,93: (0,028 + 0,36 * 4,2) - 3,5
	42,165 – 22,165: (0,61 + 3,42)
	((4: 0,128 + 14628,25) : 1,011* 0,00008 + 6,84) : 12,5
	687,8 + (88,0802 – 85,3712) : 0,045
	(3,1 * 5,3 – 14,39) : 1,7 + 0,8
	(3,8 * 1,75: 0,95 – 1,02) : 2,3 + 0,4
	((23,79: 7,8 – 6,8: 17) * 3,04 – 2,04) * 0,85
	0,15: 0,01 + (6 + 9,728: 3,2) * 2,5 – 1,4
	1,44: 3,6 + 0,8 + 3,6: 1,44* (0,1 - 0,02)
	3,45 * (11,2 + 75,6) – 0,93 * 1,26
	4,25: 0,25 – 0,06 * 82 + 0,4
	(0,237 + 45,6) * 12,01 - 11,1* (237,1 – 229,9)
	5,8 – 0,27 * 3,6 + 5,172
	12 – 5,3: (19,6: 0,35 - 0,06 * 50)
	(0,6 + 0,25 – 0,125) * 3,2 + 4,5: 100
	(15,5: 0,25 – 0,08 * 200) : 2,3 – 1,3
	(87,05 * 2,7 – 55,68:32) * 0,8: 0,02
	522,348: 87 + 2,7 * (0,84 – 0,128: 0,16)
	6400 * 0,0145 – (1272,6: 0,42 – 3000)
	(0,7: 1,4 – 0,02) : 0,012 + 1,6 * (0,548 – 0,023)
	(1,184: 3,2 + 0,832: 0,4) : 0,5 + 1,5
	4,96 ; 10 + 35,8: 100 - 0,0042
	(0,04 + 3,59) * (7,35 + 2,65) : 300

Math Trainer

Decimal actions

Option 3
	2,5 + 0,56* 28 + 0,125*15 – 0,12*7
	12,8: 4 + 76,8: 12 – 42,6: 6 – 2,4
	4,01 + 43,6: 10 – 73,2: 30 + 15,4: 100
	176,4: 100 – 0,041*40 + 13,5:50 +0,3
	(16,4 + 13,2)*3 – (10,6 + 4,8) *2 – 23,2
	(40,65 - 32,6) : 5 + (4,72 _ 2,24)*3
	4,735: 0,5 + 14,95: 1,3 + 2,121: 0,7 – 21,6
	0,01105 + 0,05 - 0,3417: 34 -_ 0,875: 125
	(5,72 – 3,21)*5 + (86,9 + 667,6) : (37,1 + 13,2)
	(0,1955 + 0,187) : 0,085 – (4,72 – 4,72)*0,157
	4,9 – (0,008 + 0,992) * (5 *0,6 – 1,4)
	(50000 – 1397,3) : (20,4 + 33,603) – 856
	3,7 *0,18 + 35,9 *0,26 – 0,109 *91
	34,98: 6,6 + 5,141: 0,53 – 0,8379: 0,057
	0,131 *470 + 26,97: 2,9 - 50,4 *1,4
	0,439 *97 – 182,75: 4,3 + 31,9 *0,43
	(20,4 – 18,23)* 4,3 + (0,40713 + 0,44176) : 0,67
	(0,357 + 7,043)*0,85 + (52 – 1,928) : 5,69
	(1,5 - 0,4732)* 35 – (0,6092 + 0,0718) : 0,75
	(139,4 + 16,6)* 0,039 - (20 – 17,54) : 2,5
	4,1819 + 0,73 *(5,375 + 2,595)
	5,0143 – 65,9*(0,0612 + 0,0058)
	(0,83 *3,7 + 9,741:51 – 0,012) : 0,325
	(67,21: 0,143 – 0,546*850 + 2,1) : 1,25
	(79* 0,63 – 9,558: 5,4 – 26,94) : 0,324
	(11,328: 16 + 7,752: 7,6) : 0,16
	13,7 – (0,53 *6,7 + 1,77*3,1 + 0,004) : 0,66
	5,3: (2,87* 0,53 – 0,043 *7,7 – 0,19)
	(3,06 – 2,97) * (5,6*0,93 – 0,84*6,2)
	(5,4*0,77 – 0,008) : (2,747: 0,67+ 0,05)

Math Trainer

Decimal actions

Option 4
	589,72:16 – 18,305:7 + 5,67: 4
	(86,9 + 667,6) : (37,1 +13,2)
	(0,93 + 0,07) : (0,93 – 0,805)
	1,35: 2,7 + 6,02 – 5,9 + 0,4: 2,5 *(4,2 – 1,075)
	((14,068 + 15,78) : (1,875 + 0,175)) : (0,325+ 0,195)
	(0,578 + 0,172)* (0,823 + 0,117) – 1,711: (4,418 + 1,382)
	(39,3 + 116,7) *0,39 – (19,01 -16,56) : 2,5
	(2,747: 0,67 + 0,05) : (0,54* 7,7 – 0,008)
	5,76*4,76: 6,12 + 81,9: 58,5*2,05
	25,6: (38,07 + 1,93) + 0,037 *10
	(3,7011: 0,73 – 9,27: 4,5 – 1,41) :1,6
	40,86: 4,5 – 0,6039: 5,49 + 0.338: 0,13
	(85,9 +667,1) : ((37 +13,2) + (11,44 – 6,42)*10
	1,224: (7 – 2,92) + 1,06*(13,5 – 3)
	(7,5* 48 – 8,2* 9,5 + 141,4) : (254,1:4,2)
	0,63*69 – 10,048: 6,4 – 19,44: 32,4 *0,8
	(3,8: 19 + 1,9: 3,8) *5,2 + 7,28: 7
	(4,9 + 1,06 – 0,98) : (0,83*0,6) : 2,4
	(28,7 *0,15) : (0,25 *0,21) + 22,5:1,25
	0,1: 0,002 + (7,91: 0,565 - 11,1: 1,48)
	(0,2028:0,24 – 0,32 *1,5) *(4,05 – 13,1625: 4,05)
	(97,44: 0,48 + 128,64: 3,2) *0,25 – 17,89
	5,4 + ((4,7 – 2,85)*1,8 + 0,0156: 0,13)
	(1,2 *0,15 + 12:100 – 1,4: 10) : 0,1
	0,545: 0,5 +2,75 *0,4 – 0,45 *3,8
	0,6 * (7,24: 0,8 – 0,968: 0,16) + 2,25 *0,04
	(6,4 *0,025 + 7,07: 3,5 – 3,68: 4) : 0,9
	2,5 *(3: 6 – 0,2: 5 + 1,2 *0,15)
	(5,508: 0,27 – 10,2 *1,3) : 0,7 + 1,3: 0,1
	1,5 + 0,5*(4,214: 0,14 – 5,436: 1,8) * 0,1

Answers

Math Trainer

Decimal actions

	Option 1	Option 2	Option 3	Option 4

When adding decimal fractions, you need to write them one under the other so that the same digits are under each other, and the comma is under the comma, and add the fractions as you add natural numbers. Add, for example, the fractions 12.7 and 3.442. The first fraction contains one decimal place, and the second contains three. To perform addition, transform the first fraction so that there are three digits after the decimal point:, then

Subtraction of decimal fractions is performed in the same way. Find the difference between the numbers 13.1 and 0.37:

When multiplying decimal fractions, it is enough to multiply the given numbers, ignoring the commas (like natural numbers), and then, as a result, separate as many digits on the right with a comma as they are after the decimal point in both factors in total.

For example, multiply 2.7 times 1.3. We have. Separate two digits on the right with a comma (the sum of the digits in the multipliers after the comma is two). As a result, we get 2.7 1.3 \u003d 3.51.

If there are fewer digits in the product than must be separated by a comma, then the missing zeros are written in front, for example:

Consider the multiplication of a decimal fraction by 10, 100, 1000, and so on. Suppose you need to multiply the fraction 12.733 by 10. We have. Separating three digits on the right with a comma, we get No. Hence,

12,733 10 \u003d 127.33. Thus, multiplying a decimal fraction by 10 is reduced to moving a comma one digit to the right.

In general, in order to multiply a decimal fraction by 10, 100, 1000, it is necessary to transfer the comma to 1, 2, 3 digits to the right in this fraction, adding, if necessary, a certain number of zeros to the fraction on the right). For example,

Dividing a decimal fraction by a natural number is performed in the same way as dividing a natural number by a natural number, and the comma in the quotient is set after the division of the integer part is completed. Let's divide 22.1 by 13:

If a whole part the dividend is less than the divisor, then the answer turns out to be zero integers, for example:

Consider now dividing a decimal fraction by a decimal. Let's divide 2.576 by 1.12. To do this, both in the dividend and in the divisor, we transfer the comma to the right by as many digits as there are after the comma in the divisor (in this example, by two). In other words, multiply the dividend and the divisor by 100 - this will not change the quotient. Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

To divide the decimal fraction by, it is necessary to move the comma to the left of the digits in this fraction (in this case, if necessary, the required number of zeros is assigned to the left). For example, .

As for natural numbers, division is not always feasible, so it is not always feasible for decimal fractions. Let's divide 2.8 by 0.09 for example.

We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as on what the decimal places are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part, we will show how the points corresponding to the fractional numbers are located on the coordinate axis.

What is decimal notation for fractional numbers

The so-called decimal notation of fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.

The decimal point is used to separate the whole part from the fractional part. As a rule, the last digit of a decimal fraction is not a zero, unless the decimal point is immediately after the first zero.

What are some examples of fractional numbers in decimal notation? It can be 34, 21, 0, 35035044, 0, 0001, 11 231 552, 9, etc.

In some textbooks, you can find the use of a period instead of a comma (5. 67, 6789. 1011, etc.) This option is considered equivalent, but it is more typical for English-language sources.

Definition of decimals

Based on the above notion of decimal notation, we can formulate the following definition of decimal fractions:

Definition 1

Decimal fractions are fractional numbers in decimal notation.

Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, a more compact notation, especially in cases where the denominator is 1000, 100, 10, etc., or a mixed number. For example, instead of 6 10 we can indicate 0, 6, instead of 25 10000 - 0, 0023, instead of 512 3 100 - 512.03.

How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be discussed in a separate material.

How to read decimals correctly

There are some rules for reading decimal notation. So, those decimal fractions, which correspond to their regular ordinary equivalents, are read in almost the same way, but with the addition of the words "zero tenths" at the beginning. So, the record 0, 14, which corresponds to 14 100, reads as "zero point fourteen hundredths."

If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have a fraction 56, 002, which corresponds to 56 2 1000, we read such an entry as "fifty six point two thousandths."

The meaning of a digit in a decimal fraction depends on where it is located (just as in the case of natural numbers). So, in decimal fraction 0, 7, seven is tenths, in 0, 0007 - ten thousandths, and in fractions 70,000, 345 it means seven tens of thousands of whole units. Thus, in decimal fractions, there is also the concept of the digit of a number.

The names of the decimal places are similar to those that exist in natural numbers. The names of those that are located after are clearly presented in the table:

Let's look at an example.

Example 1

We have decimal 43, 098. She has four in the tens, three in the ones, zero in tenths, 9 in hundredths, 8 in thousandths.

It is customary to distinguish between the digits of decimal fractions by seniority. If we move through the numbers from left to right, then we will go from the most significant digits to the least significant ones. It turns out that hundreds are older than tens, and millionths are younger than hundredths. If we take that final decimal fraction, which we gave as an example above, then in it the highest, or highest, will be the place of hundreds, and the lowest, or lowest, will be the place of 10-thousandths.

Any decimal fraction can be decomposed into separate digits, that is, represented as a sum. This action is performed in the same way as for natural numbers.

Example 2

Let's try to expand the fraction 56, 0455 into digits.

We will get:

56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005

If we remember the properties of addition, then we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.

What are final decimals

All fractions that we talked about above are final decimal fractions. This means that the number of digits after the decimal point is finite. Let's derive the definition:

Definition 1

Ending decimal fractions are a form of decimal fractions that have a finite number of digits after the decimal point.

Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231 032, 49, etc.

Any of these fractions can be converted either into a mixed number (if the value of their fractional part is different from zero), or into an ordinary fraction (with a zero integer part). We have devoted a separate material to how this is done. Here we will simply indicate a couple of examples: for example, we can reduce the final decimal fraction 5, 63 to the form 5 63 100, and 0, 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5.)

But the reverse process, i.e. writing an ordinary fraction in decimal form may not always be performed. So, 5 13 cannot be replaced by an equal fraction with a denominator of 100, 10, etc., which means that the final decimal fraction will not work out of it.

The main types of infinite decimal fractions: periodic and non-periodic fractions

We pointed out above that final fractions are called so because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.

Definition 2

Infinite decimal fractions are those that have an infinite number of digits after the decimal point.

Obviously, such numbers simply cannot be written completely, so we indicate only a part of them and then put ellipsis. This sign speaks of the endless continuation of the sequence of decimal places. Examples of infinite decimal fractions are 0, 143346732 ..., 3, 1415989032 ..., 153, 0245005 ..., 2, 66666666666 ..., 69, 748768152 .... etc.

In the "tail" of such a fraction, there can be not only sequences of numbers random at first glance, but constant repetition of the same sign or group of signs. Fractions with alternating decimal point are called periodic.

Definition 3

Periodic decimal fractions are infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the fraction period.

For example, for the fraction 3, 444444…. the period will be the number 4, and for 76, 134134134134 ... - group 134.

What is the minimum number of characters that can be left in the record of a periodic fraction? For periodic fractions, it will be enough to write the entire period once in parentheses. So, fraction 3, 444444…. it will be correct to write it down as 3, (4), and 76, 134134134134 ... - as 76, (134).

In general, records with multiple periods in brackets will have exactly the same meaning: for example, the periodic fraction 0, 677777 is the same as 0, 6 (7) and 0, 6 (77), etc. Records of the form 0, 67777 (7), 0, 67 (7777), etc. are also allowed.

To avoid mistakes, let us introduce uniformity of notation. Let's agree to write down only one period (the shortest sequence of digits), which is closest to the decimal point, and enclose it in parentheses.

That is, for the above fraction, we will consider the entry 0, 6 (7) as the main one, and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34).

If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, they will result in infinite fractions.

In principle, we can write any finite fraction as a periodic one. To do this, we just need to add infinitely many zeros to the right. What does it look like in the recording? Let's say we have a final fraction 45, 32. In periodic form, it will look like 45, 32 (0). This action is possible because adding zeros to the right of any decimal gives us an equal fraction.

Separately, we should dwell on periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9). They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. In this case, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers is easy to check by presenting them as ordinary fractions.

For example, the fraction 8, 31 (9) can be replaced with the corresponding fraction 8, 32 (0). Or 4, (9) \u003d 5, (0) \u003d 5.

Infinite decimal periodic fractions are rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.

There are also fractions that do not have an infinitely repeating sequence after the decimal point. In this case, they are called non-periodic fractions.

Definition 4

Non-periodic decimal fractions include those infinite decimal fractions in which there is no period after the decimal point, i.e. repeating group of numbers.

Sometimes non-periodic fractions look very similar to periodic ones. For example, 9, 03003000300003 ... at first glance seems to have a period, but a detailed analysis of the decimal places confirms that this is still a non-periodic fraction. You have to be very careful with such numbers.

Non-periodic fractions are irrational numbers. They are not translated into ordinary fractions.

Basic Decimal Operations

You can perform the following actions with decimal fractions: comparison, subtraction, addition, division, and multiplication. Let's analyze each of them separately.

Comparing decimal fractions can be reduced to comparing fractions that match the original decimal. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions into ordinary ones is often a laborious task. How can we quickly perform a comparison action if we need to do it while solving a problem? It is convenient to compare decimal fractions by place in the same way as we compare natural numbers. We will devote a separate article to this method.

To add some decimal fractions to others, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to the standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we must first round them to a certain digit, and then add them. The smaller the digit to which we round off, the higher the calculation accuracy. For subtraction, multiplication, and division of infinite fractions, preliminary rounding is also necessary.

Finding the difference of decimal fractions inversely to addition. In fact, with the help of subtraction, we can find such a number, the sum of which with the subtracted fraction will give us the decreasing one. We will tell you more about this in a separate article.

Multiplication of decimal fractions is performed in the same way as for natural numbers. The column calculation method is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded off before counting.

The process of dividing decimal fractions is the reverse of the process of multiplication. When solving problems, we also use column counts.

You can set an exact correspondence between the final decimal fraction and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly match the required decimal fraction.

We have already studied how to construct the points corresponding to ordinary fractions, but decimal fractions can be reduced to this form. For example, an ordinary fraction 14 10 is the same as 1, 4, so the corresponding point will be removed from the origin in the positive direction by exactly the same distance:

You can do without replacing the decimal fraction with an ordinary one, but take the method of expansion into digits as a basis. So, if we need to mark a point, the coordinate of which will be 15, 4008, then we will preliminarily represent this number as the sum of 15 + 0, 4 +, 0008. To begin with, we postpone 15 whole unit segments in the positive direction from the origin, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we get the coordinate point, which corresponds to the fraction 15, 4008.

For an infinite decimal fraction, it is better to use this method, since it allows you to get as close to the desired point as you like. In some cases, it is possible to construct an exact correspondence of an infinite fraction on the coordinate axis: for example, 2 \u003d 1, 41421. ... ... , and this fraction can be associated with a point on the coordinate ray remote from 0 by the length of the diagonal of a square, the side of which will be equal to one unit segment.

If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of the segment. Let's see how to do it correctly.

Let's say we need to get from zero to a given point on the coordinate axis (or as close as possible in the case of an infinite fraction). To do this, we gradually set aside the unit segments from the origin until we get to the desired point. After whole segments, if necessary, we measure out tenths, hundredths and smaller fractions so that the correspondence is as accurate as possible. As a result, we got a decimal fraction, which corresponds to a given point on the coordinate axis.

Above we gave a drawing with a point M. Look at it again: to get to this point, you need to measure from zero one unit segment and four tenths of it, since this point corresponds to the decimal fraction 1, 4.

If we cannot get to a point in the process of decimal measurement, it means that an infinite decimal fraction corresponds to it.

If you notice an error in the text, please select it and press Ctrl + Enter

Decimal fraction is used when you need to perform actions with non-whole numbers. This may seem irrational. But this kind of numbers greatly facilitates the mathematical operations that must be performed with them. This understanding comes with time, when their writing becomes familiar, and reading is not difficult, and the rules of decimal fractions are mastered. Moreover, all the actions are repeated already known, which are mastered with natural numbers. You just need to remember some features.

Decimal Definition

A decimal is a special representation of a non-integer number with a denominator that is divisible by 10, and the answer is obtained as one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, then it is more convenient to rewrite the number using a comma. Then the whole part will be located before it, and then the fractional part. Moreover, the recording of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the place of the denominator.

The above can be illustrated with these numbers:

9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.

Reasons why you need to use decimal fractions

Decimals were needed by mathematicians for several reasons:

Simplification of the recording. Such a fraction is located along one line without a dash between the denominator and the numerator, while the clarity does not suffer.

Simplicity in comparison. It is enough to simply correlate the numbers that are in the same positions, while with ordinary fractions it would be necessary to bring them to a common denominator.

Simplification of calculations.

Calculators are not designed for the introduction of ordinary fractions, they use decimal notation for all operations.

How to read these numbers correctly?

The answer is simple: just like an ordinary mixed number with a denominator that is a multiple of 10. The only exceptions are fractions without an integer value, then when reading you need to pronounce "zero integers".

For example, 45/1000 should be pronounced as forty five thousandths, at the same time 0.045 would sound like zero point forty-five thousandths.

A mixed number with the integer part equal to 7 and the fraction 17/100, which is written as 7.17, in both cases will be read as seven point seventeen hundredths.

The role of digits in writing fractions

To correctly mark the rank is what the mathematician demands. Decimal fractions and their meaning can change significantly if you write the number in the wrong place. However, this was true before.

To read the digits of the integer part of a decimal fraction, you just need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If the whole part sounded "tens", then after the comma it will be "tenths".

This can be clearly seen in this table.

Decimal Places Table

class	thousand			units			,	fractional part
discharge	honeycomb	dec.	units	honeycomb	dec.	units		tenth	hundredth	thousandth	ten thousandth

What is the correct way to write a mixed number as a decimal fraction?

If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert the fraction to decimal is not difficult. To do this, it is enough to rewrite all its constituent parts in a different way. The following points will help with this:

write the numerator of the fraction a little aside, at this moment the decimal point is located on the right, after the last digit;

move the comma to the left, the most important thing here is to correctly count the numbers - you need to move it by as many positions as there are zeros in the denominator;

if there are not enough of them, then zeroes should appear in empty positions;

zeros that were at the end of the numerator are no longer needed and can be crossed out;

in front of the comma, assign an integer part, if it was not there, then there will also be zero here.

Attention. You cannot cross out zeros that are surrounded by other numbers.

You can read about how to be in a situation when the denominator contains not only ones and zeros, how to convert a fraction to decimal, you can read below. This is important information that you should definitely read.

How to convert a fraction to decimal if the denominator is an arbitrary number?

Two options are possible here:

When the denominator can be represented as a number that is equal to ten to any power.

If such an operation cannot be done.

How can I check this? You need to factor the denominator. If there are only 2 and 5 in the product, then everything is fine, and the fraction is easily converted to the final decimal. Otherwise, if 3, 7 and other prime numbers appear, then the result will be infinite. For ease of use in mathematical operations, it is customary to round off such a decimal fraction. This will be discussed a little below.

Studying how such decimal fractions are obtained, grade 5. Examples here will be very helpful.

Let the denominators contain numbers: 40, 24 and 75. The factorization for them will be as follows:

• 40 \u003d 2 2 2 5;
• 24 \u003d 2 2 2 3;
• 75 \u003d 5 5 3.

In these examples, only the first fraction can be finalized.

Algorithm for converting an ordinary fraction to a final decimal

Check the prime factorization of the denominator and make sure it will consist of 2 and 5.

Add to these numbers as many 2 and 5 so that they become equal. They will give the value of the extra multiplier.

Multiply the denominator and numerator by this number. The result will be an ordinary fraction, below the line that has 10 to some extent.

If in a problem these actions are performed with a mixed number, then it must first be represented as an improper fraction. And only then proceed according to the described scenario.

Rounded decimal representation of a fraction

This way of converting a fraction to a decimal will seem even easier to someone. Because it doesn't have a lot of action. You just need to divide the value of the numerator by the denominator.

Any number with a decimal part to the right of the decimal point can be assigned an infinite number of zeros. This property should be used.

First write down the whole part followed by a comma. If the fraction is correct, then write zero.

Then it is supposed to perform division of the numerator by the denominator. So that they have the same number of digits. That is, assign the required number of zeros to the right of the numerator.

Perform long division until the required number of digits is entered. For example, if you need to round up to hundredths, then the answer should be 3. In general, there should be one more digits than you need to get in the end.

Write down the intermediate answer after the comma and round according to the rules. If the last digit is from 0 to 4, then you just need to discard it. And when it is 5-9, then the one in front of it needs to be increased by one, dropping the last one.

Back from decimal to fraction

In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary fractions, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.

For this procedure, you need to do the following:

write down the whole part, if it is equal to zero, then you do not need to write anything;

draw a fractional line;

write down the numbers from the right side above it, if zeros come first, then they need to be crossed out;

under the line, write a unit with as many zeros as the number of digits after the decimal point in the initial fraction.

This is all you need to do to convert a decimal to a fraction.

What can you do with decimal fractions?

In mathematics, these will be certain actions with decimal fractions that were previously performed for other numbers.

They are:

comparison;

addition and subtraction;

multiplication and division.

The first action, comparison, is similar to how it was done for natural numbers. To determine which is greater, you need to compare the digits of the integer part. If they turn out to be equal, then go to fractional and compare them in the same way. The number where the largest digit in the most significant digit is found will be the answer.

Adding and subtracting decimal fractions

These are perhaps the simplest steps. Because they are performed according to the rules for natural numbers.



So, in order to perform the addition of decimal fractions, they need to be written under each other, placing the commas in a column. With this notation, whole parts appear to the left of the commas, and fractional parts to the right. And now you need to add the numbers bit by bit, as is done with natural numbers, dropping a comma down. You need to start addition with the smallest digit of the fractional part of the number. If there are not enough digits in the right half, then zeros are added.

The same applies for subtraction. And here there is a rule that describes the possibility of borrowing one from the most significant bit. If there are fewer digits in the reduced fraction after the decimal point than in the subtracted fraction, then zeros are simply assigned in it.

The situation is a little more complicated with tasks where you need to perform multiplication and division of decimal fractions.

How to multiply decimal in different examples?

The rule by which decimal fractions are multiplied by a natural number is as follows:

write them down in a column, not paying attention to the comma;

multiply as if they were natural;

separate as many digits with a comma as there were in the fractional part of the original number.

A special case is an example in which a natural number is equal to 10 to any power. Then, to get an answer, you just need to move the comma to the right by as many positions as there are zeros in another factor. In other words, when multiplied by 10, the comma is shifted by one digit, by 100 - there will already be two of them, and so on. If there are not enough digits in the fractional part, then you need to write zeros in empty positions.

The rule that is used when a task needs to multiply decimal fractions by another same number:

write them under each other, ignoring the commas;

multiply as if they were natural;

separate as many digits with a comma as there were in the fractional parts of both original fractions together.

Examples are highlighted as a special case in which one of the factors is 0.1 or 0.01 and so on. In them, you need to move the comma to the left by the number of digits in the presented multipliers. That is, if it is multiplied by 0.1, then the comma is shifted by one position.

How do I split a decimal in different tasks?

Division of decimal fractions by a natural number is performed according to the following rule:

write them down for long division, as if they were natural;

divide according to the usual rule until the whole part ends;

put a comma in response;

continue dividing the fractional component until the remainder is zero;

if necessary, you can assign the required number of zeros.

If the integer part is equal to zero, then it will not be in the answer either.

Separately, there is a division into numbers equal to ten, one hundred, and so on. In such problems, you need to move the comma to the left by the number of zeros in the divisor. It happens that there are not enough digits in the whole part, then zeros are used instead. You may notice that this operation is similar to multiplying by 0.1 and similar numbers.

To perform decimal division, you need to use this rule:

turn the divisor into a natural number, and for this move the comma in it to the right to the end;

move a comma and in a divisible by the same number of digits;

proceed according to the previous scenario.

Division by 0.1 is highlighted; 0.01 and other similar numbers. In these examples, the comma is shifted to the right by the number of decimal digits. If they are over, then you need to assign the missing number of zeros. It should be noted that this action repeats division by 10 and similar numbers.



Conclusion: it's all about practice

Nothing in learning comes easily or effortlessly. It takes time and practice to master new material reliably. Mathematics is no exception.

So that the topic about decimal fractions does not cause difficulties, you need to solve as many examples with them as possible. After all, there was a time when the addition of natural numbers was perplexing. And now everything is fine.

Therefore, to paraphrase the well-known phrase: decide, decide and decide again. Then tasks with such numbers will be performed easily and naturally, like another puzzle.

By the way, puzzles are difficult to solve at first, and then you need to do the usual movements. The same is in mathematical examples: after walking the same path several times, then you will no longer think about where to turn.