How to solve decimals. Decimals, definitions, notation, examples, operations with decimals
CHAPTER III.
DECIMALS.
§ 31. Problems and examples for all operations with decimal fractions.
Follow these steps:
767. Find the quotient of division:
Follow these steps:
772. Calculate:
Find X , If:
776. The unknown number was multiplied by the difference between the numbers 1 and 0.57 and the product was 3.44. Find the 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 was 2.412. Find the unknown number.
778. Using the data from the diagram about iron smelting in the RSFSR (Fig. 36), create a problem to solve which you need to apply the actions of addition, subtraction and division.
779. 1) Length Suez Canal 165.8 km, the length of the Panama Canal is 84.7 km less than the Suez Canal, and the length of the White Sea-Baltic Canal is 145.9 km longer than the Panama Canal. What is the length of the White Sea-Baltic Canal?
2) The Moscow metro (by 1959) was built in 5 stages. The length of the first stage 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 stage, the length of the fourth stage is 9.6 km more than the third stage, and the length of the fifth stage is 11.5 km less fourth. What was the length of the Moscow metro at the beginning of 1959?
780. 1) The greatest depth of the Atlantic Ocean is 8.5 km, greatest depth Pacific Ocean at 2.3 km more depth Atlantic Ocean, and the greatest depth of the North Arctic Ocean 2 times less than the greatest depth Pacific Ocean. What is the greatest depth of the Arctic Ocean?
2) The Moskvich car consumes 9 liters of gasoline per 100 km, the Pobeda car consumes 4.5 liters more than the Moskvich, and the Volga is 1.1 times more than the Pobeda. How much gasoline does a Volga car consume per 1 km of travel? (Round 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 by horse for 1.5 hours. In total he traveled 440 km. At what speed did the student travel on the railroad if he rode horses at a speed of 10 km per hour?
2) The collective farmer had to be at a point located at a distance of 134.7 km from his home. He rode the bus for 2.4 hours at an average speed of 55 km per hour, and walked the rest of the way at a speed of 4.5 km per hour. How long did he walk?
782. 1) Over the summer, one gopher destroys about 0.12 centners of bread. In the spring, the pioneers exterminated 1,250 ground squirrels on 37.5 hectares. How much bread did the schoolchildren save for the collective farm? How much saved bread is there per 1 hectare?
2) The collective farm calculated that by destroying gophers on an area of 15 hectares of arable land, schoolchildren saved 3.6 tons of grain. How many gophers are destroyed on average per 1 hectare of land if one gopher destroys 0.012 tons of grain over the summer?
783. 1) When grinding wheat into flour, 0.1 of its weight is lost, and when baking, a bake equal to 0.4 of the weight of flour is obtained. How much baked bread will be produced from 2.5 tons of wheat?
2) The collective farm collected 560 tons of sunflower seeds. How many sunflower oil made from harvested grain if the weight of the grain is 0.7 of the weight of sunflower seeds, and the weight of the resulting oil is 0.25 of the weight of the grain?
784. 1) The yield of cream from milk is 0.16 of the weight of milk, and the yield of butter from cream is 0.25 of the weight of cream. How much milk (by weight) is required to produce 1 quintal of butter?
2) How many kilograms of porcini mushrooms must be collected to obtain 1 kg of dried mushrooms, if during preparation for drying 0.5 of the weight remains, and during drying 0.1 of the weight of the processed mushroom remains?
785. 1) The land allocated 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 allocated for the collective farm garden and for the estates of collective farmers. How much land is there on the collective farm?
2) The collective farm sowed 75% of the total sown area with grain crops, 20% with vegetables, and the remaining area with forage grasses. How much sown area did the collective farm have if it sowed 60 hectares with fodder grasses?
786. 1) How many quintals of seeds will be required to sow a field shaped like a rectangle 875 m long and 640 m wide, if 1.5 quintals of seeds are sown per 1 hectare?
2) How many quintals of seeds will be required to sow a field shaped like a rectangle if its perimeter is 1.6 km? The field width is 300 m. To sow 1 hectare, 1.5 quintals of seeds are required.
787. How many records square shape with a side of 0.2 dm will fit in a rectangle measuring 0.4 dm x 10 dm?
788. The reading room has dimensions of 9.6 m x 5 m x 4.5 m. How many seats is the reading room designed for if 3 cubic meters are needed for each person? m of air?
789. 1) What area of meadow will a tractor with a trailer of four mowers mow in 8 hours, if the working 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 the answer to the nearest 0.1 hectares.)
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 output of a three-furrow tractor plow in 10 hours. work, if the tractor speed is 5 km per hour, the grip of one body is 35 cm, and the waste of time was 0.1 of the total time spent. (Round the answer to the nearest 0.1 hectares.)
2) Find the output of a five-furrow tractor plow in 6 hours. work, if the tractor speed is 4.5 km per hour, the grip of one body is 30 cm, and the waste of time was 0.1 of the total time spent. (Round the answer to the nearest 0.1 hectares.)
791. The water consumption per 5 km of travel for a steam locomotive of a passenger train is 0.75 tons. The tender's water tank holds 16.5 tons of water. How many kilometers will the train have enough water to travel if the tank is filled to 0.9 of its capacity?
792. The siding can accommodate only 120 freight cars with an average car length of 7.6 m. How many four-axle passenger cars, each 19.2 m long, can fit on this track if 24 more freight cars are placed on this track?
793. To ensure the strength of the railway embankment, it is recommended to strengthen the slopes by sowing field grasses. For each square meter of embankment, 2.8 g of seeds are required, costing 0.25 rubles. for 1 kg. How much will it cost to sow 1.02 hectares of slopes if the cost of the work is 0.4 of the cost of the seeds? (Round the answer to the nearest 1 ruble.)
794. The brick factory was delivered to the station railway bricks. 25 horses and 10 trucks worked to transport the bricks. Each horse carried 0.7 tons per trip and made 4 trips per day. Each vehicle transported 2.5 tons per trip and made 15 trips per day. The transportation lasted 4 days. How many bricks were delivered to the station if average weight one brick 3.75 kg? (Round the answer to the nearest 1 thousand units.)
795. The flour stock was distributed among three bakeries: the first received 0.4 of the total stock, the second 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 there are 0.875 of this number, and in the first year there are 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 and 707.5;
2) three numbers: 46.5; 37.8 and 36; 0.84; 0.69 and 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 this day.
2) What is average temperature for a 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 output of the team per day.
2) To establish the standard time for manufacturing a new part, 3 lathes were supplied. The first one produced the part in 3.2 minutes, the second in 3.8 minutes, and the third in 4.1 minutes. Calculate the time standard that was set for manufacturing the part.
800. 1) The arithmetic mean of two numbers is 36.4. One of these numbers is 36.8. Find something else.
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 it was 28.4° at noon, 18.2° in the evening, and the average temperature of the day is 20.4°.
801. 1) The car traveled 98.5 km in the first two hours, and 138 km in the next three hours. How many kilometers did the average car travel per hour?
2) A test catch and weighing of yearling carp showed that out of 10 carp, 4 weighed 0.6 kg, 3 weighed 0.65 kg, 2 weighed 0.7 kg and 1 weighed 0.8 kg. What is the average weight of a yearling carp?
802. 1) For 2 liters of syrup costing 1.05 rubles. for 1 liter added 8 liters of water. How much does 1 liter of the resulting water with syrup cost?
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 does a plate of borscht cost if its volume is 0.5 liters?
803. Laboratory work “Measuring the distance between two points”,
1st appointment. Measurement with a tape measure (measuring tape). The class is divided into units of three people each. Accessories: 5-6 poles and 8-10 tags.
Progress of work: 1) points A and B are marked and a straight line is drawn between them (see task 178); 2) lay the tape measure along the hung straight line and each time mark the end of the tape measure with a tag. 2nd appointment. Measurement, steps. The class is divided into units of three people each. Each student walks the distance from A to B, counting the number of his steps. Multiplying average length of your step by the resulting number of steps, find the distance from A to B.
3rd appointment. Measuring by eye. Each student draws left hand with raised thumb(Fig. 37) and directs thumb on the pole to point B (a tree in the picture) so that the left eye (point A), thumb and point B are on the same straight line. Without changing position, close your left eye and look at your thumb with your right. Measure the resulting displacement by eye and increase it by 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 people, 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. What was the urban and rural population 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 long is each part?
2) The sum of two numbers is 100.05. One number is 97.06 more than the other. Find these numbers.
806. 1) There are 8656.2 tons of coal in three coal warehouses, in the second warehouse there are 247.3 tons of coal more than in the first, and in the third there are 50.8 tons more than in the second. How many tons of coal are in each warehouse?
2) The sum of three numbers is 446.73. The first number is less than the second by 73.17 and more than the third by 32.22. Find these numbers.
807. 1) The boat moved 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 current?
2) The steamer traveled 85.6 km along the river in 4 hours, and 46.2 km against the current in 3 hours. What is the speed of the steamboat in still water and what is the speed of the river flow?
808. 1) Two steamships delivered 3,500 tons of cargo, and one steamship delivered 1.5 times more cargo than the other. How much cargo did each ship carry?
2) The area of two rooms is 37.2 square meters. m. The area of one room is 2 times larger than the other. What is the area of each room?
809. 1) From two settlements, the distance between which is 32.4 km, a motorcyclist and a cyclist simultaneously rode 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 whose sum is 26.35, and the quotient of dividing one number by the other is 7.5.
810. 1) The plant sent three types of cargo with a total weight of 19.2 tons. The weight of the first type of cargo was three times more weight cargo of the second type, and the weight of the cargo of the third type was half as much as the weight of the cargo 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 iron ore. In March it was produced 1.3 times, in February 1.2 times more than in January. How much ore did the crew 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 greater than the length of the Moscow Canal.
2) The length of the Don River is 3.934 times greater than the length of the Moscow River. Find the length of each river if the length of the Don River is 1,467 km greater than the length of the Moscow River.
812. 1) The difference of two numbers is 5.2, and the quotient of one number divided by another is 5. Find these numbers.
2) The difference between 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, it will be 0.5 of the larger one. Find these numbers.
814. 1) The collective farm sowed 2,600 hectares of land with wheat and rye. How many hectares of land were sown with wheat and how many with rye, if 0.8 of the area sown with wheat is equal to 0.5 of the area sown with rye?
2) The collection of the two boys together amounts to 660 stamps. How many stamps does each boy's collection consist of if 0.5 of the first boy's stamps are equal to 0.6 of the second boy's collection?
815. Two students together had 5.4 rubles. After the first spent 0.75 of his money, and the second 0.8 of his money, they had the same amount of money left. How much money did each student have?
816. 1) Two steamships set out towards each other from two ports, the distance between which is 501.9 km. How long will it take them to meet if the speed of the first ship is 25.5 km per hour, and the speed of the second is 22.3 km per hour?
2) Two trains set off towards each other from two points, the distance between which is 382.2 km. How long will it take them to meet if the average speed of the first train was 52.8 km per hour, and the second one was 56.4 km per hour?
817. 1) Two cars left two cities at a distance of 462 km at the same time and met after 3.5 hours. Find the speed of each car if the speed of the first was 12 km per hour greater than the speed of the second car.
2) Of two settlements, the distance between them is 63 km, a motorcyclist and a cyclist simultaneously rode towards each other and met after 1.2 hours. Find the speed of the motorcyclist if the cyclist was traveling at a speed 27.5 km per hour 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 accurate to 1 km per hour.)
819. 1) A cyclist left A for B at an average speed of 12.4 km per hour. After 3 hours 15 minutes. another cyclist rode out from B towards him at an average speed of 10.8 km per hour. After how many hours and at what distance from A will they meet if 0.32 the distance between A and B is 76 km?
2) From cities A and B, the distance between which is 164.7 km, a truck from city A and a car from city B drove towards each other. The speed of the truck is 36 km, and the speed of the car is 1.25 times higher. The passenger car left 1.2 hours later than the truck. After how much time and at what distance from city B will the passenger car meet the truck?
820. Two ships left the same port at the same time and are heading in the same direction. The first steamer travels 37.5 km every 1.5 hours, and the second steamer travels 45 km every 2 hours. How long will it take for the first ship 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 per hour. Later, a passenger plane took off from Moscow to Leningrad and arrived in Leningrad simultaneously with the arrival of the train. average speed the plane's speed 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 traveled along the river for 5 hours, and against the current for 3 hours and covered only 165 km. How many kilometers did he walk downstream and how many against the current, if the speed of the river flow is 2.5 km per hour?
824. The train has left A and must arrive at B at a certain time; having passed half the way and doing 0.8 km in 1 minute, the train was stopped for 0.25 hours; having further increased the speed by 100 m per 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. A postman rode a bicycle from the city to the collective farm at a speed of 12.5 km per hour. 0.4 hours after this, the collective farm executive rode into the city on a horse at a speed equal to 0.6 of the postman’s speed. How long after his departure will the collective farmer meet the postman?
826. A car left city A for city B, 234 km away from A, at a speed of 32 km per hour. 1.75 hours after this, a second car left city B towards the first, the speed of which was 1.225 times greater than the speed of the first. How many hours after leaving will the second car 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 both typists to type this manuscript, working together? (Round the answer to the nearest 0.1 hour.)
2) The pool is filled with two pumps of different power. 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 if these pumps are running simultaneously? (Round answer to the nearest 0.1.)
828. 1) One team can complete an order in 8 days. The other one needs 0.5 time to complete this order. The third team can complete this order in 5 days. In how many days will the entire order be completed? working together three brigades? (Round answer to the nearest 0.1 day.)
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 it take to complete the order if three workers work together? (Round the answer to the nearest 0.1 hour.)
829. Two cars are working to clean the street. The first of them can clean the entire street in 40 minutes, the second requires 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. How long after that did the first machine finish cleaning the street?
830. 1) One of the sides 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 less than the first, and the third side is equal to half the sum of the first two sides. What is the perimeter of the triangle?
831 . 1) The base of the triangle is 4.5 cm, and its height is 1.5 cm less. Find the area of the triangle.
2) The height of the triangle is 4.25 cm, and its base is 3 times larger. Find the area of the triangle. (Round answer to the nearest 0.1.)
832. Find the area of the shaded figures (Fig. 38).
833. Which area is larger: a rectangle with sides 5 cm and 4 cm, a square with sides 4.5 cm, or a triangle whose base and height are each 6 cm?
834. The room is 8.5 m long, 5.6 m wide and 2.75 m high. The area of windows, doors and stoves is 0.1 of the total wall area of the room. How many pieces of wallpaper will be needed to cover this room if a piece of wallpaper is 7 m long and 0.75 m wide? (Round the answer to the nearest 1 piece.)
835. It is necessary to plaster and whitewash the outside of a one-story house, the dimensions of which are: length 12 m, width 8 m and height 4.5 m. The house has 7 windows measuring 0.75 m x 1.2 m each and 2 doors each measuring 0.75 m x 2.5 m. How much will the whole work cost if whitewashing and plastering is 1 sq. m. m costs 24 kopecks? (Round 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 the garden is sown with carrots, and the rest of the garden is planted with potatoes and onions, and an area 7 times larger than with onions is planted 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 of 12 m. 0.65 of the entire area of the vegetable garden is planted with potatoes, and the rest with carrots and beets, and 84 square meters are planted with beets. m more than carrots. How much land separately is there for potatoes, beets and carrots?
839. 1) The cube-shaped box was lined on all sides with plywood. How much plywood was used if the edge of the cube is 8.2 dm? (Round the answer to the nearest 0.1 sq. dm.)
2) How much paint will be needed to paint a cube with an edge of 28 cm, if per 1 sq. cm will 0.4 g of paint be used? (Answer, round to the nearest 0.1 kg.)
840. The length of a cast iron billet in the shape of a rectangular parallelepiped is 24.5 cm, width 4.2 cm and height 3.8 cm. How much do 200 cast iron billets weigh if 1 cubic. dm of cast iron weighs 7.8 kg? (Round answer to the nearest 1 kg.)
841. 1) The length of the box (with lid), shaped like 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 waste boards amount to 0.2 of the surface area that should be covered with boards? (Round 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 covered with boards. The length of the pit is 72.5 m, width 4.6 m and height 2.2 m. How many square meters of boards were used for sheathing if the waste of boards constitutes 0.2 of the surface that should be sheathed with boards? (Round the answer to the nearest 1 sq.m.)
842. 1) The length of the basement, shaped like 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 to 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 answer to the nearest 1 thousand.)
2) The length of the tank, shaped like 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 to 0.6 of its volume. How many tons of kerosene are poured into the tank if the weight of kerosene in a volume is 1 cubic meter? m equals 0.9 t? (Round answer to the nearest 0.1 t.)
843. 1) How long can it take to renew the air in a room that is 8.5 m long, 6 m wide and 3.2 m high, if through a window in 1 second. passes 0.1 cubic meters. m of air?
2) Calculate the time required to refresh the air in your room.
844. The dimensions of the concrete block for building walls are as follows: 2.7 m x 1.4 m x 0.5 m. The void makes up 30% of the volume of the block. How many cubic meters of concrete will be required to make 100 such blocks?
845. Grader-elevator (machine for digging ditches) in 8 hours. The work makes a ditch 30 cm wide, 34 cm deep and 15 km long. How many diggers does such a machine replace if one digger can remove 0.8 cubic meters? m per hour? (Round the result.)
846. The bin in the shape of a rectangular parallelepiped is 12 m long and 8 m wide. In this bin, grain is poured to a height of 1.5 m. In order to find out how much all the 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 production in the RSFSR” (Fig. 39). answer the following questions:
a) By how many million tons did steel production increase in 1959 compared to 1945?
b) How many times was the steel production in 1959 greater than the steel production in 1913? (Accurate to 0.1.)
2) Using the diagram “Cultivated areas in the RSFSR” (Fig. 40), answer the following questions:
a) By how many million hectares did the cultivated area increase in 1959 compared to 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, and paint the floor. Find out the data for drawing up an estimate (class size, cost of whitewashing 1 sq. m, cost of painting the floor 1 sq. m) from the school caretaker.
2) For planting in the garden, the school bought seedlings: 30 apple trees for 0.65 rubles. per piece, 50 cherries for 0.4 rubles. per piece, 40 gooseberry bushes for 0.2 rubles. and 100 raspberry bushes for 0.03 rubles. for a bush. Write an invoice for this purchase using the following example:
I. To divide a number by a decimal fraction, you need to move the commas in the dividend and divisor as many digits to the right as there are after the decimal point in the divisor, and then divide by the natural number.
Primary.
Perform division: 1) 16,38: 0,7; 2) 15,6: 0,15; 3) 3,114: 4,5; 4) 53,84: 0,1.
Solution.
Example 1) 16,38: 0,7.
In the divider 0,7 there is one digit after the decimal point, so let’s move the commas in the dividend and divisor one digit to the right.
Then we will need to divide 163,8 on 7 .
We divide as they divide integers. How to remove the number 8 - the first digit after the decimal point (i.e. the digit in the tenths place), so immediately put a comma in the quotient and continue dividing.
Answer: 23.4.
Example 2) 15,6: 0,15.
We move commas in the dividend ( 15,6 ) and divisor ( 0,15 ) two digits to the right, since in the divisor 0,15 there are two digits after the decimal point.
We remember that you can add as many zeros as you like to the decimal fraction on the right, and this will not change the decimal fraction.
15,6:0,15=1560:15.
We perform division of natural numbers.
Answer: 104.
Example 3) 3,114: 4,5.
Move the commas in the dividend and divisor one digit to the right and divide 31,14 on 45 By
3,114:4,5=31,14:45.
In the quotient we put a comma as soon as we remove the number 1 in the tenth place. Then we continue dividing.
To complete the division we had to assign zero to the number 9 - differences between numbers 414 And 405 . (we know that zeros can be added to the right side of a decimal fraction)
Answer: 0.692.
Example 4) 53,84: 0,1.
Move the commas in the dividend and divisor to 1 number to the right.
We get: 538,4:1=538,4.
Let's analyze the equality: 53,84:0,1=538,4. Pay attention to the comma in the dividend in in this example and a comma in the resulting quotient. We notice that the comma in the dividend has been moved to 1 number to the right, as if we were multiplying 53,84 on 10. (Watch the video “Multiplying a decimal by 10, 100, 1000, etc..") Hence the rule for dividing a decimal fraction by 0,1; 0,01; 0,001 etc.
II. To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)
Examples.
Perform division: 1) 617,35: 0,1; 2) 0,235: 0,01; 3) 2,7845: 0,001; 4) 26,397: 0,0001.
Solution.
Example 1) 617,35: 0,1.
According to the rule IIdivision by 0,1 is equivalent to multiplying by 10 , and move the comma in the dividend 1 digit to the right:
1) 617,35:0,1=6173,5.
Example 2) 0,235: 0,01.
Division by 0,01 is equivalent to multiplying by 100 , which means we move the comma in the dividend on 2 digits to the right:
2) 0,235:0,01=23,5.
Example 3) 2,7845: 0,001.
Because division by 0,001 is equivalent to multiplying by 1000 , then move the comma 3 digits to the right:
3) 2,7845:0,001=2784,5.
Example 4) 26,397: 0,0001.
Divide a decimal by 0,0001 - it's the same as multiplying it by 10000 (move the comma by 4 digits right). We get:
II. To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.
Examples.
Perform division: 1) 41,56: 10; 2) 123,45: 100; 3) 0,47: 100; 4) 8,5: 1000; 5) 631,2: 10000.
Solution.
Moving the decimal point to the left depends on how many zeros after the one are in the divisor. So, when dividing a decimal fraction by 10
we will carry over in the dividend comma to the left one digit; when divided by 100
- move the comma left two digits; when divided by 1000
convert to this decimal fraction comma three digits to the left.
In examples 3) and 4) we had to add zeros before the decimal fraction to make it easier to move the comma. However, you can assign zeros mentally, and you will do this when you learn to apply the rule well II to divide a decimal fraction by 10, 100, 1000, etc.
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We have already said that there are fractions ordinary And decimal. On this moment We've studied fractions a little. We learned that there are regular and improper fractions. We also learned that common fractions can be reduced, added, subtracted, multiplied and divided. And we also learned that there are so-called mixed numbers, which consist of an integer and a fractional part.
We haven't fully explored common fractions yet. There are many subtleties and details that should be talked about, but today we will begin to study decimal fractions, since ordinary and decimal fractions often have to be combined. That is, when solving problems you have to use both types of fractions.
This lesson may seem complicated and confusing. It's quite normal. These kinds of lessons require that they be studied, and not skimmed superficially.
Lesson contentExpressing quantities in fractional form
Sometimes it is convenient to show something in fractional form. For example, one tenth of a decimeter is written like this:
This expression means that one decimeter was divided into ten parts, and from these ten parts one part was taken:
As you can see in the figure, one tenth of a decimeter is one centimeter.
Let's consider next example. Show 6 cm and another 3 mm in centimeters in fractional form.
So, you need to express 6 cm and 3 mm in centimeters, but in fractional form. We already have 6 whole centimeters:
but there are still 3 millimeters left. How to show these 3 millimeters, and in centimeters? Fractions come to the rescue. 3 millimeters is the third part of a centimeter. And the third part of a centimeter is written as cm
A fraction means that one centimeter was divided into ten equal parts, and from these ten parts three parts were taken (three out of ten).
As a result, we have six whole centimeters and three tenths of a centimeter:
In this case, 6 shows the number of whole centimeters, and the fraction shows the number of fractional centimeters. This fraction is read as "six point three centimeters".
Fractions whose denominator contains the numbers 10, 100, 1000 can be written without a denominator. First write the whole part, and then the numerator of the fractional part. The integer part is separated from the numerator of the fractional part by a comma.
For example, let's write it without a denominator. To do this, let's first write down the whole part. The integer part is the number 6. First we write down this number:
The whole part is recorded. Immediately after writing the whole part we put a comma:
And now we write down the numerator of the fractional part. In a mixed number, the numerator of the fractional part is the number 3. We write a three after the decimal point:
Any number that is represented in this form is called decimal.
Therefore, you can show 6 cm and another 3 mm in centimeters using a decimal fraction:
6.3 cm
It will look like this:
In fact, decimals are the same as ordinary fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10000.
Like a mixed number, a decimal fraction has an integer part and a fractional part. For example, in a mixed number whole part this is 6, and the fractional part is .
In the decimal fraction 6.3, the integer part is the number 6, and the fractional part is the numerator of the fraction, that is, the number 3.
It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without an integer part. For example, a fraction is given without a whole part. To write such a fraction as a decimal, first write 0, then put a comma and write the numerator of the fraction. A fraction without a denominator will be written as follows:
Reads like "zero point five".
Converting mixed numbers to decimals
When we write mixed numbers without a denominator, we thereby convert them to decimal fractions. When converting fractions to decimals, there are a few things you need to know, which we'll talk about now.
After the whole part is written down, it is necessary to count the number of zeros in the denominator of the fractional part, since the number of zeros of the fractional part and the number of digits after the decimal point in the decimal fraction must be the same. What does it mean? Consider the following example:
At first
And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you definitely need to count the number of zeros in the denominator of the fractional part.
So, we count the number of zeros in the fractional part of a mixed number. The denominator of the fractional part has one zero. This means that in a decimal fraction there will be one digit after the decimal point and this digit will be the numerator of the fractional part of the mixed number, that is, the number 2
Thus, when converted to a decimal fraction, a mixed number becomes 3.2.
This decimal fraction reads like this:
"Three point two"
“Tenths” because the number 10 is in the fractional part of a mixed number.
Example 2. Convert a mixed number to a decimal.
Write down the whole part and put a comma:
And you could immediately write down the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the decimal point there should be as many digits as there are zeros in the denominator of the fractional part of the mixed number. And we see that the denominator of the fractional part has two zeros. This means that our decimal fraction must have two digits after the decimal point, not one.
In such cases, the numerator of the fractional part needs to be slightly modified: add a zero before the numerator, that is, before the number 3
Now you can convert this mixed number to a decimal fraction. Write down the whole part and put a comma:
And write down the numerator of the fractional part:
The decimal fraction 5.03 is read as follows:
"Five point three"
“Hundreds” because the denominator of the fractional part of a mixed number contains the number 100.
Example 3. Convert a mixed number to a decimal.
From previous examples we learned that for successful translation mixed number into a decimal fraction, the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part must be the same.
Before converting a mixed number to a decimal fraction, its fractional part needs to be slightly modified, namely, to make sure that the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part are the same.
First of all, we look at the number of zeros in the denominator of the fractional part. We see that there are three zeros:
Our task is to organize three digits in the numerator of the fractional part. We already have one digit - this is the number 2. It remains to add two more digits. They will be two zeros. Add them before the number 2. As a result, the number of zeros in the denominator and the number of digits in the numerator will be the same:
Now you can start converting this mixed number to a decimal fraction. First we write down the whole part and put a comma:
and immediately write down the numerator of the fractional part
3,002
We see that the number of digits after the decimal point and the number of zeros in the denominator of the fractional part of the mixed number are the same.
The decimal fraction 3.002 is read as follows:
"Three point two thousandths"
“Thousandths” because the denominator of the fractional part of the mixed number contains the number 1000.
Converting fractions to decimals
Common fractions with denominators of 10, 100, 1000, or 10000 can also be converted to decimals. Since an ordinary fraction does not have an integer part, first write down 0, then put a comma and write down the numerator of the fractional part.
Here also the number of zeros in the denominator and the number of digits in the numerator must be the same. Therefore, you should be careful.
Example 1.
The whole part is missing, so first we write 0 and put a comma:
Now we look at the number of zeros in the denominator. We see that there is one zero. And the numerator has one digit. This means you can safely continue the decimal fraction by writing the number 5 after the decimal point
In the resulting decimal fraction 0.5, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.5 is read as follows:
"Zero point five"
Example 2. Translate common fraction to a decimal fraction.
A whole part is missing. First we write 0 and put a comma:
Now we look at the number of zeros in the denominator. We see that there are two zeros. And the numerator has only one digit. To make the number of digits and the number of zeros the same, add one zero in the numerator before the number 2. Then the fraction will take the form . Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal fraction:
In the resulting decimal fraction 0.02, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.02 is read as follows:
“Zero point two.”
Example 3. Convert a fraction to a decimal.
Write 0 and put a comma:
Now we count the number of zeros in the denominator of the fraction. We see that there are five zeros, and there is only one digit in the numerator. To make the number of zeros in the denominator and the number of digits in the numerator the same, you need to add four zeros in the numerator before the number 5:
Now the number of zeros in the denominator and the number of digits in the numerator are the same. So we can continue with the decimal fraction. Write the numerator of the fraction after the decimal point
In the resulting decimal fraction 0.00005, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.00005 is read as follows:
“Zero point five hundred thousandths.”
Converting improper fractions to decimals
An improper fraction is a fraction in which the numerator is greater than the denominator. There are improper fractions in which the denominator contains the numbers 10, 100, 1000 or 10000. Such fractions can be converted to decimals. But before converting to a decimal fraction, such fractions must be separated into the whole part.
Example 1.
The fraction is an improper fraction. To convert such a fraction to a decimal fraction, you must first select the whole part of it. Let's remember how to isolate the whole part of improper fractions. If you have forgotten, we advise you to return to and study it.
So, let's highlight the whole part in the improper fraction. Recall that a fraction means division - in in this case dividing the number 112 by the number 10
Let's look at this picture and assemble a new mixed number, like a children's construction set. The number 11 will be the integer part, the number 2 will be the numerator of the fractional part, and the number 10 will be the denominator of the fractional part.
We got a mixed number. Let's convert it to a decimal fraction. And we already know how to convert such numbers into decimal fractions. First, write down the whole part and put a comma:
Now we count the number of zeros in the denominator of the fractional part. We see that there is one zero. And the numerator of the fractional part has one digit. This means that the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:
In the resulting decimal fraction 11.2, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
Means improper fraction when converted to a decimal fraction it becomes 11.2
The decimal fraction 11.2 is read as follows:
"Eleven point two."
Example 2. Convert improper fraction to decimal.
It is an improper fraction because the numerator is greater than the denominator. But it can be converted to a decimal fraction, since the denominator contains the number 100.
First of all, let's select the whole part of this fraction. To do this, divide 450 by 100 with a corner:
Let's collect a new mixed number - we get . And we already know how to convert mixed numbers into decimal fractions.
Write down the whole part and put a comma:
Now we count the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part. We see that the number of zeros in the denominator and the number of digits in the numerator are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:
In the resulting decimal fraction 4.50, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
This means that an improper fraction becomes 4.50 when converted to a decimal.
When solving problems, if there are zeros at the end of the decimal fraction, they can be discarded. Let's also drop the zero in our answer. Then we get 4.5
This is one of interesting features decimal fractions. It lies in the fact that the zeros that appear at the end of a fraction do not give this fraction any weight. In other words, the decimals 4.50 and 4.5 are equal. Let's put an equal sign between them:
4,50 = 4,5
The question arises: why does this happen? After all, it looks like 4.50 and 4.5 different fractions. The whole secret lies in the basic property of fractions, which we studied earlier. We will try to prove why the decimal fractions 4.50 and 4.5 are equal, but after studying the next topic, which is called “converting a decimal fraction to a mixed number.”
Converting a decimal to a mixed number
Any decimal fraction can be converted back to a mixed number. To do this, it is enough to be able to read decimal fractions. For example, let's convert 6.3 to a mixed number. 6.3 is six point three. First we write down six integers:
and next to three tenths:
Example 2. Convert decimal 3.002 to mixed number
3.002 is three whole and two thousandths. First we write down three integers
and next to it we write two thousandths:
Example 3. Convert decimal 4.50 to mixed number
4.50 is four point fifty. Write down four integers
and next fifty hundredths:
By the way, let's remember the last example from the previous topic. We said that the decimals 4.50 and 4.5 are equal. We also said that the zero can be discarded. Let's try to prove that the decimals 4.50 and 4.5 are equal. To do this, we convert both decimal fractions into mixed numbers.
When converted to a mixed number, the decimal 4.50 becomes , and the decimal 4.5 becomes
We have two mixed numbers and . Let's convert these mixed numbers to improper fractions:
Now we have two fractions and . It's time to remember the basic property of a fraction, which says that when you multiply (or divide) the numerator and denominator of a fraction by the same number, the value of the fraction does not change.
Let's divide the first fraction by 10
We got , and this is the second fraction. This means that both are equal to each other and equal to the same value:
Try using a calculator to divide first 450 by 100, and then 45 by 10. It will be a funny thing.
Converting a decimal fraction to a fraction
Any decimal fraction can be converted back to a fraction. To do this, again, it is enough to be able to read decimal fractions. For example, let's convert 0.3 to a common fraction. 0.3 is zero point three. First we write down zero integers:
and next to three tenths 0. Zero is traditionally not written down, so the final answer will not be 0, but simply .
Example 2. Convert the decimal fraction 0.02 to a fraction.
0.02 is zero point two. We don’t write down zero, so we immediately write down two hundredths
Example 3. Convert 0.00005 to fraction
0.00005 is zero point five. We don’t write down zero, so we immediately write down five hundred thousandths
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Of the many fractions found in arithmetic, those that have 10, 100, 1000 in the denominator - in general, any power of ten - deserve special attention. These fractions have a special name and notation.
A decimal is any number fraction whose denominator is a power of ten.
Examples of decimal fractions:
Why was it necessary to separate out such fractions at all? Why do they need their own recording form? There are at least three reasons for this:
- Decimals are much easier to compare. Remember: for comparison ordinary fractions they need to be subtracted from each other and, in particular, to bring the fractions to a common denominator. In decimals nothing like this is required;
- Reduce computation. Decimals add and multiply according to their own rules, and with a little practice you'll be able to work with them much faster than with regular fractions;
- Ease of recording. Unlike ordinary fractions, decimals are written on one line without loss of clarity.
Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you ask for change in the store in the amount of 2/3 of a ruble :)
Rules for writing decimal fractions
The main advantage of decimal fractions is convenient and visual notation. Namely:
Decimal notation is a form of writing decimal fractions where the integer part is separated from the fractional part by a regular period or comma. In this case, the separator itself (period or comma) is called a decimal point.
For example, 0.3 (read: “zero pointers, 3 tenths”); 7.25 (7 whole, 25 hundredths); 3.049 (3 whole, 49 thousandths). All examples are taken from the previous definition.
In writing, a comma is usually used as a decimal point. Here and further throughout the site, the comma will also be used.
To write an arbitrary decimal fraction in this form, you need to follow three simple steps:
- Write out the numerator separately;
- Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
- If the decimal point has moved, and after it there are zeros at the end of the entry, they must be crossed out.
It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, to the left of any number you can assign any number of zeros without harm to your health. It's ugly, but sometimes useful.
At first glance, this algorithm may seem quite complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:
Task. For each fraction, indicate its decimal notation:
The numerator of the first fraction is: 73. We shift the decimal point by one place (since the denominator is 10) - we get 7.3.
Numerator of the second fraction: 9. We shift the decimal point by two places (since the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange entry like “.09”.
The numerator of the third fraction is: 10029. We shift the decimal point by three places (since the denominator is 1000) - we get 10.029.
The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10,500. There are extra zeros at the end of the number. Cross them out and we get 10.5.
Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as was done in the last example. However, you should never do this with zeros inside a number (which are surrounded by other numbers). That's why we got 10.029 and 10.5, and not 1.29 and 1.5.
So, we figured out the definition and form of writing decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.
Conversion from fractions to decimals
Consider a simple numerical fraction of the form a /b. You can use the basic property of a fraction and multiply the numerator and denominator by such a number that the bottom turns out to be a power of ten. But before you do, read the following:
There are denominators that cannot be reduced to powers of ten. Learn to recognize such fractions, because they cannot be worked with using the algorithm described below.
That's it. Well, how do you understand whether the denominator is reduced to a power of ten or not?
The answer is simple: factor the denominator into prime factors. If the expansion contains only factors 2 and 5, this number can be reduced to a power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the power of ten.
Task. Check whether the indicated fractions can be represented as decimals:
Let's write out and factor the denominators of these fractions:
20 = 4 · 5 = 2 2 · 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.
12 = 4 · 3 = 2 2 · 3 - there is a “forbidden” factor 3. The fraction cannot be represented as a decimal.
640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction can be represented as a decimal.
48 = 6 · 8 = 2 · 3 · 2 3 = 2 4 · 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.
So, we’ve sorted out the denominator - now let’s look at the entire algorithm for moving to decimal fractions:
- Factor the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
- Count how many twos and fives are present in the expansion (there will be no other numbers there, remember?). Choose an additional factor such that the number of twos and fives is equal.
- Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.
Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest multiplier of all possible.
And one more thing: if the original fraction contains an integer part, be sure to convert this fraction to an improper fraction - and only then apply the described algorithm.
Task. Convert these numerical fractions to decimals:
Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, the fraction can be represented as a decimal. The expansion contains two twos and not a single five, so the additional factor is 5 2 = 25. With it, the number of twos and fives will be equal. We have:
Now let's look at the second fraction. To do this, note that 24 = 3 8 = 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.
The last two fractions have denominators 5 (prime number) and 20 = 4 · 5 = 2 2 · 5 respectively - only twos and fives are present everywhere. Moreover, in the first case, “for complete happiness” a factor of 2 is not enough, and in the second - 5. We get:
Conversion from decimals to common fractions
The reverse conversion - from decimal to regular notation - is much simpler. There are no restrictions or special checks here, so you can always convert a decimal fraction to the classic “two-story” fraction.
The translation algorithm is as follows:
- Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing is not to overdo it and do not cross out the inner zeros surrounded by other numbers;
- Count how many decimal places there are after the decimal point. Take the number 1 and add as many zeros to the right as there are characters you count. This will be the denominator;
- Actually, write down the fraction whose numerator and denominator we just found. If possible, reduce it. If the original fraction contained an integer part, we will now get an improper fraction, which is very convenient for further calculations.
Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.
Cross out the zeros on the left and the commas - we get the following numbers (these will be the numerators): 8; 3107; 225; 72008.
In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.
Decimal fractions are the same as ordinary fractions, but in so-called decimal notation. Decimal notation is used for fractions with denominators 10, 100, 1000, etc. Instead of fractions, 1/10; 1/100; 1/1000; ... write 0.1; 0.01; 0.001;... .
For example, 0.7 ( zero point seven) is a fraction 7/10; 5.43 ( five point forty three) is a mixed fraction 5 43/100 (or, which is the same, an improper fraction 543/100).
It may happen that there are one or more zeros immediately after the decimal point: 1.03 is the fraction 1 3/100; 17.0087 is the fraction 17 87/10000. General rule is this: the denominator of a common fraction must have as many zeros as there are digits after the decimal point in the decimal fraction.
A decimal fraction may end in one or more zeros. It turns out that these zeros are “extra” - they can simply be removed: 1.30 = 1.3; 5.4600 = 5.46; 3,000 = 3. Figure out why this is so?
Decimals naturally arise when dividing by “round” numbers - 10, 100, 1000, ... Be sure to understand the following examples:
27:10 = 27/10 = 2 7/10 = 2,7;
579:100 = 579/100 = 5 79/100 = 5,79;
33791:1000 = 33791/1000 = 33 791/1000 = 33,791;
34,9:10 = 349/10:10 = 349/100 = 3,49;
6,35:100 = 635/100:100 = 635/10000 = 0,0635.
Do you notice a pattern here? Try to formulate it. What happens if you multiply a decimal fraction by 10, 100, 1000?
To convert an ordinary fraction to a decimal, you need to reduce it to some “round” denominator:
2/5 = 4/10 = 0.4; 11/20 = 55/100 = 0.55; 9/2 = 45/10 = 4.5, etc.
Adding decimals is much easier than adding fractions. Addition is performed in the same way as with ordinary numbers - according to the corresponding digits. When adding in a column, the terms must be written so that their commas are on the same vertical. The comma of the sum will also be on the same vertical. The subtraction of decimal fractions is performed in exactly the same way.
If, when adding or subtracting in one of the fractions, the number of digits after the decimal point is less than in the other, then the required number of zeros should be added to the end of this fraction. You can not add these zeros, but simply imagine them in your mind.
When multiplying decimal fractions, they should again be multiplied as ordinary numbers (it is no longer necessary to write a comma under the decimal point). In the resulting result, you need to separate with a comma a number of digits equal to the total number of decimal places in both factors.
When dividing decimal fractions, you can simultaneously move the decimal point in the dividend and divisor to the right by the same number of places: this will not change the quotient:
2,8:1,4 = 2,8/1,4 = 28/14 = 2;
4,2:0,7 = 4,2/0,7 = 42/7 = 6;
6:1,2 = 6,0/1,2 = 60/12 = 5.
Explain why this is so?
- Draw a 10x10 square. Paint over some part of it equal to: a) 0.02; b) 0.7; c) 0.57; d) 0.91; e) 0.135 area of the entire square.
- What is 2.43 square? Draw it in a picture.
- Divide the number 37 by 10; 795; 4; 2.3; 65.27; 0.48 and write the result as a decimal fraction. Divide the same numbers by 100 and 1000.
- Multiply the numbers 4.6 by 10; 6.52; 23.095; 0.01999. Multiply the same numbers by 100 and 1000.
- Represent the decimal as a fraction and reduce it:
a) 0.5; 0.2; 0.4; 0.6; 0.8;
b) 0.25; 0.75; 0.05; 0.35; 0.025;
c) 0.125; 0.375; 0.625; 0.875;
d) 0.44; 0.26; 0.92; 0.78; 0.666; 0.848. - Present as a mixed fraction: 1.5; 3.2; 6.6; 2.25; 10.75; 4.125; 23.005; 7.0125.
- Express a fraction as a decimal:
a) 1/2; 3/2; 7/2; 15/2; 1/5; 3/5; 4/5; 18/5;
b) 1/4; 3/4; 5/4; 19/4; 1/20; 7/20; 49/20; 1/25; 13/25; 77/25; 1/50; 17/50; 137/50;
c) 1/8; 3/8; 5/8; 7/8; 11/8; 125/8; 1/16; 5/16; 9/16; 23/16;
d) 1/500; 3/250; 71/200; 9/125; 27/2500; 1999/2000. - Find the sum: a) 7.3+12.8; b) 65.14+49.76; c) 3.762+12.85; d) 85.4+129.756; e) 1.44+2.56.
- Think of one as the sum of two decimals. Find twenty more ways to present it this way.
- Find the difference: a) 13.4–8.7; b) 74.52–27.04; c) 49.736–43.45; d) 127.24–93.883; e) 67–52.07; e) 35.24–34.9975.
- Find the product: a) 7.6·3.8; b) 4.8·12.5; c) 2.39·7.4; d) 3.74·9.65.
