How to solve equations with fractions. Exponential solution of equations with fractions. Adding and Subtracting Decimals Solving Equations
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Lesson-tale DECIMAL FRACTIONS. SOLVING EQUATIONS
Denisova Svetlana Ivanovna
mathematic teacher
MOU " high school No. 1"
Kimry, Tver region
And he had three sisters
Ivan Tsarevich gave his sisters in marriage to the kings
copper kingdom
silver kingdom
golden kingdom
He lived without his sisters for a whole year, and he became bored. He decided to visit his sisters
and hit the road
They went out to the river, and there a huge stone blocked the road to the bridge
(y - 0.371)+ 5.44= 27.7
(0.127 + m) – 9.8= 3.2
(x + 0.379) – 1.97=1.83
If you solve them correctly, the stone will turn and clear the way
2.4 – 3x = 0.21 (2)
2.5x + 0.8x = 99 (2)
5x – 7.35 = 0.3 (3)
7.2y – 0.3y = 27.6 (3)
She had been at enmity with Koshchei for a long time and agreed to help Ivan Tsarevich, but only if his warriors solved six equations
5.8y – 2.7y = 62 (1)
0.65 + 2x = 5.9 (1)
Saying goodbye to Tsarevich Ivan, Baba Yaga told him about the power of the equation.
If you need a lock to unlock or close tightly, say the roots of the equation out loud. It will be fulfilled in an instant.
Koschey waylaid Ivan the Tsarevich and his soldiers, grabbed them and threw them into a deep dungeon. Locked with six locks.
3.5:x – 2 = 1.5 (1)
(x – 0.5) * 5 =0.4 * 2 – 0.3 * 2 (1)
y: 0.2 + 0.35 = 3.6 (2)
(0.3 + x) * 4 = 0.3 * 3 + 0.7 * 3 (2)
m: 0.12 * 0.2 = 7.2 (3)
(0.7 + x) * 5 = 0.8 * 5 + 0.6 * 5 (3)
Ivan Tsarevich said: magic words", named the roots of all equations. The dungeon doors opened. The soldiers stood in front of the gates of the Koshcheev Palace
y + 0.0015: 0.001 = 1.5
After that, Ivan Tsarevich and the beautiful Elena visited his sisters, came home and began to live, live and make good
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, 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 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 much sunflower oil will be produced from the collected grains 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 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. Water consumption per 5 km run for a steam locomotive passenger train equal to 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. For the strength of the railway embankment, it is recommended to strengthen the slopes by sowing field herbs. 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. Brickworks 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 the average weight of one brick is 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 one and a half times Furthermore, which was 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 a standard time for production new part 3 turners 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. By multiplying the average length of your step by the resulting number of steps, you find the distance from A to B.
3rd appointment. Measuring by eye. Each student draws left hand with a 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 rural population there were 9.2 million more people than the city 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. First number less than two 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 the weight of the second type of cargo, and the weight of the third type of cargo was half as much as the weight of the first and second types of cargo combined. 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 dividing one number 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 2600 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. Speed truck 36 km, and a passenger car is 1.25 times longer. The passenger car left 1.2 hours later than the truck. After how long and at what distance from city B passenger car will meet the cargo?
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 an order with a joint work of three workers? (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 less 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 total area walls 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. The outside needs to be plastered and whitewashed. cottage, 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 it be the cost of all the work if whitewashing and plastering is 1 sq. 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 is 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. Length of a cast iron billet shaped rectangular parallelepiped, is equal to 24.5 cm, width 4.2 cm and height 3.8 cm. How much do 200 cast iron blanks 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 much square meters of boards used to make a box, if the waste of boards constitutes 0.2 of the surface that should be covered with boards? (Round the answer to the nearest 0.1 sq. m.)
2) Bottom and side walls pits in the shape of a rectangular parallelepiped must be lined 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. Dimensions 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. behind the bush. Write an invoice for this purchase using the following example:
Solving equations with fractions Let's look at examples. The examples are simple and illustrative. With their help, you will be able to understand in the most understandable way.
For example, you need to solve the simple equation x/b + c = d.
An equation of this type is called linear, because The denominator contains only numbers.
The solution is performed by multiplying both sides of the equation by b, then the equation takes the form x = b*(d – c), i.e. the denominator of the fraction on the left side cancels.
For example, how to solve a fractional equation:
x/5+4=9
We multiply both sides by 5. We get:
x+20=45
x=45-20=25
Another example when the unknown is in the denominator:
Equations of this type are called fractional-rational or simply fractional.
We would solve a fractional equation by getting rid of fractions, after which this equation, most often, turns into a linear or quadratic equation, which can be solved in the usual way. You just need to consider the following points:
- the value of a variable that turns the denominator to 0 cannot be a root;
- You cannot divide or multiply an equation by the expression =0.
This is where the concept of area comes into force acceptable values(ODZ) are such values of the roots of the equation at which the equation makes sense.
Thus, when solving the equation, it is necessary to find the roots, and then check them for compliance with the ODZ. Those roots that do not correspond to our ODZ are excluded from the answer.
For example, you need to solve a fractional equation:
Based on the above rule, x cannot be = 0, i.e. ODZ in in this case: x – any value other than zero.
We get rid of the denominator by multiplying all terms of the equation by x
And we solve the usual equation
5x – 2x = 1
3x = 1
x = 1/3
Answer: x = 1/3
Let's solve a more complicated equation:
ODZ is also present here: x -2.
When solving this equation, we will not move everything to one side and bring the fractions to a common denominator. We will immediately multiply both sides of the equation by an expression that will cancel out all the denominators at once.
To reduce the denominators you need left side multiply by x+2, and the right hand by 2. This means that both sides of the equation must be multiplied by 2(x+2):
This is the most common multiplication of fractions, which we have already discussed above.
Let's write the same equation, but slightly differently
The left side is reduced by (x+2), and the right by 2. After the reduction, we obtain the usual linear equation:
x = 4 – 2 = 2, which corresponds to our ODZ
Answer: x = 2.
Solving equations with fractions not as difficult as it might seem. In this article we have shown this with examples. If you have any difficulties with how to solve equations with fractions, then unsubscribe in the comments.
Equations with fractions themselves are not difficult and are very interesting. Let's consider the types fractional equations and ways to solve them.
How to solve equations with fractions - x in the numerator
If a fractional equation is given, where the unknown is in the numerator, the solution does not require additional conditions and is solved without unnecessary hassle. General form such an equation – x/a + b = c, where x is the unknown, a, b and c – regular numbers.
Find x: x/5 + 10 = 70.
In order to solve the equation, you need to get rid of fractions. Multiply each term in the equation by 5: 5x/5 + 5x10 = 70x5. 5x and 5 are cancelled, 10 and 70 are multiplied by 5 and we get: x + 50 = 350 => x = 350 – 50 = 300.
Find x: x/5 + x/10 = 90.
This example is a slightly more complicated version of the first one. There are two possible solutions here.
- Option 1: We get rid of fractions by multiplying all terms of the equation by a larger denominator, that is, by 10: 10x/5 + 10x/10 = 90×10 => 2x + x = 900 => 3x = 900 => x=300.
- Option 2: Add the left side of the equation. x/5 + x/10 = 90. Common denominator– 10. Divide 10 by 5, multiply by x, we get 2x. Divide 10 by 10, multiply by x, we get x: 2x+x/10 = 90. Hence 2x+x = 90×10 = 900 => 3x = 900 => x = 300.
Often there are fractional equations in which the x's are located according to different sides equal sign. In such situations, it is necessary to move all the fractions with X's to one side, and the numbers to the other.
- Find x: 3x/5 = 130 – 2x/5.
- Move 2x/5 to the right with opposite sign: 3x/5 + 2x/5 = 130 => 5x/5 = 130.
- We reduce 5x/5 and get: x = 130.
How to solve an equation with fractions - x in the denominator
This type of fractional equations requires writing additional conditions. The indication of these conditions is a mandatory and integral part of the right decision. By not adding them, you run the risk, since the answer (even if it is correct) may simply not be counted.
The general form of fractional equations, where x is in the denominator, is: a/x + b = c, where x is the unknown, a, b, c are ordinary numbers. Please note that x may not be any number. For example, x cannot equal zero, since it cannot be divided by 0. This is precisely the additional condition that we must specify. This is called the range of permissible values, abbreviated as VA.
Find x: 15/x + 18 = 21.
We immediately write the ODZ for x: x ≠ 0. Now that the ODZ is indicated, we solve the equation using standard scheme, getting rid of fractions. Multiply all terms of the equation by x. 15x/x+18x = 21x => 15+18x = 21x => 15 = 3x => x = 15/3 = 5.
Often there are equations where the denominator contains not only x, but also some other operation with it, for example, addition or subtraction.
Find x: 15/(x-3) + 18 = 21.
We already know that the denominator cannot be equal to zero, which means x-3 ≠ 0. We move -3 to the right side, changing the “-” sign to “+” and we get that x ≠ 3. The ODZ is indicated.
We solve the equation, multiply everything by x-3: 15 + 18×(x – 3) = 21×(x – 3) => 15 + 18x – 54 = 21x – 63.
Move the X's to the right, numbers to the left: 24 = 3x => x = 8.
