The common denominator of several fractions is the LCM (least common multiple) of the natural numbers that are the denominators of the given fractions.

To the numerators of given fractions, you need to put additional factors equal to the ratio of the LCM and the corresponding denominator.

The numerators of given fractions are multiplied by their additional factors, the numerators of fractions with a common denominator are obtained. Action signs ("+" or "-") in the notation of fractions reduced to a common denominator are stored before each fraction. For fractions with a common denominator, the action signs are preserved in front of each reduced numerator.

Only now you can add or subtract the numerators and sign the common denominator under the result.

Attention! If in the resulting fraction the numerator and denominator have common factors, then the fraction must be reduced. It is desirable to convert an improper fraction to a mixed fraction. Leaving the result of an addition or subtraction without reducing the fraction where possible is an unfinished solution to the example!

Adding and subtracting fractions with different denominators. Rule. To add or subtract fractions with different denominators, you must first bring them to the lowest common denominator, and then perform addition or subtraction operations as with fractions with the same denominators.

Procedure for adding and subtracting fractions with different denominators

1. find the LCM of all denominators;
2. put down additional multipliers for each fraction;
3. multiply each numerator by an additional factor;
4. take the resulting products as numerators, signing a common denominator under each fraction;
5. add or subtract the numerators of fractions by signing a common denominator under the sum or difference.

Adding and subtracting fractions in the presence of letters in the numerator is also performed.

496. Find X, If:

497. 1) If you add 10 1/2 to 3/10 of an unknown number, you get 13 1/2. Find an unknown number.

2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find an unknown number.

498 *. If you subtract 10 from 3 / 4 of an unknown number and multiply the resulting difference by 5, you get 100. Find the number.

499 *. If an unknown number is increased by 2/3 of it, you get 60. What is this number?

500 *. If we add the same amount to an unknown number, and even 20 1/3, then we get 105 2/5. Find an unknown number.

501. 1) The yield of potatoes with a square-nest planting is on average 150 centners per 1 ha, and with a normal planting 3/5 of this amount. How many more potatoes can be harvested from an area of ​​15 hectares if potatoes are planted in a square-nest way?

2) An experienced worker made 18 parts in 1 hour, and an inexperienced worker 2/3 of this amount. How many more parts can an experienced worker produce in a 7-hour working day?

502. 1) Pioneers assembled within three days 56 kg of different seeds. On the first day, 3/14 of the total amount was collected, on the second, one and a half times more, and on the third day, the rest of the grain. How many kilograms of seeds did the pioneers collect on the third day?

2) When grinding wheat, it turned out: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest is bran. How much flour, semolina and bran separately did you get when grinding 3 tons of wheat?

503. 1) Three garages fit 460 cars. The number of cars that fit in the first garage is 3/4 of the number of cars that fit in the second, and in the third garage there are 1 1/2 times as many cars as in the first. How many cars fit in each garage?

2) The plant, which has three workshops, employs 6,000 workers. The number of workers in the second workshop is 1 1/2 times less than in the first, and the number of workers in the third workshop is 5/6 of the number of workers in the second workshop. How many workers are in each shop?

504. 1) First, 2/5 was poured from the tank with kerosene, then 1/3 of the total kerosene, and after that 8 tons of kerosene remained in the tank. How much kerosene was in the tank originally?

2) The cyclists raced for three days. On the first day they covered 4/15 of the entire journey, on the second day they covered 2/5, and on the third day the remaining 100 km. How far did the cyclists travel in three days?

505. 1) The icebreaker made its way through the ice field for three days. On the first day he covered 1/2 of the total distance, on the second day 3/5 of the remaining distance, and on the third day the remaining 24 km. Find the distance traveled by the icebreaker in three days.

2) Three detachments of schoolchildren planted trees for landscaping the village. The first detachment planted 7/20 of all the trees, the second 5/8 of the remaining trees, and the third the remaining 195 trees. How many trees did the three teams plant in total?

506. 1) A combine harvester harvested wheat from one plot in three days. On the first day he harvested from 5/18 of the total area of ​​the plot, on the second day from 7/13 of the remaining area, and on the third day from the remaining area of ​​30 1/2 hectares. On average, 20 centners of wheat were harvested from each hectare. How much wheat was harvested in the entire plot?

2) On the first day, the participants of the rally covered 3/11 of the entire path, on the second day 7/20 of the remaining path, on the third day 5/13 of the new remainder, and on the fourth day, the remaining 320 km. How long is the rally route?

507. 1) On the first day, the car covered 3/8 of the entire distance, on the second day 15/17 of what it passed on the first, and on the third day the remaining 200 km. How much gasoline was consumed if the car consumes 1 3/5 kg of gasoline for 10 km of travel?

2) The city consists of four districts. And in the first district live 4/13 of all the inhabitants of the city, in the second 5/6 of the inhabitants of the first district, in the third 4/11 of the inhabitants of the first; two districts combined, and the fourth district is home to 18,000 people. How much bread does the entire population of the city need for 3 days, if on average one person consumes 500 g per day?

508. 1) The tourist walked on the first day 10/31 of the entire path, on the second 9/10 of what he walked on the first day, and on the third the rest of the path, and on the third day he walked 12 km more than on the second day. How many kilometers did the tourist walk on each of the three days?

2) The car traveled all the way from city A to city B in three days. On the first day, the car covered 7/20 of the entire distance, on the second day, 8/13 of the remaining distance, and on the third day, the car covered 72 km less than on the first day. What is the distance between cities A and B?

509. 1) The executive committee allotted land to the workers of three factories for garden plots. The first plant was assigned 9/25 of the total number of plots, the second plant 5/9 of the number of plots allocated for the first, and the third - the rest of the plots. How many plots were allotted to the workers of three factories if the first plant was given 50 fewer plots than the third?

2) The plane delivered a shift of winterers to the polar station from Moscow in three days. On the first day he flew 2/5 of the entire path, on the second - 5/6 of the path he traveled on the first day, and on the third day he flew 500 km less than on the second day. How far did the plane fly in three days?

510. 1) The plant had three workshops. The number of workers in the first workshop is 2/5 of all factory workers; in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 100 more workers than in the second. How many workers are in the factory?

2) The collective farm includes residents of three neighboring villages. The number of families in the first village is 3/10 of all the families of the collective farm; in the second village the number of families is 1 1/2 times greater than in the first, and in the third village the number of families is 420 fewer than in the second. How many families are on the collective farm?

511. 1) The Artel spent in the first week 1/3 of its stock of raw materials, and in the second 1/3 of the remainder. How much raw material is left in the artel if in the first week the consumption of raw materials was 3/5 tons more than in the second week?

2) Of the imported coal for heating the house in the first month, 1/6 of it was spent, and in the second month - 3/8 of the remainder. How much coal is left for heating the house if 1 3/4 more was used in the second month than in the first month?

512. 3/5 of the entire land of the collective farm is allocated for sowing grain, 13/36 of the rest is occupied by vegetable gardens and meadows, the rest of the land is forested, and the sown area of ​​the collective farm is 217 hectares more than the forest area, 1/3 of the land allotted for sowing grain is sown with rye, and the rest is wheat. How many hectares of land did the collective farm sow with wheat and how many with rye?

513. 1) The tram route is 14 3/8 km long. During this route, the tram makes 18 stops, spending on average up to 1 1/6 minutes per stop. The average tram speed along the entire route is 12 1/2 km per hour. How long does it take for a tram to make one trip?

2) Bus route 16 km. During this route, the bus makes 36 stops of 3/4 min. each on average. The average bus speed is 30 km per hour. How long does it take for a bus to make one route?

514*. 1) It is now 6 o'clock. evenings. What part is the remaining part of the day from the past and what part of the day is left?

2) A steamboat travels downstream between two cities in 3 days. and back the same distance in 4 days. How many days will the rafts float from one city to another?

515. 1) How many boards will be used to lay the floor in a room whose length is 6 2/3 m, width h 5 1/4 m, if the length of each board is 6 2/3 m, and its width is 3/80 of the length?

2) A rectangular platform has a length of 45 1/2 m, and its width is 5/13 of the length. This area is bordered by a path 4/5 m wide. Find the area of ​​the path.

516. Find the mean arithmetic numbers:

517. 1) The arithmetic mean of two numbers 6 1 / 6 . One of the numbers 3 3 / 4 . Find another number.

2) The arithmetic mean of two numbers is 14 1 / 4 . One of these numbers is 15 5 / 6 . Find another number.

518. 1) The freight train was on the road for three hours. In the first hour he walked 36 1/2 km, in the second 40 km, and in the third 39 3/4 km. Find the average speed of the train.

2) The car traveled 81 1/2 km in the first two hours, and 95 km in the next 2 1/2 hours. How many kilometers did he walk on average per hour?

519. 1) The tractor driver completed the task of plowing the land in three days. On the first day he plowed 12 1/2 ha, on the second day 15 3/4 ha, and on the third day 14 1/2 ha. How many hectares of land did a tractor driver plow on average per day?

2) A detachment of schoolchildren, making a three-day tourist trip, was on the way on the first day 6 1 / 3 hours, on the second 7 hours. and on the third day, 4 2/3 hours. How many hours on average were students on the road every day?

520. 1) Three families live in the house. The first family for lighting the apartment has 3 light bulbs, the second 4 and the third 5 bulbs. How much should each family pay for electricity if all the lamps were the same and the total electricity bill (for the whole house) was 7 1/5 rubles?

2) The polisher rubbed the floors in the apartment where three families lived. The first family had a living area of ​​36 1/2 sq. m, the second in 24 1/2 sq. m, and the third - in 43 sq. m. For all the work was paid 2 rubles. 08 kop. How much did each family pay?

521. 1) In the garden plot, potatoes were harvested from 50 bushes, 1 1/10 kg from one bush, from 70 bushes, 4/5 kg from one bush, from 80 bushes, 9/10 kg from one bush. How many kilograms of potatoes are harvested on average from each bush?

2) A field-growing team on an area of ​​300 ha received a harvest of 20 1/2 centners of winter wheat per 1 ha, from 80 hectares 24 centners per 1 ha, and from 20 hectares - 28 1/2 centners per 1 ha. What is the average yield in a brigade from 1 hectare?

522. 1) The sum of two numbers is 7 1 / 2 . One number is greater than another by 4 4 / 5 . Find these numbers.

2) If we add the numbers expressing the width of the Tatar and Kerch Straits together, we get 11 7 / 10 km. The Tatar Strait is 3 1/10 km wider than the Kerch Strait. What is the width of each strait?

523. 1) The sum of three numbers is 35 2 / 3 . The first number is 5 1/3 greater than the second and 3 5/6 greater than the third. Find these numbers.

2) Islands New Earth, Sakhalin and Severnaya Zemlya together occupy an area of ​​196 7/10 thousand square meters. km. The area of ​​Novaya Zemlya is 44 1/10 thousand square meters. km more than the area of ​​Severnaya Zemlya and 5 1/5 thousand square meters. km larger than the area of ​​Sakhalin. What is the area of ​​each of the listed islands?

524. 1) The apartment consists of three rooms. The area of ​​the first room is 24 3/8 sq. m and is 13/36 of the entire area of ​​the apartment. The area of ​​the second room is 8 1/8 sq. m more than the area of ​​the third. What is the area of ​​the second room?

2) The cyclist during the three-day competition on the first day traveled 3 1/4 hours, which was 13/43 of the total travel time. On the second day he rode 1 1/2 hours more than on the third day. How many hours did the cyclist travel on the second day of the competition?

525. Three pieces of iron weigh together 17 1/4 kg. If the weight of the first piece is reduced by 1 1/2 kg, the weight of the second by 2 1/4 kg, then all three pieces will have the same weight. How much did each piece of iron weigh?

526. 1) The sum of two numbers is 15 1 / 5 . If the first number is reduced by 3 1/10 and the second is increased by 3 1/10, then these numbers will be equal. What is each number equal to?

2) There were 38 1/4 kg of cereal in two boxes. If 4 3/4 kg of cereals are poured from one box into another, then in both boxes there will be equal amounts of cereals. How many cereals are in each box?

527 . 1) The sum of two numbers is 17 17 / 30 . If you subtract 5 1/2 from the first number and add to the second, then the first will still be more than the second by 2 17/30. Find both numbers.

2) Two boxes contain 24 1/4 kg of apples. If 3 1/2 kg are transferred from the first box to the second, then in the first there will still be 3/5 kg more apples than in the second. How many kilograms of apples are in each box?

528 *. 1) The sum of two numbers is 8 11/14, and their difference is 2 3/7. Find these numbers.

2) The boat was moving along the river at a speed of 15 1/2 km per hour, and against the current 8 1/4 km per hour. What is the speed of the river?

529. 1) There are 110 cars in two garages, and in one of them there are 1 1/5 times more than in the other. How many cars are in each garage?

2) The living area of ​​an apartment consisting of two rooms is 47 1/2 sq. m. The area of ​​one room is 8/11 of the area of ​​the other. Find the area of ​​each room.

530. 1) An alloy consisting of copper and silver weighs 330 g. The weight of copper in this alloy is 5/28 of the weight of silver. How much silver and how much copper is in the alloy?

2) The sum of two numbers is 6 3 / 4 , and the quotient is 3 1 / 2 . Find these numbers.

531. The sum of three numbers is 22 1 / 2 . The second number is 3 1/2 times and the third is 2 1/4 times the first. Find these numbers.

532. 1) The difference of two numbers is 7; the quotient of dividing the larger number by the smaller is 5 2 / 3 . Find these numbers.

2) The difference of two numbers is 29 3/8, and their multiple ratio is 8 5/6. Find these numbers.

533. In a class, the number of absent students is 3/13 of the number of those present. How many students are in the class according to the list, if there are 20 more people present than absent?

534. 1) The difference of two numbers is 3 1 / 5 . One number is 5/7 of another. Find these numbers.

2) Father older than son for 24 years. The number of the son's years is 5/13 of the father's years. How old is the father and how old is the son?

535. The denominator of a fraction is 11 more than its numerator. What is a fraction equal to if its denominator is 3 3/4 times the numerator?

No. 536 - 537 orally.

536. 1) The first number is 1/2 of the second. How many times greater is the second number than the first?

2) The first number is 3/2 of the second. What part of the first number is the second number?

537. 1) 1/2 of the first number is equal to 1/3 of the second number. What part of the first number is the second number?

2) 2/3 of the first number is equal to 3/4 of the second number. What part of the first number is the second number? What part of the second number is the first?

538. 1) The sum of two numbers is 16. Find these numbers if 1/3 of the second number is equal to 1/5 of the first.

2) The sum of two numbers is 38. Find these numbers if 2/3 of the first number is equal to 3/5 of the second.

539 *. 1) Two boys picked 100 mushrooms together. 3/8 of the number of mushrooms picked by the first boy is numerically equal to 1/4 of the number of mushrooms picked by the second boy. How many mushrooms did each boy collect?

2) The institution employs 27 people. How many men and how many women work if 2/5 of all men are equal to 3/5 of all women?

540 *. Three boys bought a volleyball. Determine the contribution of each boy, knowing that 1/2 of the contribution of the first boy is equal to 1/3 of the contribution of the second, or 1/4 of the contribution of the third, and that the contribution of the third boy is 64 kopecks more than the contribution of the first.

541 *. 1) One number is 6 greater than another. Find these numbers if 2/5 of one number is equal to 2/3 of another.

2) The difference of two numbers is 35. Find these numbers if 1/3 of the first number is equal to 3/4 of the second number.

542. 1) The first brigade can complete some work in 36 days, and the second in 45 days. How many days will it take both teams working together to complete this task?

2) A passenger train travels the distance between two cities in 10 hours, and a freight train travels this distance in 15 hours. Both trains left these cities at the same time towards each other. In how many hours will they meet?

543. 1) A fast train travels the distance between two cities in 6 1/4 hours, and a passenger train in 7 1/2 hours. In how many hours will these trains meet if they leave both cities at the same time towards each other? (Round answer to the nearest 1 hour.)

2) Two motorcyclists left two cities at the same time towards each other. One motorcyclist can travel the entire distance between these cities in 6 hours, and another in 5 hours. In how many hours after the departure will the motorcyclists meet? (Round answer to the nearest 1 hour.)

544. 1) Three cars of different carrying capacity can carry some cargo, working separately: the first in 10 hours, the second in 12 hours. and the third in 15 hours In how many hours can they move the same cargo by working together?

2) Two trains leave two stations at the same time towards each other: the first train covers the distance between these stations in 12 1/2 hours, and the second in 18 3/4 hours. How many hours after leaving will the trains meet?

545. 1) There are two taps connected to the bath. Through one of them, the bath can be filled in 12 minutes, through the other 1 1/2 times faster. How many minutes will it take to fill 5/6 of the entire bath if both taps are opened at once?

2) Two typists must retype the manuscript. The first woman can do this job in 3 1/3 days, and the second one 1 1/2 times faster. In how many days will both typists complete the work if they work at the same time?

546. 1) The pool is filled with the first pipe in 5 hours, and through the second pipe it can be emptied in 6 hours In how many hours will the entire pool be filled if both pipes are opened at the same time?

Instruction. In an hour, the pool is filled to (1 / 5 - 1 / 6 of its capacity.)

2) Two tractors plowed the field in 6 hours. The first tractor, working alone, could plow this field in 15 hours How many hours would it take the second tractor to plow this field, working alone?

547 *. Two trains leave two stations at the same time towards each other and meet after 18 hours. after its release. How long does it take the second train to travel the distance between stations if the first train travels this distance in 1 day and 21 hours?

548 *. The pool is filled with two pipes. First, the first pipe was opened, and then after 3 3/4 hours, when half the pool was full, the second pipe was opened. After 2 1/2 hours of working together, the pool filled up. Determine the capacity of the pool if 200 buckets of water per hour were poured through the second pipe.

549. 1) A courier train left Leningrad for Moscow, which travels 1 km in 3/4 minutes. 1/2 hour after the departure of this train, a fast train left Moscow for Leningrad, the speed of which was equal to 3/4 of the speed of the courier. How far will the trains be from each other 2 1/2 hours after the departure of the courier train, if the distance between Moscow and Leningrad is 650 km?

2) From the collective farm to the city 24 km. A truck has left the collective farm and travels 1 km in 2 1/2 minutes. After 15 min. after the departure of this car from the city, a cyclist left the collective farm, at a speed half that of a truck. How long will it take for the cyclist to meet the truck after leaving?

550. 1) A pedestrian came out of one village. 4 1/2 hours after the pedestrian left, a cyclist left in the same direction, whose speed is 2 1/2 times the speed of the pedestrian. In how many hours after the pedestrian leaves, the cyclist will overtake him?

2) A fast train travels 187 1/2 km in 3 hours, and a freight train 288 km in 6 hours. 7 1/4 hours after the departure of the freight train, an ambulance leaves in the same direction. How long will it take for the fast train to overtake the freight train?

551. 1) From two collective farms, through which the road to the district center passes, two collective farmers left at the same time to the district on horseback. The first of them traveled 8 3/4 km per hour, and the second 1 1/7 times the first. The second collective farmer overtook the first in 3 4/5 hours. Determine the distance between collective farms.

2) 26 1/3 hours after the departure of the Moscow-Vladivostok train, the average speed of which is 60 km per hour, the TU-104 aircraft took off in the same direction, at a speed 14 1/6 times the speed of the train. How many hours after the flight will the plane overtake the train?

552. 1) The distance between cities along the river is 264 km. This distance the steamer traveled downstream in 18 hours, spending 1/12 of this time on stops. The speed of the river is 1 1/2 km per hour. How long would it take a steamer to travel 87 km without stopping in still water?

2) The motorboat traveled 207 km downstream in 13 1/2 hours, spending 1/9 of that time on stops. The speed of the river is 1 3/4 km per hour. How many miles can this boat travel in still water in 2 1/2 hours?

553. The boat on the reservoir covered a distance of 52 km without stopping in 3 hours and 15 minutes. Further, going along the river against the current, the speed of which is 1 3 / 4 km per hour, this boat traveled 28 1 / 2 km in 2 1 / 4 hours, making 3 equal stops in the process. How many minutes did the boat stop at each stop?

554. From Leningrad to Kronstadt at 12 noon. the next day a steamboat set out and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another steamer that left Kronstadt for Leningrad at 12:18. and walking at a speed 1 1/4 times greater than the first. At what time did the two ships meet?

555. The train had to cover a distance of 630 km in 14 hours. Having covered 2/3 of this distance, he was delayed for 1 hour and 10 minutes. At what speed must he continue his journey in order to arrive at his destination without delay?

556. At 4 o'clock 20 min. In the morning a freight train left Kyiv for Odessa at an average speed of 31 1/5 km per hour. After some time, a mail train left Odessa to meet it, the speed of which is 1 17/39 times the speed of the freight train, and met with the freight train 6 1/2 hours after its departure. At what time did the postal train leave Odessa if the distance between Kiev and Odessa is 663 km?

557*. The clock shows noon. How long does it take for the hour and minute hands to coincide?

558. 1) The factory has three workshops. The number of workers in the first workshop is 9/20 of all the workers of the plant, in the second workshop there are 1 1/2 times fewer workers than in the first, and in the third workshop there are 300 workers less than in the second. How many workers are in the factory?

2) There are three secondary schools in the city. The number of students in the first school is 3/10 of all students in these three schools; in the second school there are 1 1/2 times more students than in the first, and in the third school there are 420 students less than in the second. How many students are in the three schools?

559. 1) Two combine operators worked at the same site. After one combiner harvested 9/16 of the entire area, and the second 3/8 of the same area, it turned out that the first combiner harvested 97 1/2 hectares more than the second. On average, 32 1/2 centners of grain were threshed from each hectare. How many quintals of grain did each combine thresh?

2) Two brothers bought a camera. One had 5/8, and the second had 4/7 of the cost of the camera, and the first had 2 rubles. 25 kop. more than the second. Each paid half the cost of the apparatus. How much money does each have?

560. 1) From city A to city B, the distance between them is 215 km, a car left at a speed of 50 km per hour. At the same time, a truck left city B for city A. How many kilometers did the car travel before meeting the truck if the speed of the truck per hour was 18/25 of the speed of the car?

2) Between cities A and B 210 km. A car left town A for town B. At the same time, a truck left city B for city A. How many kilometers did the truck travel before meeting with the car if the car was moving at a speed of 48 km per hour, and the speed of the truck per hour was 3/4 of the speed of the car?

561. The collective farm harvested wheat and rye. Wheat was sown 20 hectares more than rye. The total harvest of rye amounted to 5/6 of the total harvest of wheat with a yield of 20 centners per 1 ha for both wheat and rye. The collective farm sold 7/11 of the entire harvest of wheat and rye to the state, and left the rest of the grain to meet its needs. How many trips did the two-ton trucks need to make to take out the grain sold to the state?

562. Rye and wheat flour was brought to the bakery. Weight wheat flour amounted to 3/5 of the weight of rye flour, and rye flour was brought 4 tons more than wheat. How much wheat and how much rye bread will be baked by the bakery from this flour, if the baked goods are 2/5 of the whole flour?

563. Within three days, a team of workers completed 3/4 of the entire work to repair the highway between the two collective farms. On the first day, 2 2 / 5 km of this highway was repaired, on the second day 1 1 / 2 times more than on the first, and on the third day 5 / 8 of what was repaired in the first two days together. Find the length of the highway between collective farms.

564. Fill free places in the table, where S is the area of ​​the rectangle, A- the base of the rectangle, a h-height (width) of the rectangle.

565. 1) The length of a rectangular plot of land is 120 m, and the width of the plot is 2/5 of its length. Find the perimeter and area of ​​the plot.

2) The width of the rectangular section is 250 m, and its length is 1 1/2 times the width. Find the perimeter and area of ​​the plot.

566. 1) The perimeter of a rectangle is 6 1/2 dm, its base is 1/4 dm more than the height. Find the area of ​​this rectangle.

2) The perimeter of a rectangle is 18 cm, its height is 2 1/2 cm less than the base. Find the area of ​​the rectangle.

567. Calculate the areas of the figures shown in Figure 30, dividing them into rectangles and finding the dimensions of the rectangle by measuring.

568. 1) How many sheets of dry plaster will be required to upholster the ceiling of a room whose length is 4 1/2 m and the width is 4 m, if the dimensions of the plaster sheet are 2 m x l 1/2 m?

2) How many boards 4 1/2 l long and 1/4 m wide will be required to lay a floor that is 4 1/2 m long and 3 1/2 m wide?

569. 1) A rectangular plot 560 m long and 3/4 of its length wide was sown with beans. How many seeds were required to sow the plot if 1 centner was sown per 1 hectare?

2) A wheat crop was harvested from a rectangular field at 25 centners per 1 ha. How much wheat was harvested from the whole field if the field is 800 m long and 3/8 of its length wide?

570 . 1) A rectangular plot of land, having a length of 78 3/4 m and a width of 56 4/5 m, is built up so that 4/5 of its area is occupied by buildings. Determine the area of ​​land under the buildings.

2) On a rectangular plot of land, the length of which is 9/20 km, and the width is 4/9 of its length, the collective farm proposes to plant a garden. How many trees will be planted in this garden if, on average, an area of ​​36 square meters is required for each tree?

571. 1) For normal daylight illumination of the room, it is necessary that the area of ​​\u200b\u200ball windows be at least 1/5 of the floor area. Determine if there is enough light in a room that is 5 1/2 m long and 4 m wide. Does the room have one window measuring 1 1/2 m x 2 m?

2) Using the condition of the previous problem, find out if there is enough light in your classroom.

572. 1) The barn measures 5 1/2 m x 4 1/2 m x 2 1/2 m. m of hay weighs 82 kg?

2) The woodpile has the shape of a rectangular parallelepiped, the dimensions of which are 2 1/2 m x 3 1/2 m x 1 1/2 m. What is the weight of the woodpile if 1 cu. m of firewood weighs 600 kg?

573. 1) A rectangular aquarium is filled with water up to 3/5 of the height. The length of the aquarium is 1 1/2 m, the width is 4/5 m, the height is 3/4 m. How many liters of water are poured into the aquarium?

2) The pool, having the shape of a rectangular parallelepiped, has a length of 6 1/2 m, a width of 4 m and a height of 2 m. The pool is filled with water up to 3/4 of its height. Calculate the amount of water poured into the pool.

574. A fence is to be built around a rectangular piece of land 75 m long and 45 m wide. How many cubic meters of boards should go to his device if the thickness of the board is 2 1/2 cm, and the height of the fence should be 2 1/4 m?

575. 1) What is the angle between the minute hand and the hour hand at 13:00? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23:30?

2) By how many degrees will the hour hand turn in 2 hours? 5 o'clock? 8 o'clock? 30 min.?

3) How many degrees does an arc equal to half a circle contain? 1/4 circle? 1/24 circle? 5 / 24 circles?

576. 1) Draw with a protractor: a) a right angle; b) an angle of 30°; c) an angle of 60°; d) an angle of 150°; e) an angle of 55°.

2) Measure the angles of the figure with a protractor and find the sum of all the angles of each figure (Fig. 31).

577. Run actions:

578. 1) A semicircle is divided into two arcs, one of which is 100° larger than the other. Find the magnitude of each arc.

2) A semicircle is divided into two arcs, one of which is 15° less than the other. Find the magnitude of each arc.

3) The semicircle is divided into two arcs, of which one is twice the other. Find the magnitude of each arc.

4) The semicircle is divided into two arcs, of which one is 5 times smaller than the other. Find the magnitude of each arc.

579. 1) The chart "Literacy of the population in the USSR" (Fig. 32) shows the number of literate per hundred people of the population. According to the diagram and its scale, determine the number of literate men and women for each of the indicated years.

Record the results in a table:

2) Using the data of the diagram "Soviet envoys to space" (Fig. 33), make up tasks.

580. 1) According to the sector diagram "Daily routine for a student of grade V" (Fig. 34), fill in the table and answer the questions: what part of the day is devoted to sleep? for homework? to school?

2) Build a pie chart about the mode of your day.

To express a part as a fraction of the whole, you need to divide the part by the whole.

Task 1. There are 30 students in the class, four are missing. What proportion of students are missing?

Solution:

Answer: there are no students in the class.

Finding a fraction from a number

To solve problems in which it is required to find a part of a whole, the following rule is true:

If a part of the whole is expressed as a fraction, then to find this part, you can divide the whole by the denominator of the fraction and multiply the result by its numerator.

Task 1. There were 600 rubles, this amount was spent. How much money have you spent?

Solution: to find from 600 rubles, you need to divide this amount into 4 parts, thereby we will find out how much money is one fourth:

600: 4 = 150 (p.)

Answer: spent 150 rubles.

Task 2. It was 1000 rubles, this amount was spent. How much money has been spent?

Solution: From the condition of the problem, we know that 1000 rubles consists of five equal parts. First we find how many rubles are one fifth of 1000, and then we find out how many rubles are two fifths:

1) 1000: 5 = 200 (p.) - one fifth.

2) 200 2 \u003d 400 (p.) - two fifths.

These two actions can be combined: 1000: 5 2 = 400 (p.).

Answer: 400 rubles were spent.

The second way to find a part of a whole:

To find a part of a whole, you can multiply the whole by a fraction expressing that part of the whole.

Task 3. According to the charter of the cooperative, for the validity of the reporting meeting, it must be attended by at least members of the organization. The cooperative has 120 members. With what composition can the reporting meeting be held?

Solution:

Answer: the reporting meeting can be held if there are 80 members of the organization.

Finding a number by its fraction

To solve problems in which it is required to find the whole by its part, the following rule is true:

If a part of the desired integer is expressed as a fraction, then to find this integer, you can divide this part by the numerator of the fraction and multiply the result by its denominator.

Task 1. We spent 50 rubles, this amounted to the original amount. Find the original amount of money.

Solution: from the description of the problem, we see that 50 rubles is 6 times less than the initial amount, i.e., the initial amount is 6 times more than 50 rubles. To find this amount, you need to multiply 50 by 6:

50 6 = 300 (r.)

Answer: the initial amount is 300 rubles.

Task 2. We spent 600 rubles, this amounted to the initial amount of money. Find the original amount.

Solution: we will assume that the desired number consists of three thirds. By condition, two-thirds of the number are equal to 600 rubles. First, we find one third of the initial amount, and then how many rubles are three-thirds (initial amount):

1) 600: 2 3 = 900 (p.)

Answer: the initial amount is 900 rubles.

The second way to find the whole by its part:

To find a whole by the value of its part, you can divide this value by a fraction that expresses this part.

Task 3. Line segment AB, equal to 42 cm, is the length of the segment CD. Find the length of a segment CD.

Solution:

Answer: segment length CD 70 cm

Task 4. Watermelons were brought to the store. Before lunch, the store sold, after lunch - brought watermelons, and it remains to sell 80 watermelons. How many watermelons were brought to the store in total?

Solution: first, we find out what part of the imported watermelons is the number 80. To do this, we take the total number of imported watermelons as a unit and subtract from it the number of watermelons that we managed to sell (sell):

And so, we learned that 80 watermelons are from the total number of watermelons brought. Now we will find out how many watermelons of the total amount is, and then how many watermelons are (the number of watermelons brought):

2) 80: 4 15 = 300 (watermelons)

Answer: in total, 300 watermelons were brought to the store.

Instruction

First, remember that a fraction is just a conditional notation for dividing one number by another. In addition and multiplication, dividing two integers does not always result in an integer. So call these two "divisible" numbers. The number that is being divided is the numerator, and the number that is being divided is the denominator.

To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as a slash "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.

If the numerator of a fraction is greater than its denominator, then such an "improper" fraction is usually written as a "mixed" fraction. To get a mixed fraction from an improper fraction, simply divide the numerator by the denominator and write down the resulting quotient. Then put the remainder of the division in the numerator of the fraction and write this fraction to the right of the quotient (do not touch the denominator). For example, 7/3 = 2⅓.

To add two fractions with the same denominator, simply add their numerators (leave the denominators). For example, 2/7 + 3/7 = (2+3)/7 = 5/7. Similarly, subtract two fractions (the numerators are subtracted). For example, 6/7 - 2/7 = (6-2)/7 = 4/7.

To add two fractions with different denominators, multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first. As a result, you will get the sum of two fractions with the same denominators, the addition of which is described in the previous paragraph.

For example, 3/4 + 2/3 = (3*3)/(4*3) + (2*4)/(3*4) = 9/12 + 8/12 = (9+8)/12 = 17/12 = 15/12.

If the denominators of fractions have common divisors, that is, they are divisible by the same number, select as the common denominator smallest number divisible by the first and second denominators simultaneously. So, for example, if the first denominator is 6 and the second 8, then take as a common denominator not their product (48), but the number 24, which is divisible by both 6 and 8. The numerators of the fractions are then multiplied by the quotient of dividing the common denominator by the denominator of each fraction. For example, for the denominator 6, this number will be 4 - (24/6), and for the denominator 8 - 3 (24/8). This process is more clearly seen in a specific example:

5/6 + 3/8 = (5*4)/24 + (3*3)/24 = 20/24 + 9/24 = 29/24 = 1 5/24.

Subtraction of fractions with different denominators is done in exactly the same way.

Here we will understand how subtraction of common fractions. First, we get the rule for subtracting fractions with the same denominators. Next, consider the subtraction of fractions with different denominators and give examples of subtraction with detailed decisions. After that, we will focus on subtracting a fraction from a natural number and subtracting a number from a fraction. In conclusion, we will show how the subtraction of ordinary fractions is carried out using the properties of this action.

Immediately, we note that in this article we will only talk about subtracting a smaller fraction from a larger fraction. Other cases are discussed in the article subtraction of rational numbers.

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Subtraction of fractions with the same denominators

To begin with, let's give an example that will allow us to understand how the subtraction of fractions with the same denominators.

Suppose there were five eighths of an apple on the plate, that is, 5/8 of the apple, after which two eighths were taken away. According to the meaning of subtraction (see the general idea of ​​subtraction), the specified action is described as follows: . It is clear that in this case 5−2=3 eighths of an apple remains on the plate. That is, .

The considered example illustrates rule for subtracting fractions with the same denominator: when subtracting fractions with the same denominators, the numerator of the subtrahend is subtracted from the numerator of the minuend, and the denominator remains the same.

The voiced rule with the help of letters is written as follows: . We will use this formula when subtracting fractions with the same denominators.

Consider examples of subtracting fractions with the same denominators.

Example.

Subtract the common fraction 17/15 from the common fraction 24/15.

Solution.

The denominators of the subtracted fractions are equal. The numerator of the minuend is 24 , and the numerator of the subtrahend is 17 , their difference is 7 (24−17=7, if necessary, see the subtraction of natural numbers). Therefore, subtracting fractions with the same denominators 24/15 and 17/15 gives a fraction 7/15.

Short version solution looks like this: .

Answer:

.

If possible, it is necessary to reduce the fraction and (or) extract the whole part from the improper fraction, which is obtained by subtracting fractions with the same denominators.

Example.

Compute the difference.

Solution.

We use the formula for subtracting fractions with the same denominators: .

Obviously, the numerator and denominator of the resulting fraction are divisible by 2 (see), that is, 22/12 is a reduced fraction. By reducing this fraction by 2, we arrive at the fraction 11/6.

Fraction 11/6 is incorrect (see proper and improper fractions). Therefore, it is necessary to select the whole part from it: .

So, the calculated difference of fractions with the same denominators is .

Here is the whole solution: .

Answer:

.

Subtraction of fractions with different denominators

Subtraction of fractions with different denominators is reduced to subtraction of fractions with the same denominators. To do this, it is enough to bring fractions with different denominators to a common denominator.

So to spend subtraction of fractions with different denominators, necessary:

• reduce fractions to a common denominator (usually fractions lead to the lowest common denominator);
• Subtract the resulting fractions with the same denominators.

Consider examples of subtracting fractions with different denominators.

Example.

Subtract from the common fraction 2/9 the common fraction 1/15.

Solution.

Since the denominators of the fractions to be subtracted are different, we first perform the reduction of fractions to the lowest common denominator: since LCM(9, 15)=45, then the additional factor of the fraction 2/9 is the number 45:9=5, and the additional factor of the fraction is 1/15 is the number 45:15=3 , then And .

It remains to subtract the fraction 3/45 from the fraction 10/45, we get , which gives us the required difference of fractions with different denominators.

Briefly, the solution is written as follows: .

Answer:

We should not forget about the reduction of the fraction obtained after subtraction, as well as the selection of the whole part.

Example.

Subtract the fraction 7/36 from the fraction 19/9.

Solution.

After reducing fractions with different denominators to the lowest common denominator 36, we have fractions 76/9 and 7/36. We calculate their difference: .

The resulting fraction is reducible, after its reduction by 3, we get 23/12. And this fraction is incorrect, having separated the integer part from it, we have .

Let's put together all the actions performed when subtracting the original fractions with different denominators:.

Answer:

.

Subtraction of a natural number from an ordinary fraction

Subtracting a natural number from a fraction can be reduced to the subtraction of ordinary fractions. To do this, it is enough to represent a natural number as a fraction with a denominator of 1. Let's take a look at an example solution.

Example.

Subtract the number 3 from the fraction 83/21.

Solution.

Since the number 3 is equal to the fraction 3/1, then.

Answer:

However, it is more convenient to subtract a natural number from an improper fraction by representing the fraction as a mixed number. Let's show the solution of the previous example in this way.

Subtracting a fraction from a natural number

Subtracting a fraction from a natural number can be reduced to the subtraction of ordinary fractions by representing a natural number as a fraction. Let's analyze the solution of an example illustrating this approach.

Example.

Subtract the common fraction 5/3 from the natural number 7.

Solution.

We represent the number 7 as a fraction 7/1, after which we perform the subtraction: .

Having selected the integer part from the resulting fraction, we get the final answer.

Answer:

However, there is a more rational way to subtract a fraction from a natural number. Its advantages are especially noticeable when the natural number to be reduced and the denominator of the fraction to be subtracted are large numbers. All this will be seen from the examples below.

If the subtracted fraction is correct, then the reduced natural number can be replaced by the sum of two numbers, one of which is equal to one, subtract the correct fraction from one, and then complete the calculation.

Example.

Subtract the common fraction 13/62 from the natural number 1065.

Solution.

deductible common fraction- correct. Let's replace the number 1065 with the sum 1064+1 and get . It remains to calculate the value of the resulting expression (we will talk more about the calculation of such expressions in).

Due to the properties of subtraction, the resulting expression can be rewritten as . Calculate the value of the difference in brackets, replacing the unit with a fraction 1/1 , we have . Thus, . This completes the subtraction of the fraction 13/62 from the natural number 1065.

Here is the whole solution:

And now, for comparison, let's show what numbers we would have to work with if we decided to reduce the subtraction of the original numbers to the subtraction of fractions:

Answer:

.

If the fraction to be subtracted is incorrect, then it can be replaced by a mixed number, and then subtract the mixed number from a natural number.