Now that we have learned how to add and multiply individual fractions, we can look at more complex structures. For example, what if the same problem involves adding, subtracting, and multiplying fractions?
First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:
- Exponentiation is done first - get rid of all expressions containing exponents;
- Then - division and multiplication;
- The last step is addition and subtraction.
Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.
Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:
Now let's find the value of the second expression. There are no fractions with an integer part, but there are parentheses, so first we perform addition, and only then division. Note that 14 = 7 · 2. Then:
Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:
Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.
You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:
Multistory fractions
Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.
But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:
There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:
Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:
Task. Convert multistory fractions to ordinary ones:
In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:
In the last example, the fractions were canceled before the final multiplication.
Specifics of working with multi-level fractions
There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:
- The numerator contains the single number 7, and the denominator contains the fraction 12/5;
- The numerator contains the fraction 7/12, and the denominator contains the separate number 5.
So, for one recording we got two completely different interpretations. If you count, the answers will also be different:
To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.
If you follow this rule, then the above fractions should be written as follows:
Yes, it may be unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:
Task. Find the meanings of the expressions:
So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:
Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:
Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.
I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.
Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.
Actions with fractions. In this article we will look at examples, everything in detail with explanations. We will consider ordinary fractions. We'll look at decimals later. I recommend watching the whole thing and studying it sequentially.
1. Sum of fractions, difference of fractions.
Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.
Rule: when calculating the difference between fractions with the same denominators, we obtain a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.
Formal notation for the sum and difference of fractions with equal denominators:
Examples (1):
It is clear that when ordinary fractions are given, then everything is simple, but what if they are mixed? Nothing complicated...
Option 1– you can convert them into ordinary ones and then calculate them.
Option 2– you can “work” separately with the integer and fractional parts.
Examples (2):
More:
What if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? You can also act in two ways.
Examples (3):
*Converted to ordinary fractions, calculated the difference, converted the resulting improper fraction to a mixed fraction.
*We broke it down into integer and fractional parts, got a three, then presented 3 as the sum of 2 and 1, with one represented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it in the form of a fraction with the denominator we need, then we can subtract another from this fraction.
Another example:
Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted to improper ones, then perform the necessary action. After this, if the result is an improper fraction, we convert it to a mixed fraction.
Above we looked at examples with fractions that have equal denominators. What if the denominators are different? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the basic property of the fraction is used.
Let's look at simple examples:
In these examples, we immediately see how one of the fractions can be transformed to get equal denominators.
If we designate ways to reduce fractions to the same denominator, then we will call this one METHOD ONE.
That is, immediately when “evaluating” a fraction, you need to figure out whether this approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divisible, then we perform a transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.
Now look at these examples:
This approach is not applicable to them. There are also ways to reduce fractions to a common denominator; let’s consider them.
Method TWO.
We 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:
*In fact, we reduce fractions to form when the denominators become equal. Next, we use the rule for adding fractions with equal denominators.
Example:
*This method can be called universal, and it always works. The only downside is that after the calculations you may end up with a fraction that will need to be further reduced.
Let's look at an example:
It can be seen that the numerator and denominator are divisible by 5:
Method THREE.
You need to find the least common multiple (LCM) of the denominators. This will be the common denominator. What kind of number is this? This is the smallest natural number that is divisible by each of the numbers.
Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, they are divisible by 30, 60, 90 .... The least is 30. The question is - how to determine this least common multiple?
There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15) no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, for example 51 and 119.
Algorithm. In order to determine the least common multiple of several numbers, you must:
- decompose each number into SIMPLE factors
— write down the decomposition of the BIGGER of them
- multiply it by the MISSING factors of other numbers
Let's look at examples:
50 and 60 => 50 = 2∙5∙5 60 = 2∙2∙3∙5
in the expansion of a larger number one five is missing
=> LCM(50,60) = 2∙2∙3∙5∙5 = 300
48 and 72 => 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3
in the expansion of a larger number two and three are missing
=> LCM(48.72) = 2∙2∙2∙2∙3∙3 = 144
* The least common multiple of two prime numbers is their product
Question! Why is finding the least common multiple useful, since you can use the second method and simply reduce the resulting fraction? Yes, it is possible, but it is not always convenient. Look at the denominator for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. You will agree that it is more pleasant to work with smaller numbers.
Let's look at examples:
*51 = 3∙17 119 = 7∙17
the expansion of a larger number is missing a triple
=> NOC(51,119) = 3∙7∙17
Now let's use the first method:
*Look at the difference in the calculations, in the first case there are a minimum of them, but in the second you need to work separately on a piece of paper, and even the fraction you received needs to be reduced. Finding the LOC simplifies the work significantly.
More examples:
*In the second example it is clear that the smallest number that is divisible by 40 and 60 is 120.
RESULT! GENERAL COMPUTING ALGORITHM!
— we reduce fractions to ordinary ones if there is an integer part.
- we bring fractions to a common denominator (first we look at whether one denominator is divisible by another; if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).
- Having received fractions with equal denominators, we perform operations (addition, subtraction).
- if necessary, we reduce the result.
- if necessary, then select the whole part.
2. Product of fractions.
The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:
Examples:
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 the unknown number.
2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find the 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 you increase an unknown number by 2/3 of it, you get 60. What number is this?
500 *. If you add the same amount to the unknown number, and also 20 1/3, you get 105 2/5. Find the unknown number.
501. 1) The potato yield with square-cluster planting averages 150 centners per hectare, and with conventional planting it is 3/5 of this amount. How much more potatoes can be harvested from an area of 15 hectares if potatoes are planted using the square-cluster method?
2) An experienced worker produced 18 parts in 1 hour, and an inexperienced worker produced 2/3 of this amount. How many more parts can an experienced worker produce in a 7-hour day?
502. 1) The pioneers collected 56 kg of different seeds over three days. 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 the wheat, the result was: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest was bran. How much flour, semolina and bran separately were produced when grinding 3 tons of wheat?
503. 1) Three garages can accommodate 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 the third garage has 1 1/2 times as many cars as the first. How many cars fit in each garage?
2) A factory with three workshops employs 6,000 workers. In the second workshop there are 1 1/2 times fewer workers 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 there in each workshop?
504. 1) First 2/5, then 1/3 of the total kerosene was poured from a tank with kerosene, and after that 8 tons of kerosene remained in the tank. How much kerosene was in the tank initially?
2) The cyclists raced for three days. On the first day they covered 4/15 of the entire journey, on the second - 2/5, and on the third day the remaining 100 km. How far did the cyclists travel in three days?
505. 1) The icebreaker fought its way through the ice field for three days. On the first day he walked 1/2 of the entire distance, on the second day 3/5 of the remaining distance and on the third day the remaining 24 km. Find the length of the path covered by the icebreaker in three days.
2) Three groups of schoolchildren planted trees to green the village. The first detachment planted 7/20 of all 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 entire 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 area?
2) On the first day, the rally participants covered 3/11 of the entire route, on the second day 7/20 of the remaining route, on the third day 5/13 of the new remainder, and on the fourth day the remaining 320 km. How long is the route of the rally?
507. 1) On the first day the car covered 3/8 of the entire distance, on the second day 15/17 of what it covered on the first, and on the third day the remaining 200 km. How much gasoline was consumed if a car consumes 1 3/5 kg of gasoline for 10 km?
2) The city consists of four districts. And 4/13 of all residents of the city live in the first district, 5/6 of the residents of the first district live in the second, 4/11 of the residents of the first live in the third; two districts combined, and 18 thousand people live in the fourth district. 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 journey, on the second 9/10 of what he walked on the first day, and on the third the rest of the way, 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 covered the entire route 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 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 allocated land to the workers of three factories for garden plots. The first plant was allocated 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 remaining plots. How many total plots were allocated to the workers of three factories, if the first factory was allocated 50 fewer plots than the third?
2) The plane delivered a shift of winter workers to the polar station from Moscow in three days. On the first day he flew 2/5 of the entire distance, on the second - 5/6 of the distance he covered 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 workers in 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 100 more workers than in the second. How many workers are there 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 families on 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 less than in the second. How many families are there on the collective farm?
511. 1) The artel used up 1/3 of its stock of raw materials in the first week, and 1/3 of the rest in the second week. 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, 1/6 of it was spent for heating the house in the first month, and 3/8 of the remainder in the second month. How much coal is left to heat the house if 1 3/4 more was used in the second month than in the first month?
512. 3/5 of the total land of the collective farm is allocated for sowing grain, 13/36 of the remainder is occupied by vegetable gardens and meadows, the rest of the land is forest, and the sown area of the collective farm is 217 hectares larger than the forest area, 1/3 of the land allocated 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. Along this route, the tram makes 18 stops, spending on average up to 1 1/6 minutes per stop. The average speed of the tram along the entire route is 12 1/2 km per hour. How long does it take for a tram to complete one trip?
2) Bus route 16 km. Along this route the bus makes 36 stops of 3/4 minutes each. on average each. The average bus speed is 30 km per hour. How long does a bus take for one route?
514*. 1) It’s 6 o’clock now. evenings. What part is the remaining part of the day from the past and what part of the day is left?
2) A steamer travels the distance between two cities with the current in 3 days. and back the same distance in 4 days. How many days will the rafts float downstream 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 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 its length. This area is bordered by a path 4/5 m wide. Find the area of the path.
516. Find the arithmetic mean of numbers:
517. 1) The arithmetic mean of two numbers is 6 1/6. One of the numbers is 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 covered 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 hectares, on the second day 15 3/4 hectares and on the third day 14 1/2 hectares. On average, how many hectares of land did a tractor driver plow per day?
2) A group of schoolchildren, making a three-day tourist trip, were on the road for 6 1/3 hours on the first day, 7 hours on the second. and on the third day - 4 2/3 hours. How many hours on average did schoolchildren travel every day?
520. 1) Three families live in the house. The first family has 3 light bulbs to illuminate the apartment, the second has 4 and the third has 5 light 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) A polisher was polishing the floors in an apartment where three families lived. The first family had a living area of 36 1/2 square meters. m, the second is 24 1/2 sq. m, and the third - 43 sq. m. For all the work, 2 rubles were paid. 08 kop. How much did each family pay?
521. 1) In the garden plot, potatoes were collected from 50 bushes at 1 1/10 kg per bush, from 70 bushes at 4/5 kg per bush, from 80 bushes at 9/10 kg per bush. How many kilograms of potatoes are harvested on average from each bush?
2) The field crew on an area of 300 hectares received a harvest of 20 1/2 quintals of winter wheat per 1 hectare, from 80 hectares to 24 quintals per 1 ha, and from 20 hectares - 28 1/2 quintals per 1 ha. What is the average yield in a brigade with 1 hectare?
522. 1) The sum of two numbers is 7 1/2. One number is 4 4/5 greater than the other. 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 greater than the second by 5 1/3 and greater than the third by 3 5/6. Find these numbers.
2) The islands of Novaya Zemlya, 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 larger 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 square meters. m more than the area of the third. What is the area of the second room?
2) A cyclist during a three-day competition on the first day was on the road for 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 you pour 4 3/4 kg of cereal from one box into another, then there will be equal amounts of cereal in both boxes. How much cereal is 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 it to the second, then the first will still be greater than the second by 2 17/30. Find both numbers.
2) There are 24 1/4 kg of apples in two boxes. If you transfer 3 1/2 kg 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 moved along the river at a speed of 15 1/2 km per hour, and against the current at 8 1/4 km per hour. What is the speed of the river flow?
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. 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 a larger number by a smaller number is 5 2/3. Find these numbers.
2) The difference between 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 students 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 between two numbers is 3 1/5. One number is 5/7 of another. Find these numbers.
2) The father is 24 years older than his son. The number of the son's years is equal to 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 units greater than its numerator. What is the value of a fraction 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 is the second number greater 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 collected 100 mushrooms together. 3/8 of the number of mushrooms collected by the first boy is numerically equal to 1/4 of the number of mushrooms collected by the second boy. How many mushrooms did each boy collect?
2) The institution employs 27 people. How many men work 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 more than the other. Find these numbers if 2/5 of one number are equal to 2/3 of the other.
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 team can complete some work in 36 days, and the second in 45 days. In how many days will both teams, working together, complete this job?
2) A passenger train covers the distance between two cities in 10 hours, and a freight train covers 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 covers the distance between two cities in 6 1/4 hours, and a passenger train in 7 1/2 hours. How many hours later 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 simultaneously from two cities towards each other. One motorcyclist can travel the entire distance between these cities in 6 hours, and another in 5 hours. How many hours after departure will the motorcyclists meet? (Round answer to the nearest 1 hour.)
544. 1) Three cars of different carrying capacity can transport 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 transport the same cargo, working together?
2) Two trains leave two stations simultaneously 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) Two taps are connected to the bathtub. 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 bathtub if both taps are opened at once?
2) Two typists must retype the manuscript. The first driver can complete this work in 3 1/3 days, and the second 1 1/2 times faster. How many days will it take both typists to complete the job if they work simultaneously?
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. After how many hours will the entire pool be filled if both pipes are opened at the same time?
Note. 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, working alone, to plow this field?
547 *. Two trains leave two stations simultaneously towards each other and meet after 18 hours. after his release. How long does it take the second train to cover the distance between stations if the first train covers this distance in 1 day 21 hours?
548 *. The pool is filled with two pipes. First they opened the first pipe, and then after 3 3/4 hours, when half of the pool was filled, they opened the second pipe. After 2 1/2 hours of working together, the pool was full. Determine the capacity of the pool if 200 buckets of water per hour poured through the second pipe.
549. 1) A courier train left Leningrad for Moscow and travels 1 km in 3/4 minutes. 1/2 hour after this train left Moscow, a fast train left Moscow for Leningrad, the speed of which was equal to 3/4 the speed of the express train. At what distance will the trains be from each other 2 1/2 hours after the courier train leaves, if the distance between Moscow and Leningrad is 650 km?
2) From the collective farm to the city 24 km. A truck leaves the collective farm and travels 1 km in 2 1/2 minutes. After 15 min. After this car left the city, a cyclist drove out to the collective farm, at a speed half as fast as the speed of the truck. How long after leaving will the cyclist meet the truck?
550. 1) A pedestrian came out of one village. 4 1/2 hours after the pedestrian left, a cyclist rode in the same direction, whose speed was 2 1/2 times the speed of the pedestrian. How many hours after the pedestrian leaves will the cyclist overtake him?
2) A fast train travels 187 1/2 km in 3 hours, and a freight train travels 288 km in 6 hours. 7 1/4 hours after the freight train leaves, an ambulance departs in the same direction. How long will it take the fast train to catch up with the freight train?
551. 1) From two collective farms through which the road to the regional center passes, two collective farmers rode out to the district at the same time on horseback. The first of them traveled 8 3/4 km per hour, and the second was 1 1/7 times more than the first. The second collective farmer caught up with the first after 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 was 60 km per hour, a TU-104 plane took off in the same direction, at a speed 14 1/6 times the speed of the train. How many hours after departure will the plane catch up with the train?
552. 1) The distance between cities along the river is 264 km. The steamer covered this distance downstream in 18 hours, spending 1/12 of this time stopping. The speed of the river is 1 1/2 km per hour. How long would it take a steamship to travel 87 km without stopping in still water?
2) A motor boat traveled 207 km along the river in 13 1/2 hours, spending 1/9 of this time on stops. The speed of the river is 1 3/4 km per hour. How many kilometers can this boat travel in still water in 2 1/2 hours?
553. The boat covered a distance of 52 km across the reservoir without stopping in 3 hours 15 minutes. Further, going along the river against the current, the speed of which is 1 3/4 km per hour, this boat covered 28 1/2 km in 2 1/4 hours, making 3 stops of equal duration. How many minutes did the boat wait at each stop?
554. From Leningrad to Kronstadt at 12 o'clock. The steamer left in the afternoon and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another ship that left Kronstadt for Leningrad at 12:18 p.m. and walking at 1 1/4 times the speed of 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 detained for 1 hour 10 minutes. At what speed should he continue his journey in order to reach his destination without delay?
556. At 4:20 a.m. In the morning, a freight train left Kyiv for Odessa with an average speed of 31 1/5 km per hour. After some time, a mail train came out of Odessa to meet him, the speed of which was 1 17/39 times higher than the speed of a freight train, and met 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 will it take for the hour and minute hands to coincide?
558. 1) The plant has three workshops. The number of workers in the first workshop is 9/20 of all 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 fewer workers than in the second. How many workers are there 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 fewer students than in the second. How many students are there in the three schools?
559. 1) Two combine operators worked in the same area. After one combiner harvested 9/16 of the entire plot, and the second 3/8 of the same plot, it turned out that the first combiner harvested 97 1/2 hectares more than the second. On average, 32 1/2 quintals of grain were threshed from each hectare. How many centners of grain did each combine operator thresh?
2) Two brothers bought a camera. One had 5/8, and the second 4/7 of the cost of the camera, and the first had 2 rubles worth. 25 kopecks more than the second one. Everyone paid half the cost of the device. How much money does everyone have left?
560. 1) A passenger car leaves city A for city B, the distance between them is 215 km, 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 passenger car travel before meeting the truck, if the truck's speed per hour was 18/25 the speed of the passenger car?
2) Between cities A and B 210 km. A passenger car left city A for city B. At the same time, a truck left city B for city A. How many kilometers did the truck travel before meeting the passenger car, if the passenger car was traveling at a speed of 48 km per hour, and the speed of the truck per hour was 3/4 of the speed of the passenger car?
561. The collective farm harvested wheat and rye. 20 hectares more were sown with wheat than with rye. The total rye harvest amounted to 5/6 of the total wheat harvest with a yield of 20 c 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 satisfy its needs. How many trips did the two-ton trucks need to make to remove the bread sold to the state?
562. Rye and wheat flour were brought to the bakery. The weight of wheat flour was 3/5 of the weight of rye flour, and 4 tons more rye flour was brought than wheat flour. How much wheat and how much rye bread will the bakery bake from this flour if the baked goods make up 2/5 of the total flour?
563. Within three days, a team of workers completed 3/4 of the entire work on repairing the highway between the two collective farms. On the first day, 2 2/5 km of this highway were 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 in the empty spaces 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 site.
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 site.
566. 1) The perimeter of the rectangle is 6 1/2 inch, its base is 1/4 inch greater than its height. Find the area of this rectangle.
2) The perimeter of the 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 by dividing them into rectangles and finding the dimensions of the rectangle by measurement.
568. 1) How many sheets of dry plaster will be required to cover the ceiling of a room whose length is 4 1/2 m and width 4 m, if the dimensions of the plaster sheet are 2 m x l 1/2 m?
2) How many boards 4 1/2 m long and 1/4 m wide are needed 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 harvest of 25 quintals per hectare was collected from a rectangular field. How much wheat was harvested from the entire field if the length of the field is 800 m and the width is 3/8 of its length?
570 . 1) A rectangular plot of land, 78 3/4 m long and 56 4/5 m wide, 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 plans to lay out a garden. How many trees will be planted in this garden if an average area of 36 sq.m. is required for each tree?
571. 1) For normal daylight illumination of the room, it is necessary that the area of all windows be at least 1/5 of the floor area. Determine whether there is enough light in a room whose length is 5 1/2 m and width 4 m. Does the room have one window measuring 1 1/2 m x 2 m?
2) Using the condition of the previous problem, find out whether there is enough light in your classroom.
572. 1) The barn has dimensions of 5 1/2 m x 4 1/2 m x 2 1/2 m. How much hay (by weight) will fit in this barn if it is filled to 3/4 of its height and if 1 cu. 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 cubic. m of firewood weighs 600 kg?
573. 1) A rectangular aquarium is filled with water up to 3/5 of its height. The length of the aquarium is 1 1/2 m, width 4/5 m, height 3/4 m. How many liters of water are poured into the aquarium?
2) A pool in 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 needs to be built around a rectangular piece of land, 75 m long and 45 m wide. How many cubic meters of boards should go into its construction 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 o'clock? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23:30?
2) How many degrees will the hour hand rotate 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 of a circle? 5/24 circles?
576. 1) Using a protractor, draw: a) a right angle; b) an angle of 30°; c) an angle of 60°; d) angle of 150°; e) an angle of 55°.
2) Using a protractor, measure the angles of the figure and find the sum of all the angles of each figure (Fig. 31).
577. Follow these steps:
578. 1) The semicircle is divided into two arcs, one of which is 100° larger than the other. Find the size of each arc.
2) The semicircle is divided into two arcs, one of which is 15° less than the other. Find the size of each arc.
3) The semicircle is divided into two arcs, one of which is twice as large as the other. Find the size of each arc.
4) The semicircle is divided into two arcs, one of which is 5 times smaller than the other. Find the size of each arc.
579. 1) The diagram “Population Literacy in the USSR” (Fig. 32) shows the number of literate people per hundred people of the population. Based on the data in the diagram and its scale, determine the number of literate men and women for each of the indicated years.
Write the results in the table:
2) Using the data from the diagram “Soviet envoys to Space” (Fig. 33), create tasks.
580. 1) According to the pie chart “Daily routine for a fifth grade student” (Fig. 34), fill out the table and answer the questions: what part of the day is allocated to sleep? for homework? to school?
2) Construct a pie chart about your daily routine.
This section covers operations with ordinary fractions. If it is necessary to carry out a mathematical operation with mixed numbers, then it is enough to convert the mixed fraction into an extraordinary fraction, carry out the necessary operations and, if necessary, present the final result again in the form of a mixed number. This operation will be described below.
Reducing a fraction
Mathematical operation. Reducing a fraction
To reduce the fraction \frac(m)(n) you need to find the greatest common divisor of its numerator and denominator: gcd(m,n), and then divide the numerator and denominator of the fraction by this number. If GCD(m,n)=1, then the fraction cannot be reduced. Example: \frac(20)(80)=\frac(20:20)(80:20)=\frac(1)(4)
Usually, immediately finding the greatest common divisor seems to be a difficult task, and in practice, a fraction is reduced in several stages, step by step isolating obvious common factors from the numerator and denominator. \frac(140)(315)=\frac(28\cdot5)(63\cdot5)=\frac(4\cdot7\cdot5)(9\cdot7\cdot5)=\frac(4)(9)
Reducing fractions to a common denominator
Mathematical operation. Reducing fractions to a common denominator
To bring two fractions \frac(a)(b) and \frac(c)(d) to a common denominator you need:
- find the least common multiple of the denominators: M=LMK(b,d);
- multiply the numerator and denominator of the first fraction by M/b (after which the denominator of the fraction becomes equal to the number M);
- multiply the numerator and denominator of the second fraction by M/d (after which the denominator of the fraction becomes equal to the number M).
Thus, we transform the original fractions to fractions with the same denominators (which will be equal to the number M).
For example, the fractions \frac(5)(6) and \frac(4)(9) have LCM(6,9) = 18. Then: \frac(5)(6)=\frac(5\cdot3)(6 \cdot3)=\frac(15)(18);\quad\frac(4)(9)=\frac(4\cdot2)(9\cdot2)=\frac(8)(18) . Thus, the resulting fractions have a common denominator.
In practice, finding the least common multiple (LCM) of denominators is not always a simple task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as the common denominator. For example, the fractions \frac(5)(6) and \frac(4)(9) are reduced to a common denominator N=6\cdot9:
\frac(5)(6)=\frac(5\cdot9)(6\cdot9)=\frac(45)(54);\quad\frac(4)(9)=\frac(4\cdot6)( 9\cdot6)=\frac(24)(54)
Comparison of fractions
Mathematical operation. Comparison of fractions
To compare two ordinary fractions you need:
- compare the numerators of the resulting fractions; a fraction with a larger numerator will be larger.
When comparing fractions, there are several special cases:
- From two fractions with the same denominators The fraction whose numerator is greater is greater. For example, \frac(3)(15)
- From two fractions with the same numerators The larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13)
- That fraction which simultaneously larger numerator and smaller denominator, more. For example, \frac(11)(3)>\frac(10)(8)
Attention! Rule 1 applies to any fractions if their common denominator is a positive number. Rules 2 and 3 apply to positive fractions (those with both the numerator and denominator greater than zero).
Adding and subtracting fractions
Mathematical operation. Adding and subtracting fractions
To add two fractions you need:
- bring them to a common denominator;
- add their numerators and leave the denominator unchanged.
Example: \frac(7)(9)+\frac(4)(7)=\frac(7\cdot7)(9\cdot7)+\frac(4\cdot9)(7\cdot9)=\frac(49 )(63)+\frac(36)(63)=\frac(49+36)(63)=\frac(85)(63)
To subtract another from one fraction, you need:
- reduce fractions to a common denominator;
- Subtract the numerator of the second fraction from the numerator of the first fraction and leave the denominator unchanged.
Example: \frac(4)(15)-\frac(3)(5)=\frac(4)(15)-\frac(3\cdot3)(5\cdot3)=\frac(4)(15) -\frac(9)(15)=\frac(4-9)(15)=\frac(-5)(15)=-\frac(5)(3\cdot5)=-\frac(1)( 3)
If the original fractions initially have a common denominator, then step 1 (reduction to a common denominator) is skipped.
Converting a mixed number to an improper fraction and vice versa
Mathematical operation. Converting a mixed number to an improper fraction and vice versa
To convert a mixed fraction to an improper fraction, simply sum the whole part of the mixed fraction with the fraction part. The result of such a sum will be an improper fraction, the numerator of which is equal to the sum of the product of the whole part by the denominator of the fraction with the numerator of the mixed fraction, and the denominator will remain the same. For example, 2\frac(6)(11)=2+\frac(6)(11)=\frac(2\cdot11)(11)+\frac(6)(11)=\frac(2\cdot11+ 6)(11)=\frac(28)(11)
To convert an improper fraction to a mixed number:
- divide the numerator of a fraction by its denominator;
- write the remainder of the division into the numerator and leave the denominator the same;
- write the result of the division as an integer part.
For example, the fraction \frac(23)(4) . When dividing 23:4=5.75, that is, the whole part is 5, the remainder of the division is 23-5*4=3. Then the mixed number will be written: 5\frac(3)(4) . \frac(23)(4)=\frac(5\cdot4+3)(4)=5\frac(3)(4)
Converting a Decimal to a Fraction
Mathematical operation. Converting a Decimal to a Fraction
In order to convert a decimal fraction to a common fraction, you need to:
- take the nth power of ten as the denominator (here n is the number of decimal places);
- as the numerator, take the number after the decimal point (if the integer part of the original number is not equal to zero, then take all the leading zeros as well);
- the non-zero integer part is written in the numerator at the very beginning; the zero integer part is omitted.
Example 1: 0.0089=\frac(89)(10000) (there are 4 decimal places, so the denominator has 10 4 =10000, since the integer part is 0, the numerator contains the number after the decimal point without leading zeros)
Example 2: 31.0109=\frac(310109)(10000) (in the numerator we write the number after the decimal point with all zeros: “0109”, and then before it we add the whole part of the original number “31”)
If the whole part of a decimal fraction is non-zero, then it can be converted to a mixed fraction. To do this, we convert the number into an ordinary fraction as if the whole part were equal to zero (points 1 and 2), and simply rewrite the whole part in front of the fraction - this will be the whole part of the mixed number. Example:
3.014=3\frac(14)(100)
To convert a fraction to a decimal, simply divide the numerator by the denominator. Sometimes you end up with an infinite decimal. In this case, it is necessary to round to the desired decimal place. Examples:
\frac(401)(5)=80.2;\quad \frac(2)(3)\approx0.6667
Multiplying and dividing fractions
Mathematical operation. Multiplying and dividing fractions
To multiply two ordinary fractions, you need to multiply the numerators and denominators of the fractions.
\frac(5)(9)\cdot\frac(7)(2)=\frac(5\cdot7)(9\cdot2)=\frac(35)(18)
To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second ( reciprocal fraction- a fraction in which the numerator and denominator are swapped.
\frac(5)(9):\frac(7)(2)=\frac(5)(9)\cdot\frac(2)(7)=\frac(5\cdot2)(9\cdot7)= \frac(10)(63)
If one of the fractions is a natural number, then the above rules of multiplication and division remain in force. You just need to take into account that an integer is the same fraction, the denominator of which is equal to one. For example: 3:\frac(3)(7)=\frac(3)(1):\frac(3)(7)=\frac(3)(1)\cdot\frac(7)(3)= \frac(3\cdot7)(1\cdot3)=\frac(7)(1)=7