To express a part as a fraction of the whole, you need to divide the part into the whole.
Task 1. There are 30 students in the class, four are absent. What proportion of students are absent?
Solution:
Answer: There are no students in the class.
Finding a fraction from a number
To solve problems in which you need to find a part of a whole that is fair next rule:
If a part of a 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 did you spend?
Solution: to find 600 rubles or more, we need to divide this amount into 4 parts, thereby we will find out how much money one fourth part is:
600: 4 = 150 (r.)
Answer: spent 150 rubles.
Task 2. There were 1000 rubles, this amount was spent. How much money was spent?
Solution: from the problem statement we know that 1000 rubles consists of five equal parts. First, let’s find how many rubles are one-fifth of 1000, and then we’ll find out how many rubles are two-fifths:
1) 1000: 5 = 200 (r.) - one fifth.
2) 200 · 2 = 400 (r.) - two fifths.
These two actions can be combined: 1000: 5 · 2 = 400 (r.).
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 the fraction expressing that part of the whole.
Task 3. According to the charter of the cooperative, for the reporting meeting to be valid, at least at least members of the organization must be present. The cooperative has 120 members. What composition can a reporting meeting take place?
Solution: 
Answer: the reporting meeting can take place if there are 80 members of the organization.
Finding a number by its fraction
To solve problems in which you need to find a whole from its part, the following rule applies:
If part of the desired whole is expressed as a fraction, then to find this whole, you can this part divide by the numerator of the fraction and multiply the result by its denominator.
Task 1. We spent 50 rubles, which was less than 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 original amount, i.e. the original 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, which was less than the original amount of money. Find the original amount.
Solution: We will assume that the required number consists of three thirds. According to the condition, two-thirds of the number equals 600 rubles. First, let's find one third of the original amount, and then how many rubles are three thirds (the original amount):
1) 600: 2 3 = 900 (r.)
Answer: the initial amount is 900 rubles.
The second way to find a whole from its part:
To find a whole by the value expressing its part, you can divide this value by the fraction expressing this part.
Task 3. Line segment AB, equal to 42 cm, is the length of the segment CD. Find the length of the segment CD.
Solution: 
Answer: segment length CD 70 cm.
Task 4. Watermelons were brought to the store. Before lunch, the store sold the watermelons it brought, after lunch, and there were 80 watermelons left to sell. How many watermelons did you bring to the store?
Solution: First, let’s find out what part of the brought watermelons is the number 80. To do this, let’s take the total number of watermelons brought as one and subtract from it the number of watermelons that were sold (sold):
And so, we learned that 80 watermelons make up the total number of watermelons brought. Now we find out how many watermelons from the total amount make up, and then how many watermelons make up (the number of watermelons brought):
2) 80: 4 15 = 300 (watermelons)
Answer: In total, 300 watermelons were brought to the store.
Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.
Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.
The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:
The main property of a fraction
If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,
Reducing fractions to a common denominator
To bring two fractions to a common denominator, you need:
- Multiply the numerator of the first fraction by the denominator of the second
- Multiply the numerator of the second fraction by the denominator of the first
- Replace the denominators of both fractions with their product
Operations with fractions
Addition. To add two fractions you need
- Add the new numerators of both fractions and leave the denominator unchanged
Example:
Subtraction. To subtract one fraction from another, you need
- Reduce fractions to a common denominator
- Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged
Example:
Multiplication. To multiply one fraction by another, multiply their numerators and denominators:
Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:
Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions for solving examples of this type.
How to solve examples with fractions - general rules
To solve examples with fractions of any type, be it addition, subtraction, multiplication or division, you need to know the basic rules:
- In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator at the top), you need to add their numerators and leave the denominator the same.
- In order to subtract a second fractional expression (with the same denominator) from one fraction, you need to subtract their numerators and leave the denominator the same.
- To add or subtract fractional expressions with different denominators, you need to find the lowest common denominator.
- In order to find a fractional product, you need to multiply the numerators and denominators, and, if possible, reduce.
- To divide a fraction by a fraction, you multiply the first fraction by the second fraction reversed.
How to solve examples with fractions - practice
Rule 1, example 1:
Calculate 3/4 +1/4.
According to Rule 1, if fractions have two (or more) same denominator, you just need to add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will equal 1.
Answer: 3/4 + 1/4 = 4/4 = 1.
Rule 2, example 1:
Calculate: 3/4 – 1/4
Using rule number 2, to solve this equation you need to subtract 1 from 3 and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.
Answer: 3/4 – 1/4 = 2/4 = 1/2.
Rule 3, Example 1
Calculate: 3/4 + 1/6
Solution: Using the 3rd rule, we find the lowest common denominator. The lowest common denominator is a number that is divisible by the denominators of all fractional expressions example. Thus, we need to find the minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. Divide 12 by the denominator of the first fraction, we get 3, multiply by 3, write 3 in the numerator *3 and + sign. Divide 12 by the denominator of the second fraction, we get 2, multiply 2 by 1, write 2*1 in the numerator. So, we get a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.
Answer: 11/12
Rule 3, Example 2:
Calculate 3/4 – 1/6. This example is very similar to the previous one. We do all the same steps, but in the numerator instead of the + sign, we write a minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.
Answer: 7/12
Rule 4, Example 1:
Calculate: 3/4 * 1/4
Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.
Answer: 3/16
Rule 4, Example 2:
Calculate 2/5 * 10/4.
This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are canceled.
2 cancels from 4. 10 cancels from 5. We get 1 * 2/2 = 1*1 = 1.
Answer: 2/5 * 10/4 = 1
Rule 5, Example 1:
Calculate: 3/4: 5/6
Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.
Answer: 9/10.
How to solve examples with fractions - fractional equations
Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.
Let's look at an example:
Solve the equation 15/3x+5 = 3
Let us remember that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. For this purpose, there is an OA (permissible value range).
So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3
At x = 5/3 the equation simply has no solution.
Having indicated the ODZ, in the best possible way decide given equation will get rid of fractions. To do this, we first represent all non-fractional values in the form of a fraction, in in this case number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions you need to multiply each of them by the lowest common denominator. In this case it will be (3x+5)*1. Sequencing:
- Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
- Open the brackets: 15*(3x+5) = 45x + 75.
- We do the same with the right side of the equation: 3*(3x+5) = 9x + 15.
- Equate the left and right sides: 45x + 75 = 9x +15
- Move the X's to the left, numbers to the right: 36x = – 50
- Find x: x = -50/36.
- We reduce: -50/36 = -25/18
Answer: ODZ x ≠ 5/3. x = -25/18.
How to solve examples with fractions - fractional inequalities
Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the number axis. Let's look at this example.
Sequencing:
- We equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
2. 2-x=0 => x=2 - We draw a number axis, writing the resulting values on it.
- Draw a circle under the value. There are two types of circles - filled and empty. A filled circle means that the given value is within the solution range. An empty circle indicates that this value is not included in the solution range.
- Since the denominator cannot be equal to zero, under the 2nd there will be an empty circle.
- To determine the signs, we substitute any number greater than two into the equation, for example 3. (3*3-5)/(2-3)= -4. the value is negative, which means we write a minus above the area after the two. Then substitute for X any value of the interval from 5/3 to 2, for example 1. The value is again negative. We write a minus. We repeat the same with the area located up to 5/3. We substitute any number less than 5/3, for example 1. Again, minus.
- Since we are interested in the values of x at which the expression will be greater than or equal to 0, and there are no such values (there are minuses everywhere), this inequality has no solution, that is, x = Ø (an empty set).
Answer: x = Ø
The numerator, and that which is divided by is the denominator.
To write a fraction, write its numerator first, then write under that number. horizontal line, and write the denominator below the line. The horizontal line that separates the numerator and denominator is called a fraction line. Sometimes it is depicted as an oblique "/" 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.
To calculate the product of fractions, first multiply the numerator of one fractions to the numerator is different. Write the result in the numerator of the new fractions. After this, multiply the denominators. Enter the total value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).
To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the actions, first “flip” the divisor, if it is more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by new denominator divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 ? 5 = 5; 3 ? 1 = 3).
Sources:
- Basic fraction problems
Fractional numbers can be expressed in in different forms exact value quantities. You can do the same with fractions mathematical operations, as with whole numbers: subtraction, addition, multiplication and division. To learn to decide fractions, we must remember some of their features. They depend on the type fractions, the presence of a whole part, common denominator. Some arithmetic operations after execution they require reduction of the fractional part of the result.
You will need
- - calculator
Instructions
Take a close look at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part is isolated must be converted to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given value will become the new numerator fractions. To select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. For fractions with whole part it is possible to perform actions separately first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation of separately integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.
Rewrite them using the “:” separator and continue regular division.
For getting final result Reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integers above and below the line.
note
Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.
When recording fractional numbers The dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice will be written as a fraction in the following way: 1 ½ kg rice. If the denominator of a fraction is 10, the fraction is called a decimal. In this case, the numerator (dividend) is written to the right of the whole part, separated by a comma: 1.5 kg of rice. For ease of calculation, such a fraction can always be written in in the wrong form: 1 2/10 kg potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by one integer. In this example, you can divide by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.
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 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 the wheat, the result was: 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 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 times the number of cars that fit in the second, and in the third garage it is 1 1/2 times more cars than in the first one. 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 the land 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 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. 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 more area forests, 1/3 of the land allocated for grain crops is sown with rye, and the rest with wheat. How many hectares of land did the collective farm sow with wheat and how many with rye?
513. 1) Tram route has a length of 14 3/8 km. 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) Venue rectangular shape 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 average arithmetic 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 you add up the numbers expressing the width of Tatarsky and the width 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) Amount three numbers 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) 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 area Severnaya Zemlya and 5 1/5 thousand sq. 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 more than the first. Find these numbers.
532. 1) The difference of two numbers is 7; quotient of division more for less 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) Father older than my son for 24 years. 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 vehicles 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 you open both taps 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. In how many hours would the second tractor, working alone, 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 collaboration 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 district center, two collective farmers rode out to the area on horseback at the same time. 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) Powerboat walked 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. morning a freight train left Kyiv for Odessa with average speed 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 mail train leave Odessa, if the distance between Kiev and Odessa is 663 km?
557*. The clock shows noon. After how long does the clock and minute hand will they match?
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. General fee Rye amounted to 5/6 of the total wheat harvest 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 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 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 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 condition previous task, find out if 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 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 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 is 6 1/2 m long, 4 m wide and 2 m high. 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 the arc contain? equal to half circles? 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.
