How to solve common fractions. Addition and subtraction of algebraic fractions: rules, examples. How to subtract a natural number from an ordinary fraction
To express a part in fractions of a whole, you need to divide the part into a whole.
Objective 1. There are 30 students in the class, four are missing. How many students are missing?
Solution:
Answer: there are no students in the class.
Finding the fraction of a number
To solve problems in which it is required to find a part of the whole, the following rule is true:
If part of the whole is expressed as a fraction, then to find this part, you can divide the whole by the denominator of the fraction and multiply the result by its numerator.
Objective 1. There were 600 rubles, this amount was spent. How much money did you spend?
Solution: to find from 600 rubles, you need to divide this amount into 4 parts, thereby we find out how much money is one fourth:
600: 4 = 150 (p.)
Answer: spent 150 rubles.
Objective 2. There were 1000 rubles, this amount was spent. How much money was spent?
Solution: from the condition of the problem, we know that 1000 rubles consists of five equal parts. First, we find how many rubles are one-fifth of 1000, and then find out how many rubles are two-fifths:
1) 1000: 5 = 200 (p.) - one fifth.
2) 200 2 = 400 (p.) - two-fifths.
These two actions can be combined: 1000: 5 2 = 400 (p.).
Answer: 400 rubles were spent.
The second way to find a part of a whole:
To find a part of a whole, you can multiply the whole by the fraction that expresses that part of the whole.
Objective 3. According to the charter of the cooperative, for the reporting meeting to be valid, at least members of the organization must be present at it. The cooperative has 120 members. Under what composition can the reporting meeting be held?
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 it is required to find the whole by its part, the following rule is true:
If a part of the desired integer is expressed as a fraction, then to find this whole, you can divide this part by the numerator of the fraction and multiply the result by its denominator.
Objective 1. We spent 50 rubles, which was equal to the original amount. Find the original amount of money.
Solution: from the description of the problem, we see that 50 rubles is 6 times less than the initial amount, that is, the initial amount is 6 times more than 50 rubles. To find this amount, you need to multiply 50 by 6:
50 6 = 300 (p.)
Answer: the initial amount is 300 rubles.
Objective 2. We spent 600 rubles, which was equal to the initial amount of money. Find the original amount.
Solution: we will assume that the required number consists of three third parts. By condition, two-thirds of the number is equal to 600 rubles. First, we find one third of the original amount, and then how many rubles are three thirds (original amount):
1) 600: 2 3 = 900 (p.)
Answer: the initial amount is 900 rubles.
The second way to find the whole by its part:
To find a whole by the value of the part that expresses it, you can divide this value by the fraction that expresses this part.
Objective 3. Section AB equal to 42 cm is the length of the segment CD... Find the length of a line segment CD.
Solution: 
Answer: segment length CD 70 cm.
Task 4. They brought watermelons to the store. Before lunch the store sold, after lunch - the brought watermelons, and it remains to sell 80 watermelons. How many watermelons were brought to the store in total?
Solution: first, we find out what part of the brought watermelons is the number 80. To do this, let us take the total number of brought watermelons as a unit and subtract from it the number of watermelons that we managed to sell (sell):
And so, we learned that 80 watermelons make up the total number of imported watermelons. Now we find out how many watermelons of the total amount are, and then how many watermelons are (the number of brought watermelons):
2) 80: 4 15 = 300 (watermelons)
Answer: in total, 300 watermelons were brought to the store.
Here we will figure out how subtraction of fractions... First, we get the rule for subtracting fractions with the same denominators. Next, we will consider the subtraction of fractions with different denominators and give examples of subtraction with detailed solutions. After that, we will focus on subtracting a fraction from a natural number and subtracting a number from a fraction. In conclusion, let us show how ordinary fractions are subtracted using the properties of this action.
Immediately, we note that in this article we will only talk about subtracting a smaller fraction from a larger fraction. Other cases are discussed in the article subtraction of rational numbers.
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Subtracting fractions with the same denominator
To begin with, let's give an example that will allow us to figure out how subtraction of fractions with the same denominator.
Suppose there were five eighths of an apple on the plate, that is, 5/8 of an apple, after which two eighths were taken away. In the sense of subtraction (see general understanding of subtraction), the specified action is described as follows:. It is clear that this leaves 5−2 = 3 eighths of an apple on the plate. That is, .
The considered example illustrates rule for subtracting fractions with the same denominator: when subtracting fractions with the same denominator, the numerator of the subtracted item is subtracted from the numerator of the item to be subtracted, and the denominator remains the same.
The voiced rule using letters is written as follows: 
... This formula will be used when subtracting fractions with the same denominators.
Consider examples of subtraction of fractions with the same denominator.
Example.
Subtract the fraction 17/15 from the fraction 24/15.
Solution.
The denominators of the subtracted fractions are equal. The numerator of the reduced is 24, and the numerator of the subtracted is 17, their difference is 7 (24−17 = 7, if necessary, see the subtraction of natural numbers). Therefore, subtracting fractions with the same denominator 24/15 and 17/15 gives the fraction 7/15.
A short solution looks like this: 
.
Answer:
.
If possible, it is necessary to reduce the fraction and (or) isolate the whole part from the improper fraction, which is obtained by subtracting fractions with the same denominators.
Example.
Calculate the difference.
Solution.
Let's use the formula for subtracting fractions with the same denominators: 
.
Obviously, the numerator and denominator of the resulting fraction are divisible by 2 (see), that is, 22/12 is a cancellable fraction. Having reduced this fraction by 2, we arrive at the fraction 11/6.
Fraction 11/6 is incorrect (see correct and incorrect fractions). Therefore, it is necessary to select the whole part from it:.
So, the calculated difference of fractions with the same denominators is.
Here's the whole solution: 
.
Answer:
.
Subtracting fractions with different denominators
Subtracting fractions with different denominators is reduced to subtracting fractions with the same denominator. To do this, it is enough to bring fractions with different denominators to a common denominator.
So to spend subtraction of fractions with different denominators, necessary:
- bring fractions to a common denominator (usually fractions lead to the lowest common denominator);
- subtract the resulting fractions with the same denominators.
Consider examples of subtraction of fractions with different denominators.
Example.
Subtract 1/15 from the fraction 2/9.
Solution.
Since the denominators of the subtracted fractions are different, we first perform the reduction of the fractions to the lowest common denominator: since the LCM (9, 15) = 45, then the additional factor of the fraction 2/9 is the number 45: 9 = 5, and the additional factor of the fraction is 1/15 is the number 45: 15 = 3, then 
and 
.
It remains to subtract the fraction 3/45 from the fraction 10/45, we get 
, which gives us the desired difference of fractions with different denominators.
Briefly, the solution is written as follows: 
.
Answer:
Do not forget about reducing the fraction obtained after subtracting, as well as highlighting the whole part.
Example.
Subtract 7/36 from 19/9.
Solution.
After reducing fractions with different denominators to the lowest common denominator of 36, we have the fractions 76/9 and 7/36. We calculate their difference: 
.
The resulting fraction is reducible, after reducing it by 3, we get 23/12. And this fraction is incorrect, having separated the whole part from it, we have.
Let's put together all the actions performed when subtracting the original fractions with different denominators:.
Answer:
.
Subtracting a natural number from an ordinary fraction
Subtracting a natural number from a fraction can be reduced to subtraction of ordinary fractions. To do this, it is enough to represent a natural number as a fraction with a denominator of 1. Let's take a look at the example solution.
Example.
Subtract 3 from 83/21.
Solution.
Since the number 3 is equal to the fraction 3/1, then.
Answer:
However, it is more convenient to subtract a natural number from an improper fraction by presenting the fraction as a mixed number. Let's show the solution of the previous example in this way.
Subtracting a fraction from a natural number
Subtracting a fraction from a natural number can be reduced to subtraction of ordinary fractions by representing a natural number as a fraction. Let's analyze the solution of an example illustrating this approach.
Example.
Subtract the common fraction 5/3 from the natural number 7.
Solution.
Let's represent the number 7 as a fraction 7/1, and then perform the subtraction:.
By separating the whole part from the resulting fraction, we get the final answer.
Answer:
However, there is a more rational way to subtract a fraction from a natural number. Its advantages are especially noticeable when the reduced natural number and the denominator of the subtracted fraction are large numbers. All this will be seen from the examples below.
If the fraction to be subtracted is correct, then the reduced natural number can be replaced by the sum of two numbers, one of which is equal to one, subtract the correct fraction from one, and then complete the calculations.
Example.
Subtract the fraction 13/62 from the natural number 1,065.
Solution.
The subtracted fraction is correct. Replace the number 1 065 with the sum 1 064 + 1, and we get 
... It remains to calculate the value of the resulting expression (we will talk in more detail about the calculation of such expressions in).
Due to the properties of subtraction, the resulting expression can be rewritten as 
... We calculate the value of the difference in parentheses, replacing the unit with the fraction 1/1, we have 
... In this way, . This completes the subtraction of the fraction 13/62 from the natural number 1 065.
Here's the whole solution:
And now, for comparison, we will show what numbers we would have to work with if we decided to reduce the subtraction of the original numbers to the subtraction of fractions:
Answer:
.
If the fraction to be subtracted is incorrect, then it can be replaced with a mixed number, and then subtract the mixed number from the natural number.
Almost every fifth grader is in a little shock after the first acquaintance with ordinary fractions. Not only do you need to understand the essence of fractions, but you still have to perform arithmetic operations with them. After that, the little students will systematically interrogate their teacher, find out when these fractions will end.
To avoid such situations, it is enough just to explain this difficult topic to children as easily as possible, and preferably in a playful way.
The essence of the fraction
Before learning what a fraction is, the child should get acquainted with the concept share ... This is where the associative method works best.
Imagine a whole cake that has been divided into several equal parts, say four. Then each piece of cake can be called a share. If you take one of the four pieces of cake, then it will be one fourth of the share.
The shares are different, because the whole can be divided into a completely different number of parts. The more shares in general, the less they are, and vice versa.
So that the shares could be designated, they came up with such a mathematical concept as common fraction... The fraction will allow us to write as many fractions as needed.
The constituent parts of a fraction are the numerator and denominator, which are separated by a fractional slash or a slash. Many children do not understand their meaning, therefore the essence of the fraction is not clear to them. The fractional line denotes division, there is nothing complicated here.
It is customary to write the denominator below, under the fractional line or to the right of the inscribed line. It shows the number of fractions of a whole. The numerator, it is written above the fractional bar or to the left of the slash, determines how many fractions were taken, for example the fraction 4/7. In this case, 7 is the denominator, shows that there are only 7 parts, and the numerator 4 indicates that four parts were taken out of seven.
Major shares and their writing in fractions:
In addition to the usual, there is also a decimal fraction.
Actions with fractions grade 5
In the fifth grade, they learn to perform all arithmetic operations with fractions.
All actions with fractions are performed according to the rules, and you should not hope that without learning the rule everything will turn out by itself. Therefore, you should not neglect the oral part of your math homework.
We have already understood that the notation of the decimal and the ordinary fraction is different, therefore, arithmetic operations will be performed in different ways. Actions with ordinary fractions depend on those numbers that are in the denominator, and in the decimal, after the decimal point to the right.
For fractions that have the same denominator, the addition and subtraction algorithm is very simple. We perform actions only with numerators.
For fractions with different denominators, you need to find Least Common Denominator (LCN). This is the number that will be divisible by all denominators without a remainder, and will be the smallest of such numbers if there are several of them.
To add or subtract decimal fractions, you need to write them down in a column, comma under the comma, and equalize the number of decimal places if required.
To multiply common fractions, simply find the product of the numerators and denominators. A very simple rule.
Division is performed according to the following algorithm:
- Write the dividend unchanged
- Turn division into multiplication
- Reverse the divisor (write the reciprocal to the divisor)
- Perform multiplication
Addition of fractions, explanation
Let's take a closer look at how to add fractions and decimals.
As you can see in the image above, the fraction has one-third and two-thirds with a common denominator of three. This means that you only need to add the numerators one and two, and leave the denominator unchanged. The result is a sum of three-thirds. Such an answer, when the numerator and denominator of the fraction are equal, can be written as 1, since 3: 3 = 1.
It is required to find the sum of the fractions two thirds and two ninths. In this case, the denominators are different, 3 and 9. To perform the addition, you need to choose a common one. There is a very simple way. We choose the largest denominator, this is 9. Check if it is divisible by 3. Since 9: 3 = 3 without a remainder, therefore 9 is suitable as a common denominator.
The next step is to find additional factors for each numerator. To do this, the common denominator of 9 is divided alternately by the denominator of each fraction, the resulting numbers will be added. pl. For the first fraction: 9: 3 = 3, add 3 to the numerator of the first fraction. For the second fraction: 9: 9 = 1, you can leave off the unit, since multiplying by it you get the same number.
Now we multiply the numerators by their additional factors and add the results. The resulting sum is a fraction of eight ninths.
The addition of decimal fractions follows the same rule as the addition of natural numbers. In a column, the discharge is written under the discharge. The only difference is that in decimal fractions, you need to correctly put the comma in the result. To do this, the fractions are written with a comma under the comma, and in the total, you only need to move the comma down.
Find the sum of the fractions 38, 251 and 1, 56. To make it easier to perform the actions, we level the number of decimal places to the right by adding 0.
We add fractions without paying attention to the comma. And in the resulting amount, we simply lower the comma down. Answer: 39, 811.
Subtraction of fractions, explanation
To find the difference between the two-thirds and one-third fractions, you need to calculate the difference between the numerators 2-1 = 1, and leave the denominator unchanged. In the answer, we get a difference of one third.
Find the difference between the fractions five sixths and seven tenths. Find a common denominator. We use the selection method, out of 6 and 10 the largest is 10. Check: 10: 6 is not divisible without a remainder. We add 10 more, it turns out 20: 6, it also cannot be divided without a remainder. Increase by 10 again, we got 30: 6 = 5. The common denominator is 30. The same NOZ can be found in the multiplication table.
Find additional factors. 30: 6 = 5 - for the first fraction. 30:10 = 3 - for the second. We multiply the numerators and their additional multiplies. We get a decremented 25/30 and a subtracted 21/30. Next, we subtract the numerators, and leave the denominator unchanged.
The result is a difference of 4/30. Fraction contractible. Divide it by 2. The answer is 2/15.
Division of decimal fractions grade 5
This topic discusses two options:
Decimal multiplication grade 5
Remember how you multiply natural numbers in exactly the same way and find the product of decimal fractions. First, let's figure out how to multiply a decimal fraction by a natural number. For this:
When multiplying a decimal fraction by a decimal, we act in the same way.
Mixed fractions grade 5
Five-graders like to call such fractions not mixed, but<<смешные>> it's probably easier to remember. Mixed fractions are called so because they are obtained by combining an integer and an ordinary fraction.
A mixed fraction consists of an integer and a fractional part.
When reading such fractions, the integer part is first called, then the fractional: one whole two thirds, two whole one fifth, three whole two fifths, four whole three quarters.
How do they come out, these mixed fractions? It's pretty simple. When we receive an incorrect fraction in our answer (a fraction with a numerator greater than the denominator), we must always convert it to a mixed one. It is enough to divide the numerator by the denominator. This action is called highlighting the whole part:
Converting a mixed fraction back to a wrong one is also easy:
Examples with decimal fractions grade 5 with explanation
Many questions in children are caused by examples of several actions. Let's look at a couple of such examples.
(0.4 8.25 - 2.025): 0.5 =
The first step is to find the product of the numbers 8.25 and 0.4. We carry out multiplication according to the rule. In the answer, we count three characters from right to left and put a comma.
The second action is in the same parentheses, this is the difference. Subtract 2.025 from 3.300. We write down the action in a column, comma under the comma.
The third action is division. Divide the resulting difference in the second step by 0.5. The comma is wrapped one character. The result is 2.55.
Answer: 2.55.
(0, 93 + 0, 07) : (0, 93 — 0, 805) =
The first action is the sum in brackets. We add it in a column, remember that the comma is under the comma. We get the answer 1.00.
The second action is the difference from the second parenthesis. Since the decrement has fewer decimal places than the subtracted one, we add the missing one. Subtraction result is 0, 125.
The third action is to divide the sum by the difference. The comma wraps three characters. It turned out to be a division of 1000 by 125.
Answer: 8.
Examples with common fractions with different denominators Grade 5 with explanation
In the first For example, we find the sum of the fractions 5/8 and 3/7. The common denominator will be the number 56. Find additional multipliers, divide 56: 8 = 7 and 56: 7 = 8. Add them to the first and second fractions, respectively. We multiply the numerators and their factors, we get the sum of fractions 35/56 and 24/56. Received the amount 59/56. The fraction is incorrect, we convert it to a mixed number. The rest of the examples are solved in a similar way.
Examples with fractions grade 5 for training
For convenience, convert mixed fractions to incorrect ones and follow the steps.
How to teach your child to solve fractions easily with Lego
With the help of such a constructor, you can not only develop a child's imagination well, but also explain clearly in a playful way what a fraction and a fraction are.
The picture below shows that one piece with eight circles is a whole. So, taking a puzzle with four circles, you get half, or 1/2. The picture clearly shows how to solve examples with Lego, if you count the circles on the details.
You can build turrets from a certain number of parts and label each one as in the picture below. For example, take a seven-piece turret. Each piece of the green constructor will be 1/7. If you add two more to one such part, you get 3/7. Visual explanation of the example 1/7 + 2/7 = 3/7.
To get A's in math, remember to learn the rules and practice them.
To add 2 fractions with the same denominators, it is necessary to add their numerators, and the denominatorsleave unchanged.Adding fractions, examples:
The general formula for adding ordinary fractions and subtracting fractions with the same denominator is:
Note! Check if you can reduce the fraction that you received by writing down the answer.
Adding fractions with different denominators.
The rules for adding fractions with different denominators:
- reduce fractions to the lowest common denominator (LCN). To do this, we find the smallest common multiple (LCM) of denominators;
- add the numerators of fractions, and leave the denominators unchanged;
- we reduce the fraction that we received;
- if you get an incorrect fraction, convert the improper fraction to a mixed fraction.
Examples of additions fractions with different denominators:
Addition of mixed numbers (mixed fractions).
The rules for adding mixed fractions:
- we bring the fractional parts of these numbers to the lowest common denominator (LCN);
- separately add whole parts and separately fractional parts, add up the results;
- if, when adding the fractional parts, we received an incorrect fraction, select the whole part from this fraction and add it to the resulting whole part;
- we reduce the resulting fraction.
Example additions mixed fraction:
Addition of decimal fractions.
When adding decimal fractions, the process is written in "column" (as usual column multiplication),so that the discharges of the same name are under each other without displacement. Commas are requiredwe align clearly under each other.
The rules for adding decimal fractions:
1. If necessary, equalize the number of decimal places. To do this, add zeros tothe required fraction.
2. We write down fractions so that the commas are under each other.
3. Add fractions without paying attention to the comma.
4. We put a comma in the sum under the commas, the fractions that we add.
Note! When the given decimal fractions have a different number of decimal places,then to the fraction with fewer decimal places we assign the required number of zeros, for the equation infractions are the number of decimal places.
Let's figure it out example... Find the sum of decimal fractions:
0,678 + 13,7 =
We equalize the number of decimal places in decimal fractions. Add 2 zeros to the right to the decimal fractions 13,7 .
0,678 + 13,700 =
We write down answer:
0,678 + 13,7 = 14,378
If addition of decimal fractions you have mastered it well enough, then the missing zeros can be added in the mind.
496. Find X, if:
497. 1) If you add 10 1/2 to 3/10 of the unknown number, you get 13 1/2. Find an unknown number.
2) If you subtract 10 1/2 from 7/10 of an unknown number, you get 15 2/5. Find an unknown number.
498 *. If you subtract 10 from 3/4 of the unknown number and multiply the resulting difference by 5, you get 100. Find the number.
499 *. If you increase the unknown number by 2/3 of it, you get 60. What is this number?
500 *. If you add the same amount to the unknown number, and even 20 1/3, you get 105 2/5. Find an unknown number.
501. 1) The yield of potatoes with a square-nest planting averages 150 centners per hectare, and with a conventional planting 3/5 of this amount. How much more potatoes can be harvested from an area of 15 hectares if potatoes are planted in a square-nest method?
2) An experienced worker made 18 parts in 1 hour, and an inexperienced worker made 2/3 of this amount. How many more parts will an experienced worker make 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 harvested, on the second - one and a half times more, and on the third day - the rest of the grain. How many kilograms of seeds did the pioneers collect on the third day?
2) When grinding wheat, it turned out: flour 4/5 of the total amount of wheat, semolina - 40 times less than flour, and the rest is bran. How much flour, semolina and bran separately did you get when grinding 3 tons of wheat?
503. 1) There are 460 cars in three garages. The number of cars in the first garage is 3/4 of the cars in the second, and in the third garage there are 1 1/2 times more cars than in the first. How many cars fit in each garage?
2) The plant, which has three workshops, employs 6,000 workers. In the second workshop, 1 1/2 times less work 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 the tank with kerosene, and after that 8 tons of kerosene remained in the tank. How much kerosene was in the tank initially?
2) Cyclists raced for three days. On the first day, they covered 4/15 of the entire way, on the second - 2/5, and on the third day the remaining 100 km. Which way did the cyclists go in three days?
505. 1) The icebreaker made its way through the ice field for three days. On the first day he covered 1/2 of the entire path, on the second day 3/5 of the remaining path, 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 were planting trees for landscaping the village. The first squadron planted 7/20 of all trees, the second 5/8 of the remaining trees, and the third the remaining 195 trees. How many trees have the three squads planted?
506. 1) The combine harvested wheat from one plot in three days. On the first day, he harvested 5/18 of the total area of the plot, on the second day from 7/13 of the remaining area and on the third day from the remaining area of 30 1/2 hectares. On average, 20 centners of wheat were harvested from each hectare. How much wheat was harvested in the entire plot?
2) Participants of the rally on the first day covered 3/11 of the entire path, on the second day 7/20 of the remaining path, on the third day 5/13 of the new remainder, and on the fourth day, the remaining 320 km. How long is the rally route?
507. 1) The car passed on the first day 3/8 of the entire path, on the second 15/17 of the one that passed on the first, and on the third day the remaining 200 km. How much gasoline was consumed if the car consumes 1 3/5 kg of gasoline per 10 km of travel?
2) The city consists of four districts. And in the first district 4/13 of all residents of the city live, in the second 5/6 of the inhabitants of the first district, in the third 4/11 of the inhabitants of the first; two districts combined, and the fourth district is home to 18 thousand people. How much bread does the entire population of the city need for 3 days, if on average one person consumes 500 g per day?
508. 1) The tourist walked on the first day 10/31 of the entire path, on the second 9/10 of the one that passed on the first day, and on the third the rest of the way, and on the third day he covered 12 km more than on the second day. How many kilometers did the tourist walk on each of the three days?
2) The car traveled all the way from city A to city B in three days. On the first day, the car covered 7/20 of the total distance, on the second day, 8/13 of the remaining distance, and on the third day, the car covered 72 km less than on the first day. What is the distance between cities A and B?
509. 1) The Executive Committee allotted land to the workers of three factories for garden plots. The first plant was 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 plots were allotted to the workers of the three factories, if the first factory was allotted 50 less plots than the third?
2) The plane delivered a change of winterers to the polar station from Moscow in three days. On the first day he flew 2/5 of the entire path, on the second - 5/6 of the path he covered in 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 shop is 2/5 of all workers in the plant; in the second shop there are 1 1/2 times less workers than in the first, and in the third shop 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 of the collective farm; in the second village the number of families is 1 1/2 times greater than in the first, and in the third village the number of families is 420 less than in the second. How many families are there in the collective farm?
511. 1) The Artel used up in the first week 1/3 of its stock of raw materials, and in the second 1/3 of the remainder. How much raw material remained in the artel, if in the first week the consumption of raw materials was 3/5 tons more than in the second week?
2) From the imported coal for heating the house in the first month, 1/6 of it was spent, and in the second month - 3/8 of the remainder. How much coal is left for heating the house, if in the second month it was used up 1 3/4 more than in the first month?
512. 3/5 of the entire land of the collective farm is allotted for sowing grain, 13/36 of the remainder is occupied by vegetable gardens and meadow, the rest of the land is forest, and the sown area of the collective farm is 217 hectares more than the forest area, 1/3 of the land allotted for sowing grain is sown with rye, and the rest is wheat. How many hectares of land did the collective farm sow with wheat and how many 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 make one trip?
2) The bus route is 16 km. During this route, the bus makes 36 stops of 3/4 min. on average each. The average bus speed is 30 km per hour. How long does it take for a bus to take one route?
514 *. 1) It's 6 o'clock now. evenings. What part is the rest of the day from the past and what part of the day is left?
2) A steamer downstream covers the distance between two cities in 3 days. and back the same distance in 4 days. How many days will the rafts float from one city to another?
515. 1) How many boards will be used for flooring in a room that is 6 2/3 m long and 5 1/4 m wide, if the length of each board is 6 2/3 m and its width is 3/80 of the length?
2) A rectangular platform has a length of 45 1/2 m, and its width is 5/13 of the length. This area is bordered by a 4/5 m wide track. Find the area of the track.
516. Find the arithmetic mean of numbers:
517. 1) The arithmetic mean of two numbers 6 1/6. One of the numbers 3 3/4. Find another number.
2) The arithmetic mean of two numbers 14 1/4. One of these numbers is 15 5/6. Find another number.
518. 1) The freight train was on the way 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 covered 81 1/2 km in the first two hours, and 95 km in the next 2 1/2 hours. How many kilometers did he walk on average per hour?
519. 1) The tractor driver completed the task of plowing the land in three days. On the first day he plowed 12 1/2 ha, on the second day 15 3/4 ha and on the third day 14 1/2 ha. On average, how many hectares of land have a tractor driver plowed in a day?
2) A detachment of schoolchildren, making a three-day tourist trip, was on the way on the first day 6 1/3 hours, on the second 7 hours. and on the third day - 4 2/3 hours. How many hours, on average, did the schoolchildren travel every day?
520. 1) Three families live in the house. The first family has 3 electric bulbs for lighting the apartment, the second 4 and the third 5 bulbs. How much should each family pay for electricity if all lamps were the same, and the total bill (for the whole house) for paying for electricity was 7 1/5 rubles?
2) A scrubber was scrubbing the floors of an apartment where three families lived. The first family had a living area of 36 1/2 sq. m, the second in 24 1/2 sq. m, and the third - 43 sq. m. For all the work was paid 2 rubles. 08 kopecks How much did each family pay?
521. 1) On the garden plot, potatoes were collected from 50 bushes of 1 1/10 kg per one bush, from 70 bushes of 4/5 kg per one bush, from 80 bushes of 9/10 kg per one bush. How many kilograms of potatoes are harvested on average from each bush?
2) A field-cultivation team on an area of 300 hectares received a crop of 20 1/2 centners of winter wheat per hectare, from 80 hectares to 24 centners per hectare, and from 20 hectares - 28 1/2 centners per hectare. What is the average yield per 1 ha brigade?
522. 1) The sum of two numbers is 7 1/2. One number is 4 4/5 more than the other. Find these numbers.
2) If we add the numbers expressing the width of the Tatar and the width of the Kerch straits together, we get 11 7/10 km. The Tatar Strait is 3 1/10 km wider than the Kerch Strait. What is the width of each strait?
523. 1) The sum of three numbers is 35 2/3. The first number is 5 1/3 more than the second and 3 5/6 more than the third. 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 sq. km more than the area of Severnaya Zemlya and 5 1/5 thousand square meters. km more than the area of Sakhalin. What is the area of each of the listed islands?
524. 1) The apartment consists of three rooms. The area of the first room is 24 3/8 sq. m and is 13/36 of the entire area of the apartment. The area of the second room is 8 1/8 sq. m more than the area of the third. What is the area of the second room?
2) The cyclist traveled 3 1/4 hours during the three-day competition on the first day, 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. The three pieces of iron together weigh 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 cereals from one box into the other, then the cereals will be equal in both boxes. How many cereals are in each box?
527 ... 1) The sum of the two numbers is 17 17/30. If you subtract 5 1/2 from the first number and add to the second, then the first will still be 2 17/30 more than the second. Find both numbers.
2) There are 24 1/4 kg 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 the difference is 2 3/7. Find these numbers.
2) The boat went downstream of the river at a speed of 15 1/2 km per hour, and against the current 8 1/4 km per hour. What is the speed of the river?
529. 1) There are 110 cars in two garages, and in one of them there are 1 1/5 times more than in the other. How many cars are in each garage?
2) The living area of an apartment consisting of two rooms is 47 1/2 sq. m. The area of one room is 8/11 of the area of another. 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 is in the alloy and how much copper?
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; quotient of dividing a larger number by a smaller number 5 2/3. Find these numbers.
2) The difference of two numbers is 29 3/8, and their multiple ratio is equal to 8 5/6. Find these numbers.
533. In the class, the number of absent students is equal to 3/13 of the number of those present. How many students are in the class on the list, if there are 20 more people present than absent?
534. 1) The difference of two numbers is 3 1/5. One number is 5/7 of the other. Find these numbers.
2) The father is 24 years older than his son. The number of years of the son is equal to 5/13 of the number of years of the father. How old is the father and how old is the son?
535. The denominator of the fraction is 11 units greater than its numerator. What is 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. 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 gathered 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, if two-fifths of all men are equal to three-fifths 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 greater than the other by 6. Find these numbers if 2/5 of one number is 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.
542. 1) The first team can do some work in 36 days, and the second in 45 days. How many days will both teams, working together, complete this work?
2) A passenger train travels the distance between two cities in 10 hours, and a freight train travels this distance in 15 hours. Both trains left these cities at the same time to meet 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 takes 7 1/2 hours. In how many hours will these trains meet if they leave both cities at the same time towards each other? (Round off the answer to the nearest 1 hour.)
2) Two motorcyclists left two cities at the same time towards each other. One motorcyclist can travel the entire distance between these cities in 6 hours, and another in 5 hours. How many hours after check-out will the motorcyclists meet? (Round off the answer to the nearest 1 hour.)
544. 1) Three vehicles of different carrying capacity can carry some load, working separately: the first in 10 hours, the second in 12 hours. and the third in 15 hours. How many hours can they transport the same cargo, working together?
2) Two trains leave two stations at the same time towards each other: the first train covers the distance between these stations in 12 1/2 hours, and the second in 18 3/4 hours. How many hours after leaving will the trains meet?
545. 1) Two taps are connected to the bath. Through one of them, the bath can be filled in 12 minutes, through the other 1 1/2 times faster. How many minutes will it take to fill 5/6 of the entire bath if you open both taps at once?
2) Two typists must retype the manuscript. The first ashinist can do this job in 3 1/3 days, and the second one 1 1/2 times faster. What days will both typists complete the work if they work at the same time?
546. 1) The pool is filled with the first pipe in 5 hours, and through the second pipe it can be emptied in 6 hours How many hours will the entire pool be filled if both pipes are opened at the same time?
Indication. 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 the second tractor plow this field, working alone?
547 *. Two trains leave the two stations at the same time towards each other and meet each other 18 hours later. after its release. How long does it take for the second train to cover the distance between stations, if the first train covers this distance in 1 day and 21 hours?
548 *. The pool is filled with two pipes. First, the first pipe was opened, and then after 3 3/4 hours, when half of the pool was full, the second pipe was opened. After 2 1/2 hours of working together, the pool was full. Determine the capacity of the pool if 200 buckets of water were poured in through the second pipe per hour.
549. 1) A courier train left Leningrad for Moscow, which covers 1 km in 3/4 minutes. 1/2 hour after the departure of this train from Moscow to Leningrad, a fast train left, the speed of which was equal to 3/4 the speed of the courier. How far 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 left the collective farm, which covers 1 km in 2 1/2 minutes. After 15 minutes. after this car left the city, a cyclist left for the collective farm, at a speed half that of a truck. How long does it take for a cyclist to meet a truck after leaving?
550. 1) A pedestrian came out of one village. 4 1/2 hours after the pedestrian exited, a cyclist left in the same direction, whose speed was 2 1/2 times higher than the pedestrian's speed. How many hours after the pedestrian leaves the cyclist?
2) A fast train travels 187 1/2 km in 3 hours, and a freight train 288 km in 6 hours. 7 1/4 hours after the exit of the freight train, an express train departs in the same direction. How long will it take for a fast train to catch up with a freight train?
551. 1) From the two collective farms, through which the road to the regional center passes, two collective farmers left 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 in 3 4/5 hours. Determine the distance between collective farms.
2) 26 1/3 hours after the departure of the Moscow-Vladivostok train, the average speed of which is 60 km per hour, a TU-104 plane departed in the same direction, at a speed 14 1/6 times higher than the train speed. 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 on stops. The speed of the river is 1 1/2 km per hour. How long would it take for a steamer to pass 87 km in still water without stopping?
2) The motor boat covered 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 on the reservoir covered a distance of 52 km 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 equal in time. How many minutes did the boat stop at each stop?
554. From Leningrad to Kronstadt at 12 o'clock. a day the steamer left and covered the entire distance between these cities in 1 1/2 hours. On the way, he met another steamer that left Kronstadt for Leningrad at 12 hours 18 minutes. and walking at a speed 1 1/4 times faster than the first. At what time did the meeting of both ships take place?
555. The train was supposed to travel a distance of 630 km in 14 hours. Having covered 2/3 of this distance, he was detained for 1 hour and 10 minutes. How fast should he continue on his way to arrive at his destination without delay?
556. At 4 hours 20 minutes. In the morning, a freight train left Kiev for Odessa with an average speed of 31 1/5 km per hour. After a while, a mail train left Odessa to meet him, the speed of which is 1 17/39 Times more than the speed of a freight train, and met with a freight train 6 1/2 hours after it left. At what time did the mail 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 shop is 9/20 of all workers of the plant, in the second shop there are 1 1/2 times less workers than in the first, and in the third shop there are 300 workers less 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 combiners worked on the same site. After one combine harvester harvested 9/16 of the entire plot, and the second 3/8 of the same plot, it turned out that the first harvester harvested 97 1/2 ha more than the second. On average, 32 1/2 centners of grain were threshed from each hectare. How many centners of grain did each combine harvester?
2) Two brothers bought a camera. One had 5/8, and the second had 4/7 of the cost of the camera, and the first had 2 rubles. 25 kopecks more than the second. Each paid half the cost of the apparatus. How much money does everyone have left?
560. 1) From city A to city B, the distance between which is 215 km, a passenger car left at a speed of 50 km per hour. Simultaneously with him, a truck left town B for town A. How many kilometers did a passenger car travel before meeting a truck, if the truck's speed per hour was 18/25 of the speed of a passenger car?
2) Between cities A and B 210 km. A passenger car left town A for town B. Simultaneously with her, a truck left town B for town A. How many kilometers did a truck travel before meeting a passenger car, if a passenger car was traveling at a speed of 48 km per hour, and the speed of a truck per hour was 3/4 of the speed of a passenger car?
561. The collective farm has harvested wheat and rye. Wheat was sown 20 hectares more than rye. The total harvest of rye was 5/6 of the total harvest of wheat with a yield of 20 centners per hectare for both wheat and rye. The collective farm sold 7/11 of the total harvest of wheat and rye to the state, and left the rest of the grain to meet its needs. How many trips did two-ton trucks take to transport grain sold to the state?
562. Rye and wheat flour was brought to the bakery. The weight of wheat flour was 3/5 of the weight of rye flour, and rye flour was imported 4 tons more than wheat flour. How much wheat and how much rye bread will be baked by the bakery from this flour, if the baking is 2/5 of all flour?
563. Within three days, a team of workers completed 3/4 of the entire work on the repair of the highway between the two collective farms. On the first day, 2 2/5 km of this highway was repaired, on the second day 1 1/2 times more than on the first, and on the third day 5/8 of what was repaired in the first two days together. Find the length of the highway between collective farms.
564. Fill in empty spaces in the table, where S is the area of the rectangle, a is 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 lot.
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 lot.
566. 1) The perimeter of the rectangle is 6 1/2 dm, its base is 1/4 dm greater than the 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 a 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 upholster the ceiling of a room that is 4 1/2 m long and 4 m wide if the dimensions of the plaster sheet are 2 mx l 1/2 m?
2) How many boards 4 1/2 L long and 1/4 m wide will be required for a floor that is 4 1/2 m long and 3 1/2 m wide?
569. 1) A plot of rectangular shape with a length of 560 m, and a width of 3/4 of its length, was sown with beans. How many seeds did it take to sow the plot, if 1 centner was sown on 1 hectare?
2) Wheat was harvested from a rectangular field of 25 centners per hectare. How much wheat was harvested from the whole field, if the field is 800 m long and the width is equal to 3/8 of its length?
570 ... 1) A rectangular piece 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 piece of land, the length of which is 9/20 km, and the width is 4/9 of its length, the collective farm intends to set up a garden. How many trees will be planted in this garden if an average of 36 square meters of space 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 if there is enough light in a room that is 5 1/2 m long and 4 m wide. Does the room have one 1 1/2 mx 2 m window?
2) Using the condition of the previous problem, find out if there is enough light in your class.
572. 1) The barn has dimensions of 5 1/2 mx 4 1/2 mx 2 1/2 m m of hay weighs 82 kg?
2) The woodpile of firewood has the shape of a rectangular parallelepiped, the dimensions of which are 2 1/2 mx 3 1/2 mx 1 1/2 m. What is the weight of the woodpile if 1 cubic meter. m of firewood weighs 600 kg?
573. 1) The rectangular aquarium is filled with water up to 3/5 of the height. The aquarium is 1 1/2 m long, 4/5 m wide, 3/4 m high. How many liters of water are in the aquarium?
2) The pool, which has 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 should be built around a rectangular piece of land that is 75 m long and 45 m wide. How many cubic meters of boards should go to his device if the thickness of the board is 2 1/2 cm, and the height of the fence should be 2 1/4 m?
575. 1) What is the angle of the minute and hour hands at 13 o'clock? at 15 o'clock? at 17 o'clock? at 21 o'clock? at 23 hours 30 minutes?
2) How many degrees will the hour hand turn in 2 hours? 5 hour? 8 o'clock? 30 minutes.?
3) How many degrees does an arc equal to half a circle contain? 1/4 circle? 1/24 circle? 5/24 circle?
576. 1) Draw with a protractor: a) a right angle; b) an angle of 30 °; c) an angle of 60 °; d) an angle of 150 °; e) an angle of 55 °.
2) Measure the corners of the figure with a protractor and find the sum of all the corners of each figure (fig. 31).
577. Follow the 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 the size of 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 "Literacy of the population in the USSR" (Fig. 32) shows the number of literate per hundred people of the population. According to the diagram and its scale, determine the number of literate men and women for each of the indicated years.
Record the results in the table:
2) Using the data from the diagram "Soviet Envoys to Space" (Fig. 33), draw up tasks.
580. 1) According to the pie chart "Mode of the day for a student of grade V" (Fig. 34), fill in the table and answer the questions: what part of the day is spent on sleep? homework? to school?
2) Build a pie chart about your daily routine.
