The rule for performing mathematical order. Lesson "order of actions"
This lesson discusses in detail the procedure for performing arithmetic operations in expressions without parentheses and with brackets. Students are given the opportunity, while completing assignments, to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations is different in expressions without parentheses and with parentheses, to practice applying the learned rule, to find and correct errors made when determining the order of actions.
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in in different order. Sometimes they can be swapped, sometimes not. For example, when getting ready for school in the morning, you can first do exercises, then make your bed, or vice versa. But you can’t go to school first and then put on clothes.
In mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.
We see that the meanings of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed.
Let's learn the rule for performing arithmetic operations in expressions without parentheses.
If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression contains only addition and subtraction operations. These actions are called first stage actions.
We perform the actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
This expression contains only multiplication and division operations - These are the actions of the second stage.
We perform the actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If an expression without parentheses includes not only the operations of addition and subtraction, but also multiplication and division, or both of these operations, then first perform in order (from left to right) multiplication and division, and then addition and subtraction.
Let's look at the expression.
Let's think like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's arrange the order of actions.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
In what order are arithmetic operations performed if there are parentheses in an expression?
If an expression contains parentheses, the value of the expressions in the parentheses is evaluated first.
Let's look at the expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
How should one reason to correctly establish the order of arithmetic operations in a numerical expression?
Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule(Fig. 4).
Rice. 4. Procedure
Let's practice.
Let's consider the expressions, establish the order of actions and perform calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
We will act according to the rule. The expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish a procedure. The first action is to perform the operation in parentheses, and then, in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) contains operations in parentheses, as well as multiplication and addition operations. According to the rule, we first perform the action in parentheses, then multiplication (we multiply the number 9 by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no parentheses, but there are multiplication, division and subtraction operations. We act according to the rule. First, we perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplication. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out whether the order of actions in the following expressions is correctly defined.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
Let's think like this.
37 + 9 - 6: 2 * 3 =
There are no parentheses in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the procedure is determined correctly.
Let's find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
Let's continue to talk.
The second expression contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is division, the third is addition. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the value of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. Let's check: the first action is in parentheses, the second is multiplication, the third is subtraction. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the value of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the learned rule (Fig. 5).
Rice. 5. Procedure
We don't see numerical values, so we won't be able to find the meaning of expressions, but we'll practice applying the rule we've learned.
We act according to the algorithm.
The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
Today in class we learned about the rule for the order of actions in expressions without and with brackets.
Bibliography
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Math lessons: Guidelines for the teacher. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.
- Festival.1september.ru ().
- Sosnovoborsk-soobchestva.ru ().
- Openclass.ru ().
Homework
1. Determine the order of actions in these expressions. Find the meaning of the expressions.
2. Determine in what expression this order of actions is performed:
1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the meaning of this expression.
3. Make up three expressions in which the following order of actions is performed:
1. multiplication; 2. addition; 3. subtraction
1. addition; 2. subtraction; 3. addition
1. multiplication; 2. division; 3. addition
Find the meaning of these expressions.
In the fifth century BC ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia “Achilles and the Tortoise.” Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “screw me, I’m in the house”, or rather “mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Apply mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to assure us that the banknotes of the same denomination have different numbers bills, which means they cannot be considered identical elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically recall physics: on different coins There is a different amount of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: minus sign, number four, designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
Rules for the order of performing actions in complex expressions are studied in 2nd grade, but children practically use some of them in 1st grade.
First, we consider the rule about the order of operations in expressions without parentheses, when numbers are performed either only addition and subtraction, or only multiplication and division. The need to introduce expressions containing two or more arithmetic operations of the same level arises when students become familiar with the computational techniques of addition and subtraction within 10, namely:
Similarly: 6 - 1 - 1, 6 - 2 - 1, 6 - 2 - 2.
Since to find the meanings of these expressions, schoolchildren turn to objective actions that are performed in a certain order, they easily learn the fact that arithmetic operations (addition and subtraction) that take place in expressions are performed sequentially from left to right.
Students will first encounter number expressions containing addition and subtraction operations and parentheses in the topic "Addition and Subtraction within 10." When children encounter such expressions in 1st grade, for example: 7 - 2 + 4, 9 - 3 - 1, 4 +3 - 2; in 2nd grade, for example: 70 - 36 +10, 80 - 10 - 15, 32+18 - 17; 4*10:5, 60:10*3, 36:9*3, the teacher shows how to read and write such expressions and how to find their meaning (for example, 4*10:5 read: 4 multiply by 10 and divide the resulting result at 5). By the time they study the topic “Order of Actions” in 2nd grade, students are able to find the meanings of expressions of this type. The goal of the work at this stage is to rely on the practical skills of students, to draw their attention to the order of performing actions in such expressions and to formulate the corresponding rule. Students independently solve examples selected by the teacher and explain in what order they performed them; actions in each example. Then they formulate the conclusion themselves or read from a textbook: if in an expression without parentheses only the actions of addition and subtraction (or only the actions of multiplication and division) are indicated, then they are performed in the order in which they are written (i.e., from left to right).
Despite the fact that in expressions of the form a+b+c, a+(b+c) and (a+b)+c the presence of parentheses does not affect the order of actions due to the associative law of addition, at this stage it is more advisable to orient students to that the action in parentheses is performed first. This is due to the fact that for expressions of the form a - (b + c) and a - (b - c) such a generalization is unacceptable and for students initial stage It will be quite difficult to navigate the assignment of brackets for various numerical expressions. The use of parentheses in numerical expressions containing addition and subtraction operations is further developed, which is associated with the study of such rules as adding a sum to a number, a number to a sum, subtracting a sum from a number and a number from a sum. But when first introducing parentheses, it is important to direct students to do the action in the parentheses first.
The teacher draws the children's attention to how important it is to follow this rule when making calculations, otherwise you may get an incorrect equality. For example, students explain how the meanings of the expressions are obtained: 70 - 36 +10 = 24, 60:10 - 3 = 2, why they are incorrect, what meanings these expressions actually have. Similarly, they study the order of actions in expressions with brackets of the form: 65 - (26 - 14), 50: (30 - 20), 90: (2 * 5). Students are also familiar with such expressions and can read, write and calculate their meaning. Having explained the order of actions in several such expressions, children formulate a conclusion: in expressions with brackets, the first action is performed on the numbers written in brackets. Examining these expressions, it is not difficult to show that the actions in them are not performed in the order in which they are written; to show a different order of their execution, and parentheses are used.
The following introduces the rule for the order of execution of actions in expressions without parentheses, when they contain actions of the first and second stages. Since the rules of procedure are accepted by agreement, the teacher communicates them to the children or the students learn them from the textbook. To ensure that students understand the rules introduced, along with training exercises include solutions to examples with an explanation of the order of their actions. Exercises in explaining errors in the order of actions are also effective. For example, from the given pairs of examples, it is proposed to write down only those where the calculations were performed according to the rules of the order of actions:
After explaining the errors, you can give a task: using parentheses, change the order of actions so that the expression has the specified value. For example, in order for the first of the given expressions to have a value equal to 10, you need to write it like this: (20+30):5=10.
Exercises on calculating the value of an expression are especially useful when the student has to apply all the rules he has learned. For example, the expression 36:6+3*2 is written on the board or in notebooks. Students calculate its value. Then, according to the teacher’s instructions, the children use parentheses to change the order of actions in the expression:
- 36:6+3-2
- 36:(6+3-2)
- 36:(6+3)-2
- (36:6+3)-2
Interesting, but more difficult is reverse exercise: place parentheses so that the expression has the given value:
- 72-24:6+2=66
- 72-24:6+2=6
- 72-24:6+2=10
- 72-24:6+2=69
Also interesting are the following exercises:
- 1. Arrange the brackets so that the equalities are true:
- 25-17:4=2 3*6-4=6
- 24:8-2=4
- 2. Place “+” or “-” signs instead of asterisks so that you get the correct equalities:
- 38*3*7=34
- 38*3*7=28
- 38*3*7=42
- 38*3*7=48
- 3. Place arithmetic signs instead of asterisks so that the equalities are true:
- 12*6*2=4
- 12*6*2=70
- 12*6*2=24
- 12*6*2=9
- 12*6*2=0
By performing such exercises, students become convinced that the meaning of an expression can change if the order of actions is changed.
To master the rules of the order of actions, it is necessary in grades 3 and 4 to include increasingly complex expressions, when calculating the values of which the student would apply not one, but two or three rules of the order of actions each time, for example:
- 90*8- (240+170)+190,
- 469148-148*9+(30 100 - 26909).
In this case, the numbers should be selected so that they allow actions to be performed in any order, which creates conditions for the conscious application of the learned rules.
Order of actions - Mathematics 3rd grade (Moro)
Short description:
In life you constantly do various actions: get up, wash, do exercises, have breakfast, go to school. Do you think it is possible to change this procedure? For example, have breakfast and then wash your face. Probably possible. It may not be very convenient to have breakfast if you are unwashed, but nothing bad will happen because of this. In mathematics, is it possible to change the order of operations at your discretion? No, mathematics is an exact science, so even the slightest changes in the procedure will lead to the fact that the answer of the numerical expression will become incorrect. In second grade you have already become acquainted with some rules of procedure. So, you probably remember that the order in the execution of actions is governed by brackets. They show what actions need to be completed first. What other rules of procedure are there? Is the order of operations different in expressions with and without parentheses? You will find answers to these questions in the 3rd grade mathematics textbook when studying the topic “Order of actions.” You must definitely practice applying the rules you have learned, and if necessary, find and correct errors in establishing the order of actions in numerical expressions. Please remember that order is important in any business, but in mathematics it is especially important! 
Composing an Expression with Parentheses
1. Make up expressions with brackets from the following sentences and solve them.
From the number 16, subtract the sum of the numbers 8 and 6.
From the number 34, subtract the sum of the numbers 5 and 8.
Subtract the sum of the numbers 13 and 5 from the number 39.
The difference between the numbers 16 and 3 add to the number 36
Add the difference between 48 and 28 to 16.
2. Solve the problems by first composing the correct expressions, and then solving them sequentially:
2.1. Dad brought a bag of nuts from the forest. Kolya took 25 nuts from the bag and ate them. Then Masha took 18 nuts from the bag. Mom also took 15 nuts from the bag, but put 7 of them back. How many nuts are left in the bag in the end if there were 78 of them at the beginning?
2.2. The foreman was repairing parts. At the beginning of the workday there were 38 of them. In the first half of the day he was able to repair 23 of them. In the afternoon they brought him the same amount as they had at the very beginning of the day. In the second half, he repaired another 35 parts. How many parts does he have left to repair?
3. Solve the examples correctly following the sequence of actions:
45: 5 + 12 * 2 -21:3
56 - 72: 9 + 48: 6 * 3
7 + 5 * 4 - 12: 4
18: 3 - 5 + 6 * 8
Solving expressions with parentheses
1. Solve the examples by opening the brackets correctly:
1 + (4 + 8) = | 8 - (2 + 4) = | 3 + (6 - 5) = | 59 + 25 = |
82 + 14 = | 29 + 52 = | 18 + 47 = | 39 + 53 = |
37 + 53 = | 25 + 63 = | 87 + 17 = | 19 + 52 = |
2. Solve the examples correctly following the sequence of actions:
2.1. 36: 3 + 12 * (2 - 1) : 3
2.2. 39 - (81: 9 + 48: 6) * 2
2.3. (7 + 5) * 2 - 48: 4
2.4. 18: 3 + (5 * 6) : 2 - 4
3. Solve the problems by first composing the correct expressions, and then solving them sequentially:
3.1. There were 25 packages of washing powder in the warehouse. 12 packages were taken to one store. Then the same amount was taken to the second store. After that, 3 times more packages were brought to the warehouse than before. How many packages of powder are in stock?
3.2. There were 75 tourists staying at the hotel. On the first day, 3 groups of 12 people each left the hotel, and 2 groups of 15 people each arrived. On the second day, another 34 people left. How many tourists remained in the hotel at the end of 2 days?
3.3. They brought 2 bags of clothes to the dry cleaner, 5 items in each bag. Then they took 8 things. In the afternoon they brought 18 more items for washing. And they only took 5 washed items. How many items are in the dry cleaner at the end of the day if there were 14 items at the beginning of the day?
FI _________________________________
21: 3 * 6 - (18 + 14) : 8 = | 63: (81: 9) + (8 * 7 - 2) : 6 = | 64:2: 4+ 9*7-9*1= |
37 *2 + 180: 9 – 36: 12 = | 52 * 10 – 60: 15 * 1 = | 72: 4 +58:2= |
5 *0: 25 + (72: 1 – 0) : 9 = | 21: (3 * 7) – (7* 0 + 1)*1 = | 6:6+0:8-8:8= |
91: 7 + 80: 5 – 5: 5 = | 64:4 - 3*5 +80:2= | (19*5 – 5) : 30 = |
19 + 17 * 3 – 46 = | (39+29) : 4 + 8*0= | (60-5) : 5 +80: 5= |
54 – 26 + 38: 2 = | 63: (7*3) *3= | (160-70) : 18 *1= |
200 – 80: 5 + 3 * 4 = | (29+25): (72:8)= | 72:25 + 3* 17= |
80: 16 + 660: 6 = | 3 * 290 – 800= | 950:50*1-0= |
(48: 3) : 16 * 0 = | 90-6*6+29= | 5* (48-43) +15:5*7= |
54: 9 *8 - 14: 7 * 4 = | 63: 7*4+70:7 * 5= | 24: 6*7 - 7*0= |
21: 7 * 8 + 32: 8 * 4 = | 27: 3* 5 + 26-18 *4= | 54: 6*7 - 0:1= |
45: 9 * 6 + 7 * 5 – 26 = | 28: 7 *9 + 6 * (54 – 47)= | 6*(9: 3) - 40:5 = |
21 * 1 - 56: 7 – 8 = | 9 * (64: 8) - 18:18 | 3 *(14: 2) - 63:9= |
4 * 8 + 42: 6 *5 = | 0*4+0:5 +8* (48: 8)= | 56:7 +7*6 - 5*1= |
31 * 3 - 17 – 80: 16 * 1 = | 57:19 *32 - 11 *7= | 72-96:8 +60:15 *13= |
36 + 42: 3 + 23 + 27 *0 = | 56:14 *19 - 72:18= | (86-78:13)* 4= |
650 – 50 * 4 + 900: 100 = | 630: 9 + 120 * 5 + 40= | 980 – (160 + 20) : 30= |
940 - (1680 – 1600) * 9 = | 29* 2+26 – 37:2= | 72:3 +280: (14*5)= |
300: (5 *60) * (78: 13) = | 63+ 100: 4 – 8*0= | 84:7+70:14 – 6:6= |
45: 15 – 180: 90 + 84: 7 = | 32+51 + 48:6 * 5= | 54:6 ?2 – 70:14= |
38: 2 – 48: 3 + 0 * 9 = | 30:6 * 8 – 6+3*2= | (95:19) *(68:2)= |
(300 - 8 * 7) * 10 = | 1:1 - 0*0 + 1*0 - 1*1= | (80: 4 – 60:30) *5 = |
2 * (120: 6 – 80: 20) = | 56:4+96:3- 0*7= | 20+ 20: 4 - 1*5= |
(18 + 14) : 8 – (7 *0 + 1) *1 = | (8*7-2):6 +63: (7*3)= | (50-5) : 5+21: (3*7)= |
19 + 17 * 3 – 60: 15 * 1 = | 80: 5 +3*5 +80:2= | 54: 9 *8-64:4 +16*0= |
72 * 10 - 64: 2: 4 = | 84 – 36 + 38:2 | 91:13+80:5 – 5:5 |
300 – 80: 5 + 6 * 4 = | 950:190 *1+14: 7*4= | (39+29) : 17 + 8*0= |
(120 - 30) : 18 * 1- 72: 25 = | 210:30*60-0:1= | 90-6*7+3* 17= |
240: 60 *7 – 7 * 0 = | 60:60+0:80-80:80= | 720: 40 +580:20= |
9 *7 – 9 *1 + 5 * 0: 25 = | 21: 7 * 6 +32: 4 *5= | 80:16 +66:6 -63:(81:9)= |
(19 * 5 – 5) : 30 + 70: 7 = | 15:5*7 + 63: 7 * 5= | 54: 6 * 7 - (72:1-0):9= |
3 *290 – 600 – 5 * (48 – 43) = | (300-89*7)*10 - 3?2= | (80: 4) +30*2+ 180: 9= |
30: 6 * 8 – 6 + 48: 3 + 0 *9 = | (95:19) *(68:34) - 60:30*5= | 27: 3*5 - 48:3= |
3* 290 – 800 + 950: 50 = | 80:16 +660:6*1-0= | 90-6*6+ 15:5*7= |
5*(48 - 43) + (48: 3) :16*0= | 280: (14*5) +630: 9*0= | 300: (50*6)* (78: 6)= |
If there is a question mark (?) in the examples, it should be replaced with the sign * - multiplication.
1. SOLVE EXPRESSIONS:
35: 5 + 36: 4 - 3
26 + 6 x 8 – 45: 5 24: 6 + 18 – 2 x 6
9 x 6 – 3 x 6 + 19 – 27:3
2. SOLVE EXPRESSIONS:
48: 8 + 32 – 54: 6 + 7 x 4
17 + 24: 3 x 4 – 27: 3 x 2 6 x 4: 3 + 54: 6: 3 x 6 + 2 x 9
100 – 6 x 2: 3 x 9 – 39 + 7 x 4
3. SOLVE EXPRESSIONS:
100 – 27: 3 x 6 + 7 x 4
2 x 4 + 24: 3 + 18: 6 x 9 9 x 3 – 19 + 6 x 7 – 3 x 5
7 x 4 + 35: 7 x 5 – 16: 2: 4 x 3
4. SOLVE EXPRESSIONS:
32: 8 x 6: 3 + 6 x 8 – 17
5 x 8 – 4 x 7 + 13 - 11 24: 6 + 18: 2 + 20 – 12 + 6 x 7
21: 3 – 35: 7 + 9 x 3 + 9 x 5
5. SOLVE EXPRESSIONS:
42: 7 x 3 + 2 + 24: 3 – 7 + 9 x 3
6 x 6 + 30: 5: 2 x 7 - 19 90 - 7 x 5 – 24: 3 x 5
6 x 5 – 12: 2 x 3 + 49
6. SOLVE EXPRESSIONS:
32: 8 x 7 + 54: 6: 3 x 5
50 – 45: 5 x 3 + 16: 2 x 5 8 x 6 + 23 – 24: 4 x 3 + 17
48: 6 x 4 + 6 x 9 – 26 + 13
7. SOLVE EXPRESSIONS:
42: 6 + (19 + 6) : 5 – 6 x 2
60 – (13 + 22) : 5 – 6 x 4 + 25 (27 – 19) x 4 + 18: 3 + (8 + 27) :5 -17
(82 – 74) : 2 x 7 + 7 x 4 - (63 – 27): 4
8. SOLVE EXPRESSIONS:
90 – (40 – 24: 3) : 4 x 6 + 3 x 5
3 x 4 + 9 x 6 – (27 + 9) : 4 x 5
(50 – 23) : 3 + 8 x 5 – 6 x 5 + (26 + 16) : 6
(5 x 6 – 3 x 4 + 48: 6) + (82 – 78) x 7 – 13
54: 9 + (8 + 19) : 3 – 32: 4 – 21: 7 + (42 – 14) : 4 – (44 14) : 5
9. SOLVE EXPRESSIONS:
9 x 6 – 6 x 4: (33 – 25) x 7
3 x (12 – 8) : 2 + 6 x 9 - 33 (5 x 9 - 25) : 4 x 8 – 4 x 7 + 13
9 x (2 x 3) – 48: 8 x 3 + 7 x 6 - 34
10. SOLVE EXPRESSIONS:
(8 x 6 – 36:6) : 6 x 3 + 5 x 9
7 x 6 + 9 x 4 – (2 x 7 + 54: 6 x 5) (76 – (27 + 9) + 8) : 6 x 4
(7 x 4 + 33) – 3 x 6:2
11. SOLVE EXPRESSIONS:
(37 + 7 x 4 – 17) : 6 + 7 x 5 + 33 + 9 x 3 – (85 – 67) : 2 x 5
5 x 7 + (18 + 14) : 4 – (26 – 8) : 3 x 2 – 28: 4 + 27: 3 – (17 + 31) : 6
12. SOLVE EXPRESSIONS:
(58 – 31) : 3 – 2 + (58 – 16) : 6 + 8 x 5 – (60 – 42) : 3 + 9 x 2
(9 x 7 + 56: 7) – (2 x 6 – 4) x 3 + 54: 9
13. SOLVE EXPRESSIONS:
(8 x 5 + 28: 7) + 12: 2 – 6 x 5 + (13 – 5) x 4 + 5 x 4
(7 x 8 – 14:7) + (7 x 4 + 12:6) – 10:5 + 63:9
Test “Order of arithmetic operations” (1 option)
1(1b)
2(1b)
3(1b)
4(3b)
5(2b)
6(2b)
7(1b)
8(1b)
9(3b)
10(3b)
11(3b)
12(3b)
110 – (60 +40) :10 x 8
a) 800 b) 8 c) 30
a) 3 4 6 5 2 1 4 5 6 3 2 1
3 4 6 5 1 2
5. In which of the expressions is the last action multiplication?
a) 1001:13 x (318 +466) :22
c) 10000 – (5 x 9+56 x 7) x2
6. In which of the expressions is the first action subtraction?
a) 2025:5 – (524 – 24:6) x45
b) 5870 + (90-50 +30) x8 -90
c) 5400:60 x (3600:90 -90)x5
Choose the correct answer:
9. 90 – (50- 40:5) x 2+ 30
a) 56 b) 92 c) 36
10. 100- (2x5+6 - 4x4) x2
a) 100 b) 200 c) 60
11. (10000+10000:100 +400) : 100 +100
a) 106 b) 205 c) 0
12. 150: (80 – 60:2) x 3
a) 9 b) 45 c) 1
Test "Order of Arithmetic Operations"
1(1b)
2(1b)
3(1b)
4(3b)
5(2b)
6(2b)
7(1b)
8(1b)
9(3b)
10(3b)
11(3b)
12(3b)
1. Which action in the expression will you do first?
560 – (80+20) :10 x7
a) addition b) division c) subtraction
2. What action in the same expression will you do second?
a) subtraction b) division c) multiplication
3. Choose correct option the answer to this expression:
a) 800 b) 490 c) 30
4. Choose the correct arrangement of actions:
a) 3 4 6 5 2 1 4 5 6 3 2 1
320: 8 x 7 + 9 x (240 – 60:15) c) 320: 8 x 7 + 9x (240 – 60:15)
3 4 6 5 2 1
b) 320: 8 x 7 + 9 x (240 – 60:15)
5. In which of the expressions is the last action division?
a) 1001:13 x (318 +466) :22
b) 391 x37:17 x (2248:8 – 162)
c) 10000 – (5 x 9+56 x 7) x2
6. In which of the expressions is the first action addition?
a) 2025:5 – (524 + 24 x6) x45
b) 5870 + (90-50 +30) x8 -90
c) 5400:60 x (3600:90 -90)x5
7. Choose the correct statement: “In an expression without parentheses, the actions are performed:”
a) in order b) x and: , then + and - c) + and -, then x and:
8. Choose the correct statement: “In an expression with brackets, the actions are performed:”
a) first in brackets b)x and:, then + and - c) in writing order
Choose the correct answer:
9. 120 – (50- 10:2) x 2+ 30
a) 56 b) 0 c) 60
10. 600- (2x5+8 - 4x4) x2
a) 596 b) 1192 c) 60
11. (20+20000:2000 +30) : 20 +200
a) 106 b) 203 c) 0
12. 160: (80 – 80:2) x 3
a) 120 b) 0 c) 1