Basic rules for mathematics.
To find unknown term, it is necessary to subtract the known term from the value of the sum.
To find the unknown minuend, you need to add the subtrahend to the difference.
To find the unknown subtrahend, it is necessary to subtract the value of the difference from the minuend.
To find the unknown factor, you need to divide the value of the product by the known factor.
To find the unknown dividend, you need to multiply the value of the quotient by the divisor.
To find an unknown divisor, you need to divide the dividend by the value of the quotient.
Addition action laws:
Commutative: a + b \u003d b + a (from rearranging the places of the terms, the value of the sum does not change)
Associative: (a + c) + c \u003d a + (b + c) (To add the third term to the sum of two terms, you can add the sum of the second and third terms to the first term).
The law of adding a number to 0: a + 0 = a (when adding a number to zero, we get the same number).
Multiplication laws:
Displacement: a ∙ c = c ∙ a (the value of the product does not change from the permutation of the places of factors)
Associative: (a ∙ c) ∙ c \u003d a ∙ (c ∙ c) - To multiply the product of two factors by the third factor, you can multiply the first factor by the product of the second and third factors.
Distributive law of multiplication: a ∙ (b + c) \u003d a ∙ c + b ∙ c (To multiply a number by a sum, you can multiply this number by each of the terms and add the resulting products).
Law of multiplication by 0: a ∙ 0 = 0 (multiplying any number by 0 results in 0)
Division laws:
a: 1 \u003d a (When you divide a number by 1, you get the same number)
0: a = 0 (When you divide 0 by a number, you get 0)
You can't divide by zero!
The perimeter of a rectangle is twice the sum of its length and width. Or: the perimeter of a rectangle is equal to the sum double width and double length: P \u003d (a + c) ∙ 2,
P = a ∙ 2 + b ∙ 2
Perimeter of a square equal to length side multiplied by 4 (P = a ∙ 4)
1 m = 10 dm = 100 cm 1 hour = 60 min 1t = 1000 kg = 10 q 1m = 1000 mm
1 dm = 10 cm = 100 mm 1 min = 60 sec 1 q = 100 kg 1 kg = 1000 g
1 cm = 10 mm 1 day = 24 hours 1 km = 1000 m
When performing a difference comparison, a smaller number is subtracted from a larger number; when performing a multiple comparison, a larger number is divided by a smaller one.
An equality containing an unknown is called an equation. The root of an equation is a number that, when substituted into the equation instead of x, gives the correct value. numerical equality. Solving an equation means finding its root.
The diameter divides the circle in half - into 2 equal parts. The diameter is equal to two radii.
If the expression without brackets contains the actions of the first (addition, subtraction) and the second (multiplication, division) steps, then the actions of the second step are performed first in the order, and only then the actions of the second step.
12 noon is noon. 12 o'clock at night is midnight.
Roman numerals: 1 - I, 2 - II, 3 - III, 4 - IV, 5 - V, 6 - VI, 7 - VII, 8 - VIII, 9 - IX, 10 - X, 11 - XI, 12 - XII , 13 - XIII, 14 - XIV, 15 - XV, 16 - XVI, 17 - XVII, 18 - XVIII, 19 - XIX, 20 - XX, etc.
Algorithm for solving the equation: determine what the unknown is, remember the rule, how to find the unknown, apply the rule, make a check.
To learn how to solve equations quickly and successfully, you need to start with the most simple rules and examples. First of all, you need to learn how to solve equations, on the left of which is the difference, sum, quotient or product of some numbers with one unknown, and on the right is another number. In other words, in these equations there is one unknown term and either the minuend with the subtrahend, or the divisible with a divisor, etc. It is about equations of this type that we will talk with you.
This article is devoted to the basic rules for finding factors, unknown terms, etc. All theoretical provisions We will immediately explain with specific examples.
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Finding the unknown term
Let's say we have some number of balls in two vases, say 9 . We know that there are 4 balls in the second vase. How to find the quantity in the second? Let's write this problem in mathematical form, denoting the number to be found as x. According to the original condition, this number together with 4 form 9, so we can write the equation 4 + x = 9. On the left, we got a sum with one unknown term, on the right, the value of this sum. How to find x? To do this, you need to use the rule:
Definition 1
To find the unknown term, subtract the known from the sum.
AT this case we give subtraction a meaning that is the opposite of addition. In other words, there is a certain connection between the operations of addition and subtraction, which can be expressed in literal form as follows: if a + b \u003d c, then c - a \u003d b and c - b \u003d a, and vice versa, from the expressions c - a \u003d b and c − b = a we can deduce that a + b = c .
Knowing this rule, we can find one unknown term using the known and the sum. Which term we know, the first or the second, is not important in this case. Let's see how to apply this rule on practice.
Example 1
Let's take the equation that we got above: 4 + x = 9. According to the rule, we need to subtract from the known sum, equal to 9, the known term, equal to 4. Subtract one natural number from another: 9 - 4 = 5 . We got the term we need, equal to 5.
Typically, solutions to such equations are written as follows:
- The original equation is written first.
- Next, we write down the equation that we got after we applied the rule for calculating the unknown term.
- After that, we write the equation that turned out after all the actions with numbers.
This form of writing is needed in order to illustrate the successive replacement of the original equation with equivalent ones and to display the process of finding the root. The solution to our simple equation above would be correctly written as:
4 + x = 9 , x = 9 − 4 , x = 5 .
We can check the correctness of the received answer. Let's substitute what we got into the original equation and see if the correct numerical equality comes out of it. Substitute 5 into 4 + x = 9 and get: 4 + 5 = 9 . The equality 9 = 9 is correct, which means that the unknown term was found correctly. If the equality turned out to be wrong, then we should go back to the solution and double-check it, since this is a sign of a mistake. As a rule, most often this is a computational error or the application of an incorrect rule.
Finding the unknown subtrahend or minuend
As we mentioned in the first paragraph, there is a certain relationship between the processes of addition and subtraction. With its help, you can formulate a rule that will help you find the unknown minuend when we know the difference and the subtrahend, or the unknown subtrahend through the minuend or the difference. We write these two rules in turn and show how to apply them to solve problems.
Definition 2
To find the unknown minuend, add the minuend to the difference.
Example 2
For example, we have an equation x - 6 = 10 . Reduced unknown. According to the rule, we need to add the subtracted 6 to the difference 10, we get 16. That is, the original minuend is sixteen. Let's write the solution in its entirety:
x − 6 = 10 , x = 10 + 6 , x = 16 .
Let's check the result by adding the resulting number to the original equation: 16 - 6 = 10. Equality 16 - 16 will be correct, which means that we have calculated everything correctly.
Definition 3
To find the unknown subtrahend, subtract the difference from the minuend.
Example 3
Let's use the rule to solve the equation 10 - x = 8 . We do not know what is being subtracted, so we need to subtract the difference from 10, i.e. 10 - 8 = 2. Hence, the required subtrahend is equal to two. Here is the entire solution entry:
10 - x = 8 , x = 10 - 8 , x = 2 .
Let's check for correctness by substituting a deuce in the original equation. Let's get the correct equality 10 - 2 = 8 and make sure that the value we found will be correct.
Before moving on to other rules, we note that there is a rule for transferring any terms from one part of the equation to another with the sign reversed. All of the above rules are fully consistent with it.
Finding the unknown multiplier
Let's look at two equations: x 2 = 20 and 3 x = 12. In both, we know the value of the product and one of the factors, we need to find the second. To do this, we need to use another rule.
Definition 4
To find the unknown factor, you need to divide the product by the known factor.
This rule is based on a sense that is the opposite of multiplication. There is the following relationship between multiplication and division: a b = c when a and b are not equal to 0, c: a = b, c: b = c and vice versa.
Example 4
Calculate the unknown factor in the first equation by dividing the known quotient 20 by the known factor 2 . We carry out the division natural numbers and we get 10 . Let's write down the sequence of equalities:
x 2 = 20 x = 20: 2 x = 10 .
We substitute the ten in the original equality and we get that 2 10 \u003d 20. The value of the unknown multiplier was done correctly.
Let us clarify that if one of the factors is zero, this rule cannot be applied. So, we cannot solve the equation x 0 = 11 with its help. This notation doesn't make sense because the solution is to divide 11 by 0 , and division by zero is undefined. More about similar cases we told in the article devoted to linear equations.
When we apply this rule, we are essentially dividing both sides of the equation by a different factor than 0 . Exists separate rule, according to which such a division can be carried out, and it will not affect the roots of the equation, and what we wrote about in this paragraph is fully consistent with it.
Finding an unknown dividend or divisor
Another case we need to consider is finding the unknown dividend if we know the divisor and the quotient, and also finding the divisor when the quotient and the dividend are known. We can formulate this rule with the help of the connection between multiplication and division already mentioned here.
Definition 5
To find the unknown dividend, multiply the divisor by the quotient.
Let's see how this rule applies.
Example 5
Let's use it to solve the equation x: 3 = 5 . We multiply the known quotient and the known divisor among ourselves and get 15, which will be the divisible we need.
Here short entry the whole solution:
x: 3 = 5, x = 3 5, x = 15.
The check shows that we calculated everything correctly, because when dividing 15 by 3, it really turns out 5. True numerical equality is evidence of the correct decision.
This rule can be interpreted as multiplying the right and left sides of the equation by the same number other than 0. This transformation does not affect the roots of the equation in any way.
Let's move on to the next rule.
Definition 6
For finding unknown divisor divide the dividend by the quotient.
Example 6
Let's take a simple example - Equation 21: x = 3 . To solve it, we divide the known divisible 21 by the quotient 3 and get 7. This will be the desired divisor. Now we make the decision correctly:
21:x=3, x=21:3, x=7.
Let's make sure the result is correct by substituting the seven in the original equation. 21: 7 = 3, so the root of the equation was calculated correctly.
It's important to note that this rule only applies when the quotient is non-zero, otherwise we'd have to divide by 0 again. If the quotient is zero, two options are possible. If the dividend is also zero and the equation looks like 0: x = 0 , then the value of the variable will be any, that is given equation It has infinite number roots. But an equation with a quotient equal to 0, with a dividend other than 0, will not have solutions, since there are no such divisor values. An example would be equation 5: x = 0, which does not have any root.
Consistent application of rules
Often in practice there are more challenging tasks, in which the rules for finding terms, minuends, subtrahends, factors, divisibles and quotients must be applied sequentially. Let's take an example.
Example 7
We have an equation like 3 x + 1 = 7 . We calculate the unknown term 3 x , subtracting one from 7. We end up with 3 · x = 7 − 1 , then 3 · x = 6 . This equation is very easy to solve: divide 6 by 3 and get the root of the original equation.
Here is a shorthand for solving yet another equation (2 x − 7): 3 − 5 = 2:
(2 x - 7) : 3 - 5 = 2 , (2 x - 7) : 3 = 2 + 5 , (2 x - 7) : 3 = 7 , 2 x - 7 = 7 3 , 2 x − 7 = 21 , 2 x = 21 + 7 , 2 x = 28 , x = 28: 2 , x = 14 .
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| P. | AT. | FROM. |
236m?(236+95)m?(H.-108)m
On the main question tasks How many meters of fabric did the store sell in 3 days? we cannot answer right away, because we do not know how many meters of fabric the store sold on Tuesday and Wednesday. Knowing that on Monday, the store sold 236 m of fabric, and on Tuesday - 95 m more than on Monday, we can find how many meters of fabric the store sold on Tuesday by adding, we are prompted by the words __ more. By knowing how many meters of fabric the store sold on Tuesday, we can find how many meters of fabric they sold on Wednesday. The task statement says: on Tuesday - 95 m more than on Monday and 108 m more than on Wednesday . This is an indirect condition, the word suggests and . So Wednesday 108 m less than on Tuesday. We find the action of subtraction, we are prompted by the words __ less. Knowing how much fabric the store sold on Tuesday and Wednesday, we can answer the main question of the problem How many meters of fabric did the store sell in 3 days? the action of addition to find the whole is to add the parts (add 3 parts). The problem is solved in three steps ...
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Addition:
Subtraction: add subtract difference.
Multiplication:
Division: multiply divide to private.
Learn the names of action components and the rules for finding unknown components:
Addition: term, term, sum. To find the unknown term, subtract the known term from the sum.
Subtraction: minuend, subtrahend, difference. To find the minuend, you need to subtrahend add difference. To find the subtrahend, you need from the minuend subtract difference.
Multiplication: multiplier, multiplier, product. To find the unknown factor, you need to divide the product by the known factor.
Division: divisible, divisor, quotient. To find the dividend, you need a divisor multiply to private. To find the divisor, you need the dividend divide to private.
- Makarenko Inna Alexandrovna
- 30.09.2016
Material Number: DB-225492
Publication Certificate this material the author can download in the "Achievements" section of his site.
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How to Find the Unknown Term Subtracted Reduced Rule
A numeric expression is a composite of certain rules a record that uses numbers, signs arithmetic operations and brackets.
Example: 7 (15 - 2) - 25 3 + 1.
To find value of a numeric expression, which does not contain brackets, you must perform from left to right, in order, first all the operations of multiplication and division, and then all the operations of addition and subtraction.
If there are parentheses in the numeric expression, then the actions in them are performed first.
An algebraic expression is a notation composed according to certain rules that uses letters, numbers, arithmetic signs, and brackets.
Example: a + b + ; 6 + 2 (n - 1).
If in algebraic expression substitute numbers instead of a letter, then we will move from an algebraic expression to a numerical one: for example, if we substitute the number 25 instead of the letter n in the expression 6 + 2 (n - 1), we get 6 + 2 (25 - 1).
In this way,
6 + 2 (n - 1) is an algebraic expression;
6 + 2 (25 - 1) - numeric expression;
54 is the value of the numeric expression.
An equation is an equality of expressions containing a letter, if the task is to find this letter. The letter itself in this case is called unknown. The value of the unknown, when substituting into the equation, the correct numerical equality is obtained, is called the root of the equation.
Example:
x + 9 = 16 - equation; x is unknown.
For x \u003d 7, 7 + 9 \u003d 16, the numerical equality is correct, which means that 7 is the root of the equation.
solve the equation— it means to find all its roots or to prove that they do not exist.
When solving the simplest equations, the laws of arithmetic operations and the rules for finding the components of actions are used.
Rules for finding action components:
- To find the unknown term, it is necessary to subtract the known term from the sum.
- To find minuend, it is necessary to add the difference to the subtrahend.
- To find subtrahend, it is necessary to subtract the difference from the reduced.
If you subtract the difference from the minuend, you get the subtrahend.
These rules are the basis for preparing for solving equations that, in primary school are solved based on the rule for finding the corresponding unknown component of equality.
Solve equation 24-x-19.
The subtrahend is unknown in the equation. To find the unknown subtrahend, you need to subtract the difference from the reduced: x \u003d 24 - 19, x \u003d 5.
In a stable mathematics textbook, the operations of addition and subtraction are studied simultaneously. Some alternative textbooks (I.I. Arginskaya, N.B. Istomina) first study addition and then subtraction.
An expression of the form 3+5 is called sum .
The numbers 3 and 5 in this entry are called terms .
An entry like 3+5=8 is called equality . The number 8 is called the value of the expression. Since the number 8 in this case is the result of summation, it is also often called amount.
Find the sum of numbers 4 and 6 (Answer: the sum of the numbers 4 and 6 is 10).
Expressions like 8-3 are called difference.
The number 8 is called reduced , and the number 3 is subtractable.
The value of the expression - the number 5 can also be called difference.
Find the difference between the numbers 6 and 4. (Answer: the difference between the numbers 6 and 4 is 2.)
Since the names of the components of the addition and subtraction actions are entered by agreement (children are told these names and they need to be remembered), the teacher actively uses tasks that require recognition of the action components and the use of their names in speech.
7. Among these expressions, find those in which the first term (reduced, subtracted) is 3:
8. Make an expression in which the second term (reduced, subtracted) is equal to 5. Find its value.
9. Select examples in which the sum is 6. Underline them in red. Choose examples where the difference is 2. Highlight them in blue.
10. What is the name of the number 4 in the expression 5-4? What is the number 5 called? Find the difference. Write another example where the difference is the same number.
11. Reduced 18, subtracted 9. Find the difference.
12. find the difference between the numbers 11 and 7. Name the minuend, the subtrahend.
In grade 2, children get acquainted with the rules for checking the results of addition and subtraction:
Addition can be checked by subtraction:
57 + 8 = 65. Check: 65 - 8 = 57
One term was subtracted from the sum, another term was obtained. So the addition is correct.
This rule is applicable to checking the action of addition in any concenter (when checking calculations with any numbers).
Subtraction can be checked by addition:
63-9=54. Check: 54+9=63
The subtrahend was added to the difference, and the minuend was obtained. So the subtraction is correct.
This rule also applies to testing the operation of subtraction with any numbers.
In 3rd grade, children are introduced to the rules for the relationship of the components of addition and subtraction, which are a generalization of the child's ideas about how to check addition and subtraction:
If you subtract one term from the sum, you get another term.
Finding subtrahend, minuend and difference for first graders
Long road to the world of knowledge starts with the first examples, simple equations and tasks. In our article, we will consider the subtraction equation, which, as you know, consists of three parts: minuend, subtrahend, difference.
Now let's look at the rules for calculating each of these components using simple examples.
To do young mathematicians understanding the basics of science is easier and more accessible, let's represent these complex and frightening terms with the names of numbers in the equation. After all, each person has a name by which they turn to him in order to ask something, tell something, exchange information. The teacher in the class, calling the student to the board, looks at him and calls him by name. So we, looking at the numbers in the equation, can very easily understand what number is called. And then turn to the number in order to correctly solve the equation or even find the lost number, more on that later.
It is interesting: bit terms- what is this?
But, without knowing anything about the numbers in the equation, let's get to know them first. To do this, we give an example: the equation 5−3= 2. The first and most big number 5 after we subtract 3 from it becomes smaller, decreases. Therefore, in the world of mathematics, it is called so - Reduced. The second number 3, which we subtract from the first, is also easy to recognize and remember - it is Subtrahendable. Looking at the third number 2, we see the difference between the Reduced and the Subtracted - this is the Difference, what we got as a result of the subtraction. Like this.
How to find the unknown
We met three brothers:
But there are times when some of the numbers are lost or simply unknown. What to do? Everything is very simple - in order to find such a number, we need to know only two other values, as well as a few rules of mathematics, and, of course, be able to use them. Let's start with the easiest situation, when we need to find the Difference.
This is interesting: what is a circle chord in geometry, definition and properties.
How to find the difference
Let's imagine that we bought 7 apples, gave 3 apples to our sister and kept some for ourselves. Decreasing is our 7 apples, the number of which has decreased. The deductible is those 3 apples we gave. The difference is the number of apples left. What can be done to find out this number? Solve the equation 7−3= 4. Thus, although we gave 3 apples to our sister, we still have 4 left.
The rule for finding the minuend
Now we know what to do if lost.
How to find subtrahend
Consider what to do if lost. Imagine that we bought 7 apples, brought them home and went for a walk, and when we returned, there were only 4 left. In this case, the number of apples that someone ate in our absence will be subtracted. Let's denote this number as the letter Y. We get the equation 7-Y=4. To find the unknown subtrahend, you need to know a simple rule and do the following - subtract the Difference from the Reduced, that is, 7 -4 \u003d 3. Our unknown value was found, this is 3. Hooray! Now we know how much was eaten.
Just in case, we can check our progress and substitute the subtrahend found in original example. 7−3= 4. The difference has not changed, which means we did everything right. There were 7 apples, ate 3, left 4.
The rules are very simple, but to be sure and not forget anything, you can do this - come up with an easy and understandable subtraction example for yourself and, solving other examples, look for unknown values, simply by substituting numbers and easily find the correct answer. For example, 5−3= 2. We already know how to find both the minuend 5 and the minuend 3, so solving more complex equation, say 25-X= 13, we can recall our simple example and understand that to find the unknown Subtrahend, we only need to subtract the number 13 from 25, that is, 25 -13= 12.
Well, now we got acquainted with subtraction, its main participants.
We can distinguish them from each other, find if they are unknown and solve any equations with their participation. Let this knowledge help and be useful to you at the beginning of an interesting and exciting journey to the country of Mathematics. Good luck!
Compound problems for finding the minuend, subtrahend and difference
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On the this lesson Students will be introduced to compound problems for finding the minuend, subtrahend, and difference. Several compound tasks (in several steps) will be considered in which it will be necessary to find the difference, subtracted and reduced.
Let's revisit the definition of compound tasks.
Compound tasks are tasks in which the answer to the main question of the task requires the performance of several actions.
Let's remember the components of which action is the minuend and the subtrahend. These are subtraction components. What action results in difference? And the difference is also the result of subtraction.
Problem 1 solution
Task 1
Rice. 2. Scheme of task 1
From the diagram in Fig. 2 we can see that we know the whole - these are 90 roses. The whole in this problem is the minuend, which consists of two parts: the subtrahend and the difference. We see that what is subtracted is not yet known to us, but we can recognize it. We can find out how many roses are in three bouquets. And the unknown in this problem is the difference, we will find it by the second action.
First we need to find out how many roses are in the three bouquets. The bouquets were the same, each bouquet had 9 roses. So, in order to find out how many roses are in three bouquets, you need to repeat 9 three times, that is, multiply 9 by 3.
How many roses are left? We are looking for difference. To find the difference, subtract the minuend from the minuend. From the number of roses that were brought to the store -90 - subtract the number of roses that are in the bouquets - 27. So, there are 63 roses left.
In problem 1, we found the difference. Such tasks are called tasks to find the difference.
Problem 2 solution
Task 2
Rice. 4. Scheme of task 2
From the diagram in Fig. 4 clearly shows that the parts are known to us. We don't yet know how many textbooks are on the shelves, but we can figure it out. We know how many textbooks have not yet been put on the shelves 8. But we do not know the whole . In this case, the integer is the minuend. So we start problem of finding the reduced.
Let's remember the rule for finding the minuend if we know the subtrahend and the difference. To find the minuend, we must add the subtrahend to the difference. But what we subtract is not yet known, we will find out.
If there are 15 textbooks on each shelf and there are 4 such shelves, then we can find out how many textbooks are on the shelves. To do this, we multiply the number of textbooks on one shelf - 15 - by the number of shelves - 4. And we determine that there are 60 books on four shelves.
And we have eight textbooks left, they have not yet been put on the shelves. How do we know how many books were brought to the library in total? To the number of textbooks that are on the shelves - 60 - we add the number of textbooks that are left - 8 - and find out that in total school library 68 books were brought.
Problem 3 solution
You have already got acquainted with the problems of finding the difference and finding the minuend. Let's determine what is unknown in Problem 3.
Task 3
Let's find out what is unknown in this problem.
Rice. 6. Scheme for problem 3
From the diagram in Fig. 6 it can be seen that we know the integer - this is the number of barrels that Winnie the Pooh had - 10. The integer in our problem is the reduced one that we know. The part that he gave to the Rabbit is not yet known to us, and this is the main question of the problem. We also know that Winnie the Pooh placed the remaining barrels of honey on two shelves, 3 barrels on each shelf. We don't yet know how many kegs are on the shelves, but we can figure it out.
In this problem, the subtrahend is unknown. For to find the subtrahend, you need from the minuend, which we know , subtract the difference, which is still unknown to us. We will start solving the problem by finding the difference.
Winnie the Pooh has 3 barrels on two shelves. How to find out how many kegs are on the shelves? To do this, you need the number of barrels on one shelf - 3 - repeat, that is, multiply by 2, since there were two shelves.
So, out of 10 barrels, 6 are on the shelves, and the rest were presented by Winnie the Pooh to the Rabbit. How to find out how many barrels of honey Winnie the Pooh gave the Rabbit? To do this, we will use the rule, subtract the difference from the minuend, and we will have our subtrahend, which is equal to 4. This means that Winnie the Pooh gave 4 barrels of honey to his friend Rabbit.
Today at the lesson we got acquainted with a new type of problems and learned how to reason in order to solve them correctly. In the next lesson, we will solve compound problems for difference and multiple comparison.
Bibliography
- Alexandrova E.I. Maths. Grade 2 – M.: Bustard, 2004.
- Bashmakov M.I., Nefyodova M.G. Maths. Grade 2 – M.: Astrel, 2006.
- Dorofeev G.V., Mirakova T.I. Maths. Grade 2 – M.: Enlightenment, 2012.
Homework
What are called composite tasks? Which action components are the minuend and the subtrahend?
The hedgehog collected 28 apples. He gave 9 of them to the hedgehog and a few more to the squirrel. How many apples did the hedgehog give to the squirrel if he had 12 apples left?
There were pickles in the jar. They ate 12 cucumbers at breakfast, and 21 at lunch. How many cucumbers were in the jar if there were 15 cucumbers left in it?
Tourists walked 5 km on the first day, 3 km on the second day. How many km do they have to walk if they have 2 km to go?
