To learn how to quickly and successfully solve equations, you need to start with the most simple rules and examples. First of all, you need to learn how to solve equations that have a difference, sum, quotient, or product of some numbers with one unknown on the left, and another number on the right. In other words, in these equations there is one unknown term and either a minuend with a subtrahend, or a dividend with a divisor, etc. It is about equations of this type that we will talk to you.
This article is devoted to the basic rules that allow you to find factors, unknown terms, etc. We will immediately explain all theoretical principles using specific examples.
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Finding the unknown term
Let's say we have a certain number of balls in two vases, for example, 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 that needs to be found as x. According to the original condition, this number together with 4 form 9, which means we can write the equation 4 + x = 9. On the left we have a sum with one unknown term, on the right we have 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, you need to subtract the known term from the sum.
In this case, we give subtraction a meaning that is the opposite of addition. In other words, there is a certain connection between the actions of addition and subtraction, which can be expressed literally as follows: if a + b = c, then c − a = b and c − b = a, and vice versa, from the expressions c − a = b and c − b = a, we can deduce that a + b = c.
Knowing this rule, we can find one unknown term using the known term and the sum. Which term we know, the first or the second, in this case does not matter. Let's see how to apply this rule in practice.
Example 1
Let's take the equation that we got above: 4 + x = 9. According to the rule, we need to subtract from a known sum equal to 9 a known term equal to 4. Let's subtract one natural number from another: 9 - 4 = 5. We got the term we needed, 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 resulted after we applied the rule for calculating the unknown term.
- After this, we write the equation that was obtained after all the manipulations with numbers.
This form of notation is needed to illustrate the sequential 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 the unknown term was found correctly. If the equality turned out to be incorrect, then we should go back to the solution and recheck it, since this is a sign of an error. As a rule, most often this is a computational error or the application of an incorrect rule.
Finding an unknown subtrahend or minuend
As we already mentioned in the first paragraph, there is a certain connection between the processes of addition and subtraction. With its help, we can formulate a rule that will help us find an unknown minuend when we know the difference and the subtrahend, or an unknown subtrahend through the minuend or the difference. Let's write these two rules in turn and show how to apply them to solve problems.
Definition 2
To find the unknown minuend, you need to add the subtrahend to the difference.
Example 2
For example, we have the equation x - 6 = 10. Unknown minuend. According to the rule, we need to add the subtracted 6 to the difference of 10, we get 16. That is, the original minuend is equal to sixteen. Let's write down the entire solution:
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. The equality 16 - 16 will be correct, which means we have calculated everything correctly.
Definition 3
To find the unknown subtrahend, you need to subtract the difference from the minuend.
Example 3
Let's use the rule to solve the equation 10 - x = 8. We don’t know the subtrahend, so we need to subtract the difference from 10, i.e. 10 - 8 = 2. This means that the required subtrahend is equal to two. Here's the entire solution:
10 - x = 8, x = 10 - 8, x = 2.
Let's check for correctness by substituting a two into 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 replacing the sign with the opposite one. All the above rules fully comply with it.
Finding an unknown factor
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 an unknown factor, you need to divide the product by the known factor.
This rule is based on a meaning that is the opposite of the meaning of multiplication. There is the following connection 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
Let's calculate the unknown factor in the first equation by dividing the known quotient 20 by the known factor 2. We carry out division natural numbers and we get 10. Let us write down the sequence of equalities:
x · 2 = 20 x = 20: 2 x = 10.
We substitute the ten into the original equality and get that 2 · 10 = 20. The value of the unknown multiplier was performed correctly.
Let us clarify that if one of the multipliers is zero, this rule cannot be applied. Thus, we cannot solve the equation x · 0 = 11 with its help. This notation makes no sense, since to solve it you need to divide 11 by 0, and division by zero is not defined. We talked about such cases in more detail in the article devoted to linear equations.
When we apply this rule, we are essentially dividing both sides of the equation by a factor other than 0. There is a 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 completely consistent with it.
Finding an unknown dividend or divisor
Another case that we need to consider is finding the unknown dividend if we know the divisor and the quotient, as well as finding the divisor when the quotient and the dividend are known. We can formulate this rule using the connection between multiplication and division already mentioned here.
Definition 5
To find the unknown dividend, you need to multiply the divisor by the quotient.
Let's see how this rule is applied.
Example 5
Let's use it to solve the equation x: 3 = 5. We multiply the known quotient and the known divisor together and get 15, which will be the dividend we need.
Here's a summary of the entire solution:
x: 3 = 5, x = 3 5, x = 15.
Checking shows that we calculated everything correctly, because when dividing 15 by 3, it actually turns out to be 5. Correct numerical equality is evidence of a correct solution.
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
To find an unknown divisor, you need to divide the dividend by the quotient.
Example 6
Let's take a simple example - equation 21: x = 3. To solve it, divide the known dividend 21 by the quotient 3 and get 7. This will be the required divisor. Now let’s formalize the solution correctly:
21: x = 3, x = 21: 3, x = 7.
Let's make sure the result is correct by substituting seven into the original equation. 21: 7 = 3, so the root of the equation was calculated correctly.
It is important to note that this rule only applies to cases where the quotient is not equal to zero, because otherwise we will again have to divide by 0. If zero is private, two options are possible. If the dividend is also equal to zero and the equation looks like 0: x = 0, then the value of the variable will be any, that is, this equation has an infinite number of roots. But an equation with a quotient equal to 0 and a dividend different from 0 will not have solutions, since such values of the divisor do not exist. An example would be equation 5: x = 0, which does not have any roots.
Consistent application of rules
Often in practice there are more complex problems in which the rules for finding addends, minuends, subtrahends, factors, dividends and quotients must be applied sequentially. Let's give an example.
Example 7
We have an equation of the form 3 x + 1 = 7. We calculate the unknown term 3 x by subtracting one from 7. We end up with 3 x = 7 − 1, then 3 x = 6. This equation is very simple to solve: divide 6 by 3 and get the root of the original equation.
Here is a short summary of the solution to 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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Addition:
Subtraction: add subtract difference.
Multiplication:
Division: multiply divide to quotient.
Learn the names of action components and the rules for finding unknown components:
Addition: term, term, sum. To find the unknown term, you need to subtract the known term from the sum.
Subtraction: minuend, subtrahend, difference. To find the minuend, you need to go to the subtrahend add difference. To find the subtrahend, you need from the minuend subtract difference.
Multiplication: multiplier, multiplier, product. To find an unknown factor, you need to divide the product by the known factor.
Division: dividend, divisor, quotient. To find the dividend you need a divisor multiply to quotient. To find the divisor, you need the dividend divide to quotient.
- Makarenko Inna Alexandrovna
- 30.09.2016
Material number: DB-225492
The author can download the certificate of publication of this material in the “Achievements” section of his website.
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How to find an unknown term subtrahend minuend rule
A numerical expression is a record compiled according to certain rules that uses numbers, arithmetic symbols and parentheses.
Example: 7 · (15 – 2) – 25 · 3 + 1.
To find value of numeric expression, which does not contain parentheses, 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 a numeric expression, then the actions in them are performed first.
An algebraic expression is a record compiled according to certain rules that uses letters, numbers, arithmetic signs and parentheses.
Example: a + b + ; 6 + 2 · (n – 1).
If in algebraic expression instead of a letter we substitute numbers, then we will move from an algebraic expression to a numerical one: for example, if in the expression 6 + 2 · (n - 1) instead of the letter n we substitute the number 25, we get 6 + 2 · (25 - 1).
Thus,
6 + 2 · (n - 1) - algebraic expression;
6 + 2 · (25 - 1) - numerical expression;
54 is the value of the numeric expression.
An equation is the 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 substituted into the equation, the correct numerical equality is obtained, is called root of the equation.
Example:
x + 9 = 16 - equation; x is unknown.
When x = 7, 7 + 9 = 16, the numerical equality is correct, which means 7 is the root of the equation.
Solve the equation- this means finding all its roots or proving that they do not exist.
When solving the simplest equations, the laws of arithmetic operations and the rules for finding components of actions are used.
Rules for finding action components:
- To find the unknown term, you need to subtract the known term from the sum.
- To find minuend, you need to add the difference to the subtrahend.
- To find subtrahend, you need to subtract the difference from the minuend.
If you subtract the difference from the minuend, you get the subtrahend.
These rules are the basis for preparing to solve equations that primary school are solved based on the rule for finding the corresponding unknown component of an equality.
Solve the equation 24-x-19.
The subtrahend in the equation is unknown. To find the unknown subtrahend, you need to subtract the difference from the minuend: x = 24 – 19, x = 5.
In a stable mathematics textbook, the operations of addition and subtraction are taught simultaneously. In some alternative textbooks (I.I. Arginskaya, N.B. Istomina), addition is first studied, and then subtraction.
An expression of the form 3+5 is called amount .
The numbers 3 and 5 in this entry are called terms .
A notation of the form 3+5=8 is called equality . The number 8 is called the meaning of the expression. Since the number 8 in this case is obtained as a 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 of the form 8-3 are called difference.
The number 8 is called reducible , and the number 3 is deductible.
The meaning 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 addition and subtraction actions are introduced by agreement (children are told these names and need to remember them), the teacher actively uses tasks that require recognizing the components of actions and using their names in speech.
7. Among these expressions, find those in which the first term (minued, subtracted) is equal to 3:
8. Make up an expression in which the second term (minued, subtracted) is equal to 5. Find its value.
9. Choose examples in which the sum is 6. Underline them in red. Choose examples in which the difference is 2. Highlight them in blue.
10. What is the number 4 called in the expression 5-4? What is the number 5 called? Find the difference. Make up another example in which the difference is equal to the same number.
11. Minuend 18, subtrahend 9. Find the difference.
12. Find the difference between the numbers 11 and 7. Name the minuend and subtrahend.
In grade 2, children become familiar with the rules for checking the results of addition and subtraction operations:
Addition can be checked by subtraction:
57+8 = 65. Check: 65 – 8 =57
Subtract one term from the sum and get another term. This means the addition was done correctly.
This rule applies to checking the action of addition in any concentration (when checking calculations with any numbers).
Subtraction can be checked by addition:
63-9=54. Check: 54+9=63
We added the subtrahend to the difference and got the minuend. This means that the subtraction was performed correctly.
This rule also applies to testing the operation of subtraction with any numbers.
In 3rd grade, children are introduced to rules for the relationship between the components of addition and subtraction, which are a generalization of the child’s ideas about ways to check addition and subtraction:
If you subtract one term from the sum, you get another term.
Finding subtrahends, minuends and differences for first graders
The long road to the world of knowledge starts with the first examples, simple equations and tasks. In our article we will look at the subtraction equation, which, as is known, consists of three parts: the minuend, the subtrahend, and the difference.
Now let's look at the rules for calculating each of these components using simple examples.
To make understanding the basics of science easier and more accessible for young mathematicians, let's imagine these complex and frightening terms as the names of the numbers in the equation. After all, every person has a name by which they are addressed in order to ask about something, tell something, or exchange information. The teacher in the classroom, calling a 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 which number is called what. And then turn to the number in order to correctly solve the equation or even find the lost number, more on that later.
This is interesting: bit terms - what are they?
But without knowing anything about the numbers in the equation, let's get to know them first. To do this, let's give an example: equation 5−3= 2. The first and most big number 5 after we subtract 3 from it becomes smaller, decreases. That's why in the world of mathematics they call it that way - Reducible. The second number 3, which we subtract from the first, is also easy to recognize and remember - it is Subtractable. Looking at the third number 2, we see the difference between the Minuend and the Subtrahend - this is the Difference, what we got as a result of the subtraction. Like this.
How to find unknowns
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 only need to know two other values, as well as several 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 chord of a circle 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. The diminished is our 7 apples, the number of which has decreased. The subtracted is the 3 apples we gave. The difference is the number of apples remaining. What can I do to find out this amount? Solve the equation 7−3= 4. Thus, although we gave 3 apples to our sister, we still have 4 left.
Minuend search rule
Now let's find out what to do if lost.
How to find the subtrahend
Let's consider what to do, if the deductible is lost. Let's imagine that we bought 7 apples, brought them home and went for a walk, and when we returned, there were only 4 left. The subtract in this case will be the number of apples that someone ate in our absence. Let's denote this number as the letter Y. The equation will be 7-Y=4. To find the unknown subtrahend, you need to know a simple rule and do the following - subtract the Difference from the Minuend, that is, 7 -4 = 3. Our unknown value has been found, this is 3. Hurray! Now we know how much was eaten.
Just in case, we can check our progress and substitute the found Subtrahend into the original example. 7−3= 4. The difference has not changed, which means we did everything right. There were 7 apples, 3 were eaten, 4 remained.
The rules are very simple, but to be sure and not forget anything, you can do this - come up with an easy and understandable example for subtraction for yourself and, solving other examples, look for unknown values, simply plugging in the numbers and easily finding the correct answer. For example, 5−3= 2. We already know how to find both the minuend of 5 and the subtrahend of 3, so when solving a more complex equation, say 25-X= 13, we can remember our simple example and understand that in order to find the unknown Subtractable, you just need to subtract the number 13 from 25, that is, 25 -13= 12.
Well, now we are familiar with subtraction and its main participants.
We know how to distinguish them from each other, find if they are unknown and solve any equations involving them. Let this knowledge help and be useful to you at the beginning of an interesting and exciting journey to the land of Mathematics. Good luck!
Composite problems for finding the minuend, subtrahend and difference
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In this lesson, students will become familiar with compound problems of finding the minuend, subtrahend, and difference. Several compound problems (in several steps) will be considered, in which you will need to find the difference, the subtrahend and the minuend.
Let's review the definition of compound tasks.
Composite problems are problems in which the answer to main question a task requires several actions.
Let's remember the components of which action are the minuend and the subtrahend. These are the components of subtraction. What action results in the difference? And the difference is also the result of subtraction.
Solution to problem 1
Problem 1
Rice. 2. Scheme of problem 1
From the diagram in Fig. 2 we can see that the whole is known to us - these are 90 roses. The integer in this problem is the minuend, which consists of two parts: the subtrahend and the difference. We see that what is being subtracted is still unknown to us, but we can find it out. We can find out how many roses are in three bouquets. And the unknown in this problem is the difference, we will find it with the second action.
First we need to find out how many roses are in three bouquets. The bouquets were the same, each bouquet had 9 roses. This means that in order to find out how many roses are in three bouquets, you need to repeat 9 three times, that is, 9 multiplied by 3.
How many roses are left? We are looking for difference. In order to find the difference, you need to subtract the subtrahend from the minuend. From the number of roses that were brought to the store - 90 - we subtract the number of roses in the bouquets - 27. This means that there are 63 roses left.
In problem 1 we found the difference. Such tasks are called problems to find the difference.
Solution to problem 2
Problem 2
Rice. 4. Scheme of problem 2
From the diagram in Fig. 4 it is clearly visible 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 don’t know the whole . The whole in this case is the minuend. So we start problem of finding the minuend.
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 we don’t yet know what is being subtracted, so 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.
We still have eight textbooks left, they have not yet been put on the shelves. How do we find out how many books were brought to the library? To the number of textbooks that are on the shelves - 60 - we add the number of textbooks that remain - 8 - and we find out that in total school library 68 books were brought.
Solution to Problem 3
You have already become familiar with the problems of finding the difference and finding the minuend. Let's determine what is unknown in problem 3.
Problem 3
Let's find out what is unknown in this problem.
Rice. 6. Scheme for task 3
From the diagram in Fig. 6 it is clear that we know the integer - this is the number of barrels that Winnie the Pooh had - 10. The integer in our problem is the minuend 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 task. We also know that Winnie the Pooh placed the remaining barrels of honey on two shelves, 3 barrels on each shelf. We don't know how many kegs are on the shelves yet, but we can figure it out.
In this problem the subtrahend is unknown. For that to find the subtrahend, you need from the minuend, which we know , subtract the difference, which is still unknown to us. We will begin solving the problem by finding the difference.
Winnie the Pooh has 3 barrels on two shelves. How do you know how many barrels 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.
This means that out of 10 barrels, 6 are on the shelves, and the rest were given to the Rabbit by Winnie the Pooh. How can you find out how many barrels of honey Winnie the Pooh gave to the Rabbit? To do this, we will use the rule, subtract the difference from the minuend, and we will be left with our subtrahend, which is equal to 4. This means that Winnie the Pooh gave 4 barrels of honey to his friend Rabbit.
Today in class we were introduced to a new type of problem and learned how to reason in order to solve it correctly. In the next lesson we will solve compound problems involving difference and multiple comparison.
Bibliography
- Alexandrova E.I. Mathematics. 2nd grade. – M.: Bustard, 2004.
- Bashmakov M.I., Nefedova M.G. Mathematics. 2nd grade. – M.: Astrel, 2006.
- Dorofeev G.V., Mirakova T.I. Mathematics. 2nd grade. – M.: Education, 2012.
Homework
What tasks are called compound tasks? The components of which action are minuend and 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. We ate 12 cucumbers at breakfast, and 21 at lunch. How many cucumbers were in the jar if there were 15 cucumbers left in it?
The tourists walked 5 km on the first day, and 3 km on the second day. How many total kilometers do they have to walk if they have 2 kilometers left to walk?
Basic rules for mathematics.
To find the unknown term, you need to subtract the known term from the sum value.
To find the unknown minuend, you need to add the subtrahend to the difference value.
To find the unknown subtrahend, you need to subtract the difference value from the minuend.
To find an unknown factor, you need to divide the product value by the known factor
To find the unknown dividend, you need to multiply the quotient by the divisor.
To find an unknown divisor, you need to divide the dividend by the value of the quotient.
Laws of addition:
Commutative: a + b = b + a (the value of the sum does not change from rearranging the places of the terms)
Combinative: (a + b) + c = a + (b + c) (To add a 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 for adding a number with 0: a + 0 = a (when adding a number with zero, we get the same number).
Multiplication laws:
Commutative: a ∙ b = b ∙ a (the value of the product does not change from rearranging the places of the factors)
Combinative: (a ∙ b) ∙ c = a ∙ (b ∙ 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) = 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 (when any number is multiplied by 0, the result is 0)
Laws of division:
a: 1 = a (When a number is divided by 1, the same number is obtained)
0: a = 0 (When 0 is divided by a number, the result is 0)
You can't divide by zero!
The perimeter of a rectangle is equal to twice the sum of its length and width. Or: the perimeter of a rectangle is equal to the sum of twice the width and twice the length: P = (a + b) ∙ 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 c 1m = 1000 mm
1 dm = 10 cm = 100 mm 1 min = 60 sec 1 c = 100 kg 1 kg = 1000 g
1 cm = 10 mm 1 day = 24 hours 1 km = 1000 m
When performing a differential comparison, the smaller number is subtracted from a larger number; when performing a multiple comparison, the larger number is divided by the smaller number.
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, produces a true 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 an expression without parentheses contains actions of the first (addition, subtraction) and second (multiplication, division) stages, then the actions of the second stage are performed first in order, and only then the actions of the second stage.
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.