To find the unknown. Divisible divisor private rule

Often you can find such equations in which the divisor is unknown. For example 350: X = 50, where 350 is the dividend, X is the divisor, and 50 is the quotient. To solve these examples, it is necessary to perform a certain set of actions with the numbers that are known.

You will need

- pencil or pen;
- a sheet of paper or a notebook.

Instruction

Imagine that one woman had a number of children. She bought 30 sweets at the store. Returning home, the lady divided the sweets equally among the children. Thus, each child received 5 sweets for dessert. Question: How many children did the woman have?
Write a simple equation where the unknown, i.e. X is the number of children, 5 is the number of sweets each child received, and 30 is the number of sweets that were purchased. So you should get an example: 30: X = 5. In this mathematical expression 30 is called the dividend, X is the divisor, and the resulting quotient is 5.
Now start solving. We know that to find a divisor, you need to divide the dividend by the quotient. It turns out: X \u003d 30: 5; 30: 5 \u003d 6; X \u003d 6.
Make a test by substituting the resulting number into the equation. So, 30: X = 5, you have found an unknown divisor, i.e. X \u003d 6, thus: 30: 6 \u003d 5. The expression is true, and from this it follows that the equation is solved correctly. Of course, when solving examples in which prime numbers, checking is optional. But when the equations are two-digit, three-digit, four-digit, etc. numbers, be sure to check yourself. After all, it does not take much time, but gives absolute confidence in the result.

Instruction

Most often, you need to decompose the number into prime factors. These are numbers that divide the original number without a remainder, and at the same time they themselves can be divided without a remainder only by itself and one (for such numbers 2, 3, 5, 7, 11, 13, 17, etc.). Moreover, no regularity was found in the series. Take them from a special table or find them using an algorithm called the "sieve of Eratosthenes."

Numbers with more than two divisors are called composite numbers. What numbers can be composite?
As numbers divisible by 2, then all are even numbers, Besides numbers 2 will be composite. Indeed, when dividing 2: 2, the two is divisible by itself, that is, it has only two divisors (1 and 2) and is a prime number.

Let's see if even has numbers any other dividers. Divide it first by 2. From the commutativity of the multiplication operation, it is obvious that the resulting quotient will also be a divisor numbers. Then, if the resulting quotient is an integer, divide again by 2 this quotient. Then the resulting new quotient y = (x:2):2 = x:4 will also be a divisor of the original numbers. Similarly, and 4 will be a divisor of the original numbers.

Continuing this chain, we generalize the rule: we sequentially divide first and then the resulting quotient by 2 until either quotient becomes equal to an odd number. In this case, all the resulting quotients will be divisors of this numbers. In addition, the divisors of this numbers will and numbers 2^k where k = 1...n, where n is the number of steps in this chain. Example: 24:2 = 12, 12:2 = 6, 6:2 = 3 - odd number. Therefore, 12, 6 and 3 - dividers numbers 24. There are 3 steps in this chain, therefore, the divisors numbers 24 will also numbers 2^1 = 2 (already known from the parity numbers 24), 2^2 = 4 and 2^3 = 8. Thus, numbers 1, 2, 3, 4, 6, 8, 12 and 24 will be divisors numbers 24.

However, not for all even numbers this can give everything. dividers numbers. Consider, for example, the number 42. 42:2 = 21. However, as you know, numbers 3, 6 and 7 will also be divisors numbers 42.
There are divisibility numbers. Let's consider the most important of them:
Sign of divisibility by 3: when the sum of the digits numbers is divisible by 3 without a remainder.
Sign of divisibility by 5: when the last digit numbers 5 or 0.
Divisibility by 7: when the result of subtracting twice the last digit from this numbers without the last digit is divisible by 7.
Sign of divisibility by 9: when the sum of the digits numbers is divisible by 9 without a remainder.
The sign of divisibility by 11: when the sum of the digits occupying odd places is either equal to the sum of the digits occupying even places, or from it to a number divisible by 11.
There are also signs of divisibility by 13, 17, 19, 23 and others numbers.

For both even and odd numbers, you need to use the signs of division by a particular number. By dividing the number, you should determine dividers the resulting private, etc. (the chain is similar to the chain of even numbers when divided by 2, described above).

Sources:

Signs of divisibility

Of the four main mathematical operations division is the most resource-intensive operation. It can be done manually (column), on calculators various designs, as well as using a slide rule.

Instruction

To divide one number by another by a column, write the dividend first, then the divisor. Place between them vertical line. Draw a horizontal line under the divider. Consistently, as if deleting from the lower digits, get a number that is greater than the divisor. By successively multiplying the numbers from 0 to 9 by a divisor, find the largest of numbers, smaller than obtained at the previous stage. Write this number as the first digit of the quotient. Write the result of multiplying this number by the divisor under the dividend with a shift by one digit to the right. Subtract, and with its result, perform the same actions until you find all the digits of the quotient. Determine the location of the comma by subtracting the order of the divisor from the order of the dividend.

If the numbers are not divisible by each other, two situations are possible. In the first of them, one digit or a combination of several digits will be repeated indefinitely. Then it is pointless to continue the calculation - it is enough to take this digit or a chain of digits into a period. In the second situation, any regularity in the particular will not succeed. Then stop dividing, having achieved the desired accuracy of the result, and round the last one.

To divide one number by another using a calculator with arithmetic (both simple and engineering), press the reset button, enter the dividend, press the divide button, enter the divisor, and then press the equals button. On a calculator with a formula notation, divide in the same way, taking into account the fact that the key with the equal sign can carry, for example, Enter or Exe. Modern appliances of this type are two-line: typed on the top line, and the result is displayed on the bottom more big numbers. Using the Ans key, this result can be used in the next calculation. In all cases, the result is automatically rounded within the calculator's digit grid.

On a reverse polish calculator, first press the reset button, then enter the dividend and press the Enter key (it may have an upward arrow instead). The number will be in the stack cell. Now enter the divisor and press the division key. The number from the stack will be divided by the number that was previously displayed on the indicator.

slide rule use when little precision is required. Remove from both numbers, and then from each of them take two senior digits. On the A scale, find the divisor, and then combine it with the divisor on the B scale. Then find the last unit - right above it on the A scale will be located private. Determine the location of the comma in it in the same way as the column.

Sources:

Column division order
private numbers are

Schoolchildren often come across the following wording among math assignments: "find the least common multiple of numbers." This must be learned to do in order to fulfill various activities with fractions with different denominators.

Finding the least common multiple: basic concepts

To understand how to calculate the LCM, you should first determine the meaning of the term "multiple".

A multiple of A is a natural number that is divisible by A without remainder. Thus, 15, 20, 25, and so on can be considered multiples of 5.

The divisors of a particular number can be limited quantity, but there are an infinite number of multiples.

common multiple natural numbers- a number that is divisible by them without a remainder.

The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is evenly divisible by all these numbers.

To find the NOC, you can use several methods.

For small numbers, it is convenient to write out in a line all the multiples of these numbers until a common one is found among them. Multiples denote in the record capital letter TO.

For example, multiples of 4 can be written like this:

K(4) = (8,12, 16, 20, 24, ...)

K(6) = (12, 18, 24, ...)

So, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This entry is performed as follows:

LCM(4, 6) = 24

Greatest overall divider is the maximum number that each of the proposed numbers can be divisible by. This term is often used to abbreviate complex fractions, where both the numerator and denominator must be divided by the same number. Sometimes it is possible to determine the greatest common divider by eye, however, in most cases, in order to find it, you need to spend a number of mathematical operations.

You will need

To do this, you will need a piece of paper or a calculator.

Instruction

Spread out each complex number to the product of primes or factors. For example, 60 and 80, where 60 is equal to 2*2*3*5, and 80 is 2*2*2*2*5, it can be written more simply using . AT this case will look like two in the second multiplied by five and three, and the second is the product of two in the fourth and five.

Now write down the common for both numbers. In our version, these are two and five. However, in other cases, this number can be one, two or three digits, and even . Next, you need to work. Choose the smallest of each of the factors. In the example, this is two to the second power and five to the first.

At the end, you just need to multiply the resulting numbers. In our case, everything is extremely simple: two times five equals 20. Thus, the number 20 can be called the largest common divisor for 60 and 80.

How to find the unknown minuend, subtrahend?

The connection between addition and subtraction of numbers, which we already mentioned in the previous paragraph, allows us to obtain a rule for finding an unknown minuend through a known subtrahend and difference, as well as a rule for finding an unknown subtrahend through a known minuend and difference. We will formulate them in turn, and immediately give the solution of the corresponding equations.

To find the unknown minuend, you need to add the subtrahend to the difference.

For example, consider the equation x−2=5 . It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.

If you omit the explanations, then the solution is written as follows:
x−2=5 ,
x=5+2 ,
x=7 .

For self-control, we will perform a check. We substitute the found reduced into the original equation, and we obtain the numerical equality 7−2=5. It is correct, therefore, we can be sure that we have correctly determined the value of the unknown minuend.

You can move on to finding the unknown subtrahend. It is found by adding next rule: to find the unknown subtrahend, it is necessary to subtract the difference from the minuend.

We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9 , we have 9−4=5 . Thus, the required subtrahend is equal to five.

Here is a short version of the solution to this equation:
9−x=4 ,
x=9−4 ,
x=5 .

It remains only to check the correctness of the found subtrahend. Let's make a check, for which we substitute the found value 5 instead of x into the original equation, and we get the numerical equality 9−5=4. It is correct, therefore the value of the subtrahend that we found is correct.

And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.

To find the unknown factor, you need to...

Let's take a look at the equations x 3=12 and 2 y=6 . In them unknown number is the factor on the left side, and the product and the second factor are known. To find the unknown factor, you can use the following rule: to find unknown multiplier, it is necessary to divide the product by a known factor.

This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c , in which a≠0 and b≠0, it follows that c:a=b and c:b=c , and vice versa.

For example, let's find the unknown factor of the equation x·3=12 . According to the rule, we need to divide famous work 12 by a known multiplier of 3 . Let's do : 12:3=4 . So the unknown factor is 4 .

Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12 ,
x=12:3 ,
x=4 .

It is also desirable to check the result: we substitute the found value instead of the letter in the original equation, we get 4 3 \u003d 12 - the correct numerical equality, so we correctly found the value of the unknown factor.

And one more thing: acting according to the studied rule, we actually perform the division of both parts of the equation by a non-zero known multiplier. In grade 6, it will be said that both parts of the equation can be multiplied and divided by the same non-zero number, this does not affect the roots of the equation.

How to find the unknown dividend, divisor?

As part of our topic, it remains to figure out how to find the unknown dividend with a known divisor and quotient, as well as how to find an unknown divisor with a known dividend and quotient. The relationship between multiplication and division already mentioned in the previous paragraph allows you to answer these questions.

To find the unknown dividend, you need to multiply the quotient by the divisor.

Let's consider its application with an example. Solve the equation x:5=9 . To find the unknown divisible of this equation, it is necessary, according to the rule, to multiply the known quotient 9 by the known divisor 5, that is, we perform the multiplication of natural numbers: 9 5 \u003d 45. Thus, the desired dividend is 45.

Let's show short note solutions:
x:5=9 ,
x=9 5 ,
x=45 .

The check confirms that the value of the unknown dividend is found correctly. Indeed, when substituting the number 45 into the original equation instead of the variable x, it turns into the correct numerical equality 45:5=9.

Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.

Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.

Consider an example. Find the unknown divisor from equation 18:x=3 . To do this, we need to divide the known dividend 18 by the known quotient 3, we have 18:3=6. Thus, the required divisor is equal to six.

The solution can also be formulated as follows:
18:x=3 ,
x=18:3 ,
x=6 .

Let's check this result for reliability: 18:6=3 is the correct numerical equality, therefore, the root of the equation is found correctly.

It is clear that this rule can only be used when the quotient is non-zero so as not to encounter division by zero. When the quotient is zero, two cases are possible. If in this case the dividend is equal to zero, that is, the equation has the form 0:x=0 , then this equation satisfies any non-zero value of the divisor. In other words, the roots of such an equation are any numbers that are not equal to zero. If at zero the partial dividend is different from zero, then for any values of the divisor, the original equation does not turn into the correct numerical equality, that is, the equation has no roots. To illustrate, we present the equation 5:x=0 , it has no solutions.

Sharing Rules

Consistent application of the rules for finding the unknown term, minuend, subtrahend, multiplier, dividend and divisor allows solving equations with a single variable more than complex type. Let's deal with this with an example.

Consider the equation 3 x+1=7 . First, we can find the unknown term 3 x , for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6 . Now it remains to find the unknown factor by dividing the product of 6 by the known factor 3 , we have x=6:3 , whence x=2 . So the root of the original equation is found.

To consolidate the material, we present short solution one more 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 ,
2x−7=21 ,
2x=21+7 ,
2x=28 ,
x=28:2 ,
x=14 .

Bibliography.

Mathematics.. 4th grade. Proc. for general education institutions. At 2 o'clock, Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova and others]. - 8th ed. - M.: Education, 2011. - 112 p.: ill. - (School of Russia). - ISBN 978-5-09-023769-7.
Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.

Equations, solving equations

solving equations

3+x=8,
x=8−3,
x=5.

make a check

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x−2=5,
x=5+2,
x=7.

9−x=4,
x=9−4,
x=5.

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How to find the divisor

x 3=12,
x=123,
x=4.

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x5=9,
x=9 5,
x=45.

The solution can also be formulated as follows:
18x=3,
x=183,
x=6.

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(2 x−7)3−5=2,
(2 x−7)3=2+5,
(2 x−7)3=7,
2 x−7=7 3,
2x−7=21,
2x=21+7,
2x=28,
x=282,
x=14.

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Mathematics.
Mathematics

Division. Division with remainder

Definition of division

To divide the number a by the number b means to find such a new number by which b must be multiplied to get a.

This implies the following definition of action: division is called such arithmetic operation, by means of which, given the product of two numbers and one of them (a known factor), another number (an unknown factor) is found.

When dividing this work called divisible, this factor is divider, and the desired factor is private.

Hence it is clear that division is the inverse of multiplication.

The division of the number a by the number b can be written in two ways:

1) or 2), and each of these equalities means that when dividing a number a per number b in the quotient, a natural number q is obtained.

Division with remainder

When requiring that the quotient be an integer, dividing the number a per number b maybe not always.

For example, when you cannot divide 23 by 4, because there is no such integer that you can multiply 4 by and get a product equal to 23.

But you can specify the largest integer, when multiplied by 4, the integer closest to 23 is obtained. This number is 5. When multiplying 5 by 4, we get 20.

The difference between the dividend 23 and 20 is 3 - called the remainder of the division.

The division itself in such cases is called division with remainder.

The case when an integer is obtained in the quotient and there will be no remainder is called division without a remainder or by whole division, the quotient is called complete private or simply private.

If the division of the number a by the number b results in an incomplete quotient q and the remainder r, then it is written as follows.

When dividing with a remainder, an incomplete quotient is called largest number, which, when multiplied by a divisor, gives a product that does not exceed the dividend. The difference between the dividend and this product is called the remainder.

This implies, that there should always be a remainder when dividing less divisor , because if the remainder were equal to or greater than the divisor, then the quotient would not then be the largest possible number. If the remainder is subtracted from the dividend, then the resulting difference ( a - r) is divided by the given divisor b without a remainder, and in the quotient the number will still turn out q.

In terms of division, the difference is .

Hence: (in the sense of division).

The last equality shows that in the case of division with a remainder the dividend is equal to the divisor times the quotient plus the remainder.

Note. Further, the expression: one number is divisible by another without a remainder (completely)- replace with the expression: one number is divisible by another.

Number a in this case is called multiple of b.

Related information:

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Equations, solving equations

Finding an unknown term, multiplier, etc., rules, examples, solutions

Long way to develop skills solving equations starts with solving the very first and relatively simple equations. By such equations we mean equations, on the left side of which is the sum, difference, product or quotient of two numbers, one of which is unknown, and on the right side there is a number. That is, these equations contain an unknown term, minuend, subtrahend, multiplier, dividend, or divisor. The solution of such equations will be discussed in this article.

To find the unknown term, you need to...

Zhenya and Kolya decided to eat apples, for which they began to knock them off the apple tree. Zhenya got 3 apples, and at the end of the process the boys had 8 apples. How many apples did Kolya knock down?

To translate this typical task into mathematical language, we denote the unknown number of apples that Kolya knocked down by x. Then by condition 3 Zhenya's apples and x Kolins together make 8 apples. The last phrase corresponds to an equation of the form 3+x=8. On the left side of this equation is the sum containing the unknown term, on the right side is the value of this sum - the number 8. So how to find the unknown term x that interests us?

There is a rule for this: To find the unknown term, subtract the known term from the sum..

This rule is explained by the fact that subtraction is given a meaning opposite to that of addition. In other words, there is a relationship between addition and subtraction of numbers, which is expressed as follows: from the fact that a+b=c it follows that c−a=b and c−b=a, and vice versa, from c−a=b, as well as from c−b=a it follows that a+b=c.

The voiced rule allows one known term and a known sum to determine another unknown term. It does not matter which of the terms is unknown, the first or the second. Let's consider its application with an example.

Let's go back to our equation 3+x=8. According to the rule, we need to subtract the known term 3 from the known sum 8. That is, we subtract natural numbers: 8−3=5, so we found the unknown term we need, it is equal to 5.

Accepted next form records of the solution of similar equations:

first write down the original equation,
below is the equation obtained after applying the rule for finding the unknown term,
finally, even lower, write down the equation obtained after performing operations with numbers.

The meaning of this form of writing is that the original equation is successively replaced equivalent equations, from which the root of the original equation eventually becomes obvious. They talk about this in detail in algebra lessons in grade 7, but for now let's draw up a solution to our grade 3 level equation:
3+x=8,
x=8−3,
x=5.

In order to verify the correctness of the received answer, it is desirable make a check. To do this, the resulting equation root must be substituted into the original equation and see if this gives the correct numerical equality.

So, we substitute the number 5 instead of x into the original equation 3 + x = 8, we get 3 + 5 = 8 - this equality is correct, therefore, we correctly found the unknown term. If during the check we received an incorrect numerical equality, then this would indicate to us that we had incorrectly solved the equation. The main reasons for this may be either the application of the wrong rule, or computational errors.

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How to find the unknown minuend, subtrahend?

To find the unknown minuend, you need to add the subtrahend to the difference.

For example, consider the equation x−2=5. It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.

If you omit the explanations, then the solution is written as follows:
x−2=5,
x=5+2,
x=7.

For self-control, we will perform a check. Substitute in the original equation found minuend, while we obtain the numerical equality 7−2=5. It is correct, therefore, we can be sure that we have correctly determined the value of the unknown minuend.

You can move on to finding the unknown subtrahend. It is found by adding according to the following rule: to find the unknown subtrahend, it is necessary to subtract the difference from the minuend.

We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9, we have 9−4=5. Thus, the required subtrahend is equal to five.

Here is a short version of the solution to this equation:
9−x=4,
x=9−4,
x=5.

It remains only to check the correctness of the found subtrahend. Let's make a check, for which we substitute the found value 5 instead of x in the original equation, and we get the numerical equality 9−5=4. It is correct, therefore the value of the subtrahend that we found is correct.

And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with the opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.

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To find the unknown factor, you need to...

Let's take a look at the equations x 3=12 and 2 y=6. In them, the unknown number is the factor on the left side, and the product and the second factor are known. To find the unknown factor, you can use the following rule: to find the unknown factor, you need to divide the product by the known factor.

This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c, in which a≠0 and b≠0, it follows that ca=b and cb=c, and vice versa.

For example, let's find the unknown factor of the equation x·3=12. According to the rule, we need to divide the known product 12 by the known factor 3. Let's divide the natural numbers: 123=4. So the unknown factor is 4.

Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12,
x=123,
x=4.

Separately, you need to pay attention to the fact that the voiced rule cannot be used to find an unknown factor when the other factor is zero. For example, this rule is not suitable for solving the equation x·0=11. Indeed, if in this case we adhere to the rule, then in order to find an unknown factor, we need to divide the product 11 by another factor equal to zero, and we cannot divide by zero. We will discuss these cases in detail when we talk about linear equations.

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How to find the unknown dividend, divisor?

To find the unknown dividend, you need to multiply the quotient by the divisor.

Let's consider its application with an example. Let's solve the equation x5=9. To find the unknown divisible of this equation, according to the rule, it is necessary to multiply the known quotient 9 by the known divisor 5, that is, we perform the multiplication of natural numbers: 9 5 \u003d 45. Thus, the desired dividend is 45.

Let's show a short notation of the solution:
x5=9,
x=9 5,
x=45.

Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.

Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.

Consider an example. Find the unknown divisor from the equation 18x=3. To do this, we need to divide the known dividend 18 by the known quotient 3, we have 183=6. Thus, the required divisor is equal to six.

The solution can also be formulated as follows:
18x=3,
x=183,
x=6.

Let's check this result for reliability: 186=3 - the correct numerical equality, therefore, the root of the equation is found correctly.

It is clear that this rule can only be applied when the quotient is different from zero, so as not to encounter division by zero. When the quotient is zero, two cases are possible. If in this case the dividend is equal to zero, that is, the equation has the form 0x=0, then this equation satisfies any non-zero value of the divisor. In other words, the roots of such an equation are any numbers that are not equal to zero. If, when the quotient is equal to zero, the dividend is different from zero, then for any values of the divisor, the original equation does not turn into a true numerical equality, that is, the equation has no roots. To illustrate, we present the equation 5x=0, it has no solutions.

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Sharing Rules

Consistent application of the rules for finding the unknown term, minuend, subtrahend, multiplier, dividend and divisor allows solving equations with a single variable of a more complex form. Let's deal with this with an example.

Consider the equation 3 x+1=7. First, we can find the unknown term 3 x, for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6. Now it remains to find the unknown factor by dividing the product of 6 by the known factor of 3, we have x=63, whence x=2. So the root of the original equation is found.

To consolidate the material, we present a brief solution of 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,
2x−7=21,
2x=21+7,
2x=28,
x=282,
x=14.

Top of page

Mathematics.. 4th grade. Proc. for general education institutions. At 2 h. Ch. 1 / .- 8th ed. — M.: Enlightenment, 2011. — 112 p.: ill. - (School of Russia). — ISBN 978-5-09-023769-7.
Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. — M.: Mnemozina, 2007. — 280 p.: ill. ISBN 5-346-00699-0.

Equations, solving equations

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

To translate this typical problem into mathematical language, let's denote the unknown number of apples that Kolya knocked down as x. Then by condition 3 Zhenya's apples and x Kolins together make 8 apples. The last phrase corresponds to an equation of the form 3+x=8. On the left side of this equation is the sum containing the unknown term, on the right side is the value of this sum - the number 8. So how to find the unknown term x that interests us?

There is a rule for this: To find the unknown term, subtract the known term from the sum..

The following form of writing the solution of such equations is adopted:

first write down the original equation,
below is the equation obtained after applying the rule for finding the unknown term,
finally, even lower, write down the equation obtained after performing operations with numbers.

The meaning of this form of writing is that the original equation is successively replaced by equivalent equations, from which the root of the original equation eventually becomes obvious. They talk about this in detail in algebra lessons in grade 7, but for now let's draw up a solution to our grade 3 level equation:
3+x=8,
x=8−3,
x=5.

So, we substitute the number 5 instead of x into the original equation 3 + x = 8, we get 3 + 5 = 8 - this equality is correct, therefore, we correctly found the unknown term. If during the check we received an incorrect numerical equality, then this would indicate to us that we had incorrectly solved the equation. The main reasons for this may be either the application of the wrong rule, or computational errors.

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How to find the unknown minuend, subtrahend?

To find the unknown minuend, you need to add the subtrahend to the difference.

For example, consider the equation x−2=5. It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.

If you omit the explanations, then the solution is written as follows:
x−2=5,
x=5+2,
x=7.

We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9, we have 9−4=5. Thus, the required subtrahend is equal to five.

Here is a short version of the solution to this equation:
9−x=4,
x=9−4,
x=5.

And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with the opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.

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To find the unknown factor, you need to...

Let's take a look at the equations x 3=12 and 2 y=6. In them, the unknown number is the factor on the left side, and the product and the second factor are known.

How to find a quotient divisor I write rules that are not memorable

To find the unknown factor, you can use the following rule: to find the unknown factor, you need to divide the product by the known factor.

This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c, in which a≠0 and b≠0, it follows that ca=b and cb=c, and vice versa.

Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12,
x=123,
x=4.

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How to find the unknown dividend, divisor?

To find the unknown dividend, you need to multiply the quotient by the divisor.

Let's show a short notation of the solution:
x5=9,
x=9 5,
x=45.

Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.

Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.

The solution can also be formulated as follows:
18x=3,
x=183,
x=6.

Let's check this result for reliability: 186=3 - the correct numerical equality, therefore, the root of the equation is found correctly.

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Sharing Rules

Consider the equation 3 x+1=7. First, we can find the unknown term 3 x, for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6. Now it remains to find the unknown factor by dividing the product of 6 by the known factor of 3, we have x=63, whence x=2. So the root of the original equation is found.

To consolidate the material, we present a brief solution of 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,
2x−7=21,
2x=21+7,
2x=28,
x=282,
x=14.

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Mathematics.. 4th grade. Proc. for general education institutions. At 2 h. Ch. 1 / .- 8th ed. — M.: Enlightenment, 2011. — 112 p.: ill. - (School of Russia). — ISBN 978-5-09-023769-7.
Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. — M.: Mnemozina, 2007. — 280 p.: ill. ISBN 5-346-00699-0.

Equations, solving equations

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

There is a rule for this: To find the unknown term, subtract the known term from the sum..

The following form of writing the solution of such equations is adopted:

first write down the original equation,
below is the equation obtained after applying the rule for finding the unknown term,
finally, even lower, write down the equation obtained after performing operations with numbers.

So, we substitute the number 5 instead of x into the original equation 3 + x = 8, we get 3 + 5 = 8 - this equality is correct, therefore, we correctly found the unknown term. If during the check we received an incorrect numerical equality, then this would indicate to us that we had incorrectly solved the equation. The main reasons for this may be either the application of the wrong rule, or computational errors.

Top of page

How to find the unknown minuend, subtrahend?

To find the unknown minuend, you need to add the subtrahend to the difference.

For example, consider the equation x−2=5. It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.

If you omit the explanations, then the solution is written as follows:
x−2=5,
x=5+2,
x=7.

We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9, we have 9−4=5. Thus, the required subtrahend is equal to five.

Here is a short version of the solution to this equation:
9−x=4,
x=9−4,
x=5.

And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with the opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.

Top of page

To find the unknown factor, you need to...

This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c, in which a≠0 and b≠0, it follows that ca=b and cb=c, and vice versa.

Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12,
x=123,
x=4.

Top of page

How to find the unknown dividend, divisor?

To find the unknown dividend, you need to multiply the quotient by the divisor.

Let's show a short notation of the solution:
x5=9,
x=9 5,
x=45.

Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.

Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.

The solution can also be formulated as follows:
18x=3,
x=183,
x=6.

Let's check this result for reliability: 186=3 - the correct numerical equality, therefore, the root of the equation is found correctly.

dividend divisor private rule

Top of page

Sharing Rules

Consider the equation 3 x+1=7. First, we can find the unknown term 3 x, for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6. Now it remains to find the unknown factor by dividing the product of 6 by the known factor of 3, we have x=63, whence x=2. So the root of the original equation is found.

To consolidate the material, we present a brief solution of 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,
2x−7=21,
2x=21+7,
2x=28,
x=282,
x=14.

Top of page

Mathematics.. 4th grade. Proc. for general education institutions. At 2 h. Ch. 1 / .- 8th ed. — M.: Enlightenment, 2011. — 112 p.: ill. - (School of Russia). — ISBN 978-5-09-023769-7.
Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. — M.: Mnemozina, 2007. — 280 p.: ill. ISBN 5-346-00699-0.

Equations, solving equations

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

There is a rule for this: To find the unknown term, subtract the known term from the sum..

The following form of writing the solution of such equations is adopted:

first write down the original equation,
below is the equation obtained after applying the rule for finding the unknown term,
finally, even lower, write down the equation obtained after performing operations with numbers.

So, we substitute the number 5 instead of x into the original equation 3 + x = 8, we get 3 + 5 = 8 - this equality is correct, therefore, we correctly found the unknown term. If during the check we received an incorrect numerical equality, then this would indicate to us that we had incorrectly solved the equation. The main reasons for this may be either the application of the wrong rule, or computational errors.

Top of page

How to find the unknown minuend, subtrahend?

To find the unknown minuend, you need to add the subtrahend to the difference.

For example, consider the equation x−2=5. It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.

If you omit the explanations, then the solution is written as follows:
x−2=5,
x=5+2,
x=7.

We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9, we have 9−4=5. Thus, the required subtrahend is equal to five.

Here is a short version of the solution to this equation:
9−x=4,
x=9−4,
x=5.

And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with the opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.

Top of page

To find the unknown factor, you need to...

This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c, in which a≠0 and b≠0, it follows that ca=b and cb=c, and vice versa.

Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12,
x=123,
x=4.

What is the dividend, divisor, quotient and remainder (examples)?

Indeed, if in this case we adhere to the rule, then in order to find an unknown factor, we need to divide the product 11 by another factor equal to zero, and we cannot divide by zero. We will discuss these cases in detail when we talk about linear equations.

Top of page

How to find the unknown dividend, divisor?

To find the unknown dividend, you need to multiply the quotient by the divisor.

Let's show a short notation of the solution:
x5=9,
x=9 5,
x=45.

Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.

Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.

The solution can also be formulated as follows:
18x=3,
x=183,
x=6.

Let's check this result for reliability: 186=3 - the correct numerical equality, therefore, the root of the equation is found correctly.

Top of page

Sharing Rules

Consider the equation 3 x+1=7. First, we can find the unknown term 3 x, for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6. Now it remains to find the unknown factor by dividing the product of 6 by the known factor of 3, we have x=63, whence x=2. So the root of the original equation is found.

To consolidate the material, we present a brief solution of 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,
2x−7=21,
2x=21+7,
2x=28,
x=282,
x=14.

Top of page

Mathematics.. 4th grade. Proc. for general education institutions. At 2 h. Ch. 1 / .- 8th ed. — M.: Enlightenment, 2011. — 112 p.: ill. - (School of Russia). — ISBN 978-5-09-023769-7.
Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. — M.: Mnemozina, 2007. — 280 p.: ill. ISBN 5-346-00699-0.

Portal for the student. Self-training

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Instruction

Finding the least common multiple: basic concepts

How to find the unknown minuend, subtrahend?

To find the unknown factor, you need to...

How to find the unknown dividend, divisor?

Sharing Rules

How to find the divisor

Division. Division with remainder

What is a quotient of numbers

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

How to find the unknown minuend, subtrahend?

To find the unknown factor, you need to...

How to find the unknown dividend, divisor?

Sharing Rules

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

How to find the unknown minuend, subtrahend?

To find the unknown factor, you need to...

How to find a quotient divisor I write rules that are not memorable

How to find the unknown dividend, divisor?

Sharing Rules

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

How to find the unknown minuend, subtrahend?

To find the unknown factor, you need to...

How to find the unknown dividend, divisor?

dividend divisor private rule

Sharing Rules

Finding an unknown term, multiplier, etc., rules, examples, solutions

To find the unknown term, you need to...

How to find the unknown minuend, subtrahend?

To find the unknown factor, you need to...

What is the dividend, divisor, quotient and remainder (examples)?

How to find the unknown dividend, divisor?

Sharing Rules

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