In this article, we will take a close look at division with remainder. Let's start with a general idea about this action, then find out the meaning of dividing natural numbers with a remainder, and introduce the necessary terms. Then we outline the range of problems solved by dividing natural numbers with a remainder. In conclusion, let's dwell on all kinds of connections between the dividend, the divisor, the incomplete quotient and the remainder of the division.

Page navigation.

Answer:

The dividend is 79.

It should also be noted that checking the result of dividing natural numbers with a remainder is carried out by checking the validity of the resulting equality a=b·c+d .

Finding the remainder if the dividend, divisor and incomplete quotient are known

In its meaning, the remainder d is the number of elements that remains in the original set after the exclusion from its a elements b times c elements each. Therefore, by virtue of the sense of multiplication of natural numbers and the sense of subtraction of natural numbers, the equality d=a−b c. Thus, the remainder d of dividing a natural number a by a natural number b is equal to the difference between the dividend a and the product of the divisor b and the incomplete quotient c.

The resulting connection d=a−b·c allows you to find the remainder when the dividend, divisor and incomplete quotient are known. Let's consider an example solution.

Consider a simple example:
15:5=3
In this example, we divided the natural number 15 completely 3, no remainder.

Sometimes a natural number cannot be completely divided. For example, consider the problem:
There were 16 toys in the closet. There were five children in the group. Each child took the same number of toys. How many toys does each child have?

Solution:
Divide the number 16 by 5 by a column and get:

We know that 16 times 5 is not divisible. The nearest smaller number that is divisible by 5 is 15 with a remainder of 1. We can write the number 15 as 5⋅3. As a result (16 - dividend, 5 - divisor, 3 - partial quotient, 1 - remainder). Got formula division with remainder which can be done solution verification.

a= b⋅ c+ d
a - divisible
b - divider,
c - incomplete quotient,
d - remainder.

Answer: Each child will take 3 toys and one toy will remain.

Remainder of the division

The remainder must always be less than the divisor.

If the remainder is zero when dividing, then the dividend is divisible. completely or no remainder per divisor.

If, when dividing, the remainder is greater than the divisor, this means that the number found is not the largest. There is a larger number that will divide the dividend and the remainder will be less than the divisor.

Questions on the topic “Division with remainder”:
Can the remainder be greater than the divisor?
Answer: no.

Can the remainder be equal to the divisor?
Answer: no.

How to find the dividend by the incomplete quotient, divisor and remainder?
Answer: we substitute the values of the incomplete quotient, divisor and remainder into the formula and find the dividend. Formula:
a=b⋅c+d

Example #1:
Perform division with a remainder and check: a) 258:7 b) 1873:8

Solution:
a) Divide in a column:

258 - divisible,
7 - divider,
36 - incomplete quotient,
6 - remainder. Remainder less than divisor 6<7.

7⋅36+6=252+6=258

b) Divide in a column:

1873 - divisible,
8 - divider,
234 - incomplete quotient,
1 is the remainder. Remainder less than divisor 1<8.

Substitute in the formula and check whether we solved the example correctly:
8⋅234+1=1872+1=1873

Example #2:
What remainders are obtained when dividing natural numbers: a) 3 b) 8?

Answer:
a) The remainder is less than the divisor, therefore less than 3. In our case, the remainder can be 0, 1 or 2.
b) The remainder is less than the divisor, therefore, less than 8. In our case, the remainder can be 0, 1, 2, 3, 4, 5, 6 or 7.

Example #3:
What is the largest remainder that can be obtained by dividing natural numbers: a) 9 b) 15?

Answer:
a) The remainder is less than the divisor, therefore, less than 9. But we need to indicate the largest remainder. That is, the nearest number to the divisor. This number is 8.
b) The remainder is less than the divisor, therefore, less than 15. But we need to indicate the largest remainder. That is, the nearest number to the divisor. This number is 14.

Example #4:
Find the dividend: a) a: 6 \u003d 3 (rem. 4) b) c: 24 \u003d 4 (rem. 11)

Solution:
a) Solve using the formula:
a=b⋅c+d
(a is the dividend, b is the divisor, c is the partial quotient, d is the remainder.)
a:6=3(rest.4)
(a is the dividend, 6 is the divisor, 3 is the incomplete quotient, 4 is the remainder.) Substitute the numbers in the formula:
a=6⋅3+4=22
Answer: a=22

b) Solve using the formula:
a=b⋅c+d
(a is the dividend, b is the divisor, c is the partial quotient, d is the remainder.)
s:24=4(rest.11)
(c is the dividend, 24 is the divisor, 4 is the incomplete quotient, 11 is the remainder.) Substitute the numbers in the formula:
c=24⋅4+11=107
Answer: s=107

Task:

Wire 4m. must be cut into pieces of 13 cm. How many of these pieces will there be?

Solution:
First you need to convert meters to centimeters.
4m.=400cm.
You can divide by a column or in your mind we get:
400:13=30(rest 10)
Let's check:
13⋅30+10=390+10=400

Answer: 30 pieces will turn out and 10 cm of wire will remain.

Division with a remainder takes place in the third grade of elementary school. The topic is quite difficult for a child to understand and requires him to have an almost perfect knowledge of the multiplication table. But all mathematical knowledge improves with practice, and therefore, solving tasks, the child with each example will complete it faster and with fewer errors. Our simulator involves practicing the skill of fast division with a remainder.

How to divide with a remainder

1. We determine that the division is with a remainder (does not divide completely).

34:6 is not resolved without a remainder

2. We select the nearest smaller number to the first (divisible), which is divisible by the second (divisor).

The closest number to 34 that is divisible by 6 is 30

3. Perform the division of this number by the divisor.

4. We write the answer (private).

5. To find the remainder, subtract from the first number (divisible) the number that was selected. We write down the rest. When dividing with a remainder, the remainder must always be less than the divisor.

34-30=4 (rest 4) 4<6 Ответ: 34:6=5 (ост.4)

We check the division like this:

We multiply the answer by the divisor (second number) and add the remainder to the answer. If the dividend is obtained (the first number), then the division was performed correctly.

5*6+4=34 The division is correct.

Large numbers are easily and simply divided by a column. In this case, in the corner under the divisor, we will write an integer, and at the very bottom there will be a remainder that is less than the divisor.

If, when dividing with a remainder, the dividend is less than the divisor, then their partial quotient is zero, and the remainder is equal to the dividend.

For example:

6: 10 = 0 (rest 6)
14: 112 = 0 (rest 14)

The following video shows how to divide large numbers with a remainder by a column:

Download training cards for division with a remainder

Save the card sheet to your computer and print on A4. One sheet is enough for 5 days of working out the division with the remainder. It has 5 columns with examples. You can even cut the sheet into 5 pieces. Above each column is a cloud, a smiley and a sun, let the child evaluate his work when he finishes the column.

It is easy to teach a child to divide by a column. It is necessary to explain the algorithm of this action and consolidate the material covered.

According to the school curriculum, children begin to explain division by a column already in the third grade. Students who grasp everything “on the fly” quickly understand this topic
But, if the child fell ill and missed the lessons of mathematics, or he did not understand the topic, then the parents must explain the material to the child on their own. It is necessary to convey information to him as clearly as possible.
Moms and dads during the educational process of the child must be patient, showing tact in relation to their child. In no case should you yell at a child if something does not work out for him, because this way you can discourage him from all the desire to study

Important: In order for a child to understand the division of numbers, he must thoroughly know the multiplication table. If the kid does not know multiplication well, he will not understand division.

During home extra classes, cheat sheets can be used, but the child must learn the multiplication table before proceeding to the topic “Division”.

So how do you explain to a child column division:

Try to explain in small numbers first. Take counting sticks, for example, 8 pieces
Ask the child how many pairs are in this row of sticks? Correct - 4. So, if you divide 8 by 2, you get 4, and if you divide 8 by 4, you get 2
Let the child divide by himself another number, for example, a more complex one: 24:4
When the baby has mastered the division of prime numbers, then you can proceed to the division of three-digit numbers into single-digit

Division is always given to children a little more difficult than multiplication. But diligent additional classes at home will help the baby understand the algorithm of this action and keep up with their peers at school.

Start simple - division by a single digit:

Important: Calculate in your mind so that the division turns out without a remainder, otherwise the child may get confused.

For example, 256 divided by 4:

Draw a vertical line on a sheet of paper and divide it in half on the right side. Write the first number on the left, and the second on the right above the line.
Ask the baby how many fours fit in a two - not at all
Then we take 25. For clarity, separate this number from above with a corner. Again ask the child how many fours fit in twenty-five? That's right, six. We write the number "6" in the lower right corner under the line. The child must use the multiplication table for the correct answer.
Write down the number 24 under 25, and underline to write down the answer - 1
Ask again: how many fours can fit in a unit - not at all. Then we demolish the number "6" to one
It turned out 16 - how many fours fit in this number? Correct - 4. We write down "4" next to "6" in the answer
Under 16 we write 16, underline and it turns out “0”, which means we divided correctly and the answer turned out to be “64”

Written division by two digits

When the child has mastered the division by a single number, you can move on. Written division by a two-digit number is a little more complicated, but if the baby understands how this action is performed, then it will not be difficult for him to solve such examples.

Important: Again, start explaining with simple steps. The child will learn to correctly select numbers and it will be easy for him to divide complex numbers.

Perform together this simple action: 184:23 - how to explain:

First we divide 184 by 20, it turns out approximately 8. But we do not write the number 8 in the answer, since this is a trial number
Check if 8 fits or not. We multiply 8 by 23, it turns out 184 - this is exactly the number that we have in the divisor. The answer will be 8

Important: For the child to understand, try taking 9 instead of the eight, let him multiply 9 by 23, it turns out 207 - this is more than we have in the divisor. The number 9 does not suit us.

So gradually the baby will understand the division, and it will be easy for him to divide more complex numbers:

Divide 768 by 24. Determine the first digit of the private - we divide 76 not by 24, but by 20, it turns out 3. We write 3 in response under the line to the right
Under 76 we write down 72 and draw a line, write down the difference - it turned out 4. Is this figure divisible by 24? No - we demolish 8, it turns out 48
Is 48 divisible by 24? That's right - yes. It turns out 2, we write this figure in response
It turned out 32. Now you can check whether we performed the division action correctly. Multiply in a column: 24x32, it turns out 768, then everything is correct

If the child has learned to divide by a two-digit number, then you need to move on to the next topic. The algorithm for dividing by a three-digit number is the same as the algorithm for dividing by a two-digit number.

For example:

Divide 146064 by 716. First we take 146 - ask the child if this number is divisible by 716 or not. That's right - no, then we take 1460
How many times will the number 716 fit in the number 1460? Correct - 2, so we write this figure in the answer
We multiply 2 by 716, it turns out 1432. We write this figure under 1460. It turns out the difference is 28, we write under the line
Demolition 6. Ask the child - 286 is divisible by 716? That's right - no, so we write 0 in the answer next to 2. We demolish another number 4
We divide 2864 by 716. We take 3 each - a little, 5 each - a lot, which means we get 4. We multiply 4 by 716, we get 2864
Write 2864 under 2864 for a difference of 0. Answer 204

Important: To check the correctness of the division, multiply together with the child in a column - 204x716 = 146064. The division is correct.

It's time for the child to explain that division can be not only whole, but also with a remainder. The remainder is always less than or equal to the divisor.

Division with a remainder should be explained with a simple example: 35:8=4 (remainder 3):

How many eights fit in 35? Correct - 4. Remains 3
Is this number divisible by 8? That's right - no. So the remainder is 3.

After that, the child should learn that you can continue the division by adding 0 to the number 3:

The answer is the number 4. After it, we write a comma, since adding zero indicates that the number will be with a fraction
It turned out 30. Divide 30 by 8, it turns out 3. We write in response, and under 30 we write 24, underline and write 6
We carry the number 0 to the number 6. Divide 60 by 8. Take 7 each, it turns out 56. Write under 60 and write down the difference 4
We add 0 to the number 4 and divide by 8, it turns out 5 - we write it down in response
We subtract 40 from 40, we get 0. So, the answer is: 35:8=4.375

Tip: If the child does not understand something, do not be angry. Let a couple of days go by and try to explain the material again.

Mathematics lessons at school will also reinforce knowledge. Time will pass and the kid will quickly and easily solve any division examples.

The algorithm for dividing numbers is as follows:

Make an estimate of the number that will be in the answer
Find the first incomplete dividend
Determine the number of digits in a quotient
Find the digits in each digit of the quotient
Find the remainder (if any)

According to this algorithm, division is performed both by single-digit numbers and by any multi-digit number (two-digit, three-digit, four-digit, and so on).

When studying with a child, often ask him examples for making an estimate. He must quickly calculate the answer in his mind. For example:

1428:42
2924:68
30296:56
136576:64
16514:718

To consolidate the result, you can use the following division games:

"Puzzle". Write five examples on a piece of paper. Only one of them should be with the correct answer.

Condition for the child: Among several examples, only one is solved correctly. Find him in a minute.

Video: Arithmetic game for kids addition subtraction division multiplication

Video: Educational cartoon Mathematics Learning by heart the multiplication and division tables by 2

Video: Introduction to division | Fun MATH for kids

Video: Dividing a two-digit number by a single one

When the child additionally studies at home, he consolidates the material covered at school. Thanks to this, it is easier for him to learn and he will not lag behind his peers. Therefore, help your children, study at home with them. and the baby will succeed!

Video: Long division part 1

Video: Long division part 2

Video: Long division part 3

Video: Long division part 4

Video: Long division part 5

What does 3rd grade do in math? Division with remainder, examples and tasks - that's what is studied in the lessons. Division with a remainder and the algorithm for such calculations will be discussed in the article.

Peculiarities

Consider the topics included in the program that Grade 3 is studying. Division with a remainder is a special section of mathematics. What is it about? If the dividend is not evenly divisible by the divisor, then the remainder remains. For example, we divide 21 by 6. It turns out 3, but the remainder remains 3.

In cases where, during the division of natural numbers, the remainder is equal to zero, they say that the division was made by an integer. For example, if 25 is divided by 5, the result is 5. The remainder is zero.

Solution of examples

In order to perform division with a remainder, a specific notation is used.

Let's give examples in mathematics (Grade 3). Division with a remainder can be left out. It is enough to write in a line: 13:4=3 (remainder 1) or 17:5=3 (remainder 2).

Let's analyze everything in more detail. For example, when 17 is divided by three, the integer five is obtained, in addition, the remainder is two. What is the procedure for solving such an example for division with a remainder? First you need to find the maximum number up to 17, which can be divided without a remainder by three. The largest will be 15.

Next, 15 is divided by the number three, the result of the action will be the number five. Now we subtract the number we found from the divisible, that is, subtract 15 from 17, we get two. Mandatory action is the reconciliation of the divisor and the remainder. After verification, the response of the action taken is necessarily recorded. 17:3=15 (remainder 2).

If the remainder is greater than the divisor, the action was not performed correctly. It is according to this algorithm that class 3 division with a remainder performs. The examples are first analyzed by the teacher on the blackboard, then the children are invited to test their knowledge by conducting independent work.

Multiplication example

One of the most difficult topics that grade 3 faces is division with a remainder. Examples can be complex, especially when additional column calculations are required.

Let's say you need to divide the number 190 by 27 to get the minimum remainder. Let's try to solve the problem using multiplication.

We select a number that, when multiplied, will give a figure as close as possible to the number 190. If we multiply 27 by 6, we get the number 162. Subtract the number 162 from 190, the remainder will be 28. It turned out to be more than the original divisor. Therefore, the number six is not suitable for our example as a multiplier. Let's continue the solution of the example, taking the number 7 for multiplication.

Multiplying 27 by 7, we get the product 189. Next, we will check the correctness of the solution, for this we subtract the result obtained from 190, that is, subtract the number 189. The remainder will be 1, which is clearly less than 27. This is how complex expressions are solved at school (Grade 3, division with remainder). Examples always include a response record. The whole mathematical expression can be formulated as follows: 190:27=7 (remainder 1). Similar calculations can be made in a column.

This is how class 3 division with a remainder performs. The examples given above will help to understand the algorithm for solving such problems.

Conclusion

In order for primary school students to form the correct computational skills, the teacher, during mathematics classes, must pay attention to explaining the algorithm of the child’s actions when solving tasks for division with a remainder.

According to the new federal state educational standards, special attention is paid to an individual approach to learning. The teacher should select tasks for each child, taking into account his individual abilities. At each stage of teaching the rules of division with a remainder, the teacher must carry out intermediate control. It allows him to identify the main problems that arise with the assimilation of the material for each student, timely correct knowledge and skills, eliminate emerging problems, and get the desired result.