Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in Everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.
Division is an interesting operation, as we will see in this article!
Dividing numbers
So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.
It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.
So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.
Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.
Division with remainder
What is division with a remainder? This is the same division, only the result is not an even number, as shown above.
For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).
For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).
Division by 3 and 9
A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (depending on what you need).
If the answer is obtained without a remainder, then the number will be divided without a remainder.
For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.
For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.
Multiplication and division
Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.
Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. IN in this case verification is done by multiplying the answer by the divisor.
Division 3 class
In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:
Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?
Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?
Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?
Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?
Division 4th grade
The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:
Column division
What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4, can be divided, and the answer is clear - 4. That 512:8 in the mind is not easy for a child. And tell us about the solution technique similar examples- our task.
Let's look at an example, 512:8.
1 step. Let's write the dividend and divisor as follows:
The quotient will ultimately be written under the divisor, and the calculations under the dividend.
Step 2. We start dividing from left to right. First we take the number 5: 
Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:
Now 51 is greater than 8. This is an incomplete quotient.
Step 4. We put a dot under the divisor.
Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:
Step 6. We begin the division operation. Largest number, divisible by 8 without a remainder to 51 – 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
Step 7. Then write the number exactly below the number 51 and put a “-” sign:
Step 8. Then we subtract 48 from 51 and get the answer 3.
* 9 step*. We take down the number 2 and write it next to the number 3:
Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.
So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.
Division of three digits
Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.
Division of fractions
Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):
As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.
Dividing numbers into classes
Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.
Division of natural numbers
Division natural numbers– this is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers. 
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Division presentation
Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!
Examples for division
Easy level
Average level
Difficult level
Games for developing mental arithmetic
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve skills oral counting in an interesting playful way.
Game "Guess the operation"
The game “Guess the Operation” develops thinking and memory. The main point games need to be selected mathematical sign so that the equality is true. There are examples on the screen, look carefully and put the right sign"+" or "-" so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
Game "Simplification"
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
Game "Piggy Bank"
The Piggy Bank game develops thinking and memory. The main point of the game is to choose which piggy bank to use more money.In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.
Game "Fast addition reload"
The game “Fast addition reboot” develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to the given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three numbers and press them. If you answered correctly, then you score points and continue playing.
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Let's consider the concept of division in the problem:
There were 12 apples in the basket. Six children sorted the apples. Each child got the same number of apples. How many apples does each child have?
Solution:
We need 12 apples to divide among six children. Let's write down problem 12:6 mathematically.
Or you can say it differently. What number must the number 6 be multiplied by to get the number 12? Let's write the problem in the form of an equation. We don’t know the number of apples, so let’s denote them as the variable x.
To find the unknown x we need 12:6=2
Answer: 2 apples for each child.
Let's take a closer look at the example 12:6=2:
The number 12 is called divisible. This is the number that is being divided.
The number 6 is called divider. This is the number that is divided by.
And the result of dividing the number 2 is called private. The quotient shows how many times the dividend is greater than the divisor.
In literal form, the division looks like this:
a:b=c
a– divisible,
b- divider,
c– private.
So what is division?
Division- this is the inverse action of one factor, we can find another factor.
Division is checked by multiplication, that is:
a:
b=
c, check with⋅b=
a
18:9=2, check 2⋅9=18
Unknown multiplier.
Let's consider the problem:
Each package contains 3 pieces of Christmas balls. To decorate the Christmas tree we need 30 balls. How many packages of Christmas balls do we need?
Solution:
x – unknown number of packages of balls.
3 – pieces in one package of balloons.
30 – total balls.
x⋅3=30 we need to take 3 so many times to get a total of 30. x is an unknown factor. That is, To find the unknown you need to divide the product by the known factor.
x=30:3
x=10.
Answer: 10 packs of balloons.
Unknown dividend.
Let's consider the problem:
Each package contains 6 colored pencils. There are 3 packs in total. How many pencils were there in total before they were put into packages?
Solution:
x – total pencils,
6 pencils in each package,
3 – packs of pencils.
Let's write the equation of the problem in division form.
x:6=3
x is the unknown dividend. To find the unknown dividend, you need to multiply the quotient by the divisor.
x=3⋅6
x=18
Answer: 18 pencils.
Unknown divisor.
Let's look at the problem:
There were 15 balls in the store. During the day, 5 customers came to the store. Buyers bought an equal number of balloons. How many balloons did each customer buy?
Solution:
x – the number of balls that one buyer bought,
5 – number of buyers,
15 – number of balls.
Let's write the equation of the problem in division form:
15:x=5
x – in this equation is an unknown divisor. To find the unknown divisor, we divide the dividend by the quotient.
x=15:5
x=3
Answer: 3 balls for each buyer.
Properties of dividing a natural number by one.
Division rule:
Any number divided by 1 results in the same number.
7:1=7
a:1=
a
Properties of dividing a natural number by zero.
Let's look at an example: 6:2=3, you can check whether we divided correctly by multiplying 2⋅3=6.
If we are 3:0, then we will not be able to check, because any number multiplied by zero will be zero. Therefore, recording 3:0 makes no sense.
Division rule:
You cannot divide by zero.
Properties of dividing zero by a natural number.
0:3=0 this entry makes sense. If we divide anything into three parts, we get nothing.
0:
a=0
Division rule:
When dividing 0 by any natural number not equal to zero, the result will always be 0.
The property of dividing identical numbers.
3:3=1
a:
a=1
Division rule:
When dividing any number by itself that is not equal to zero, the result will be 1.
Questions on the topic “Division”:
In the entry a:b=c, what is quotient here?
Answer: a:b and c.
What is private?
Answer: the quotient shows how many times the dividend is greater than the divisor.
At what value of m is the entry 0⋅m=5?
Answer: when multiplied by zero, the answer will always be 0. The entry does not make sense.
Is there such an n such that 0⋅n=0?
Answer: Yes, the entry makes sense. When any number is multiplied by 0, it will be 0, so n is any number.
Example #1:
Find the value of the expression: a) 0:41 b) 41:41 c) 41:1
Answer: a) 0:41=0 b) 41:41=1 c) 41:1=41
Example #2:
For what values of the variables is the equality true: a) x:6=8 b) 54:x=9
a) x – in this example is divisible. To find the dividend, you need to multiply the quotient by the divisor.
x – unknown dividend,
6 – divisor,
8 – quotient.
x=8⋅6
x=48
b) 54 – dividend,
x is a divisor,
9 – quotient.
To find an unknown divisor, you need to divide the dividend by the quotient.
x=54:9
x=6
Task #1:
Sasha has 15 marks, and Misha has 45 marks. How many times more stamps does Misha have than Sasha?
Solution:
The problem can be solved in two ways. First way:
15+15+15=45
It takes 3 numbers 15 to get 45, therefore, Misha has 3 times more marks than Sasha.
Second way:
45:15=3
Answer: Misha has 3 times more stamps than Sasha.
MATHEMATICS
5 CLASS
DIVISION OF NATURAL NUMBERS.
Plan - summary of the lesson “Division of natural numbers”.
Item: mathematics
Class: 5
Lesson topic: Division of natural numbers.
Lesson number in topic: Lesson 4 out of 7
Basic tutorial: Mathematics. 5th grade: textbook for
educational institutions / N.Ya.Vilenkin, V.I.Zhokhov, A.S.Chesnokov, S.I.Shvartsburd. – 25th edition, erased. – M.: Mnemosyne, 2009
The purpose of the lesson: create conditions for reproducing and adjusting the necessary knowledge and skills, analyzing tasks and methods of performing them; completing tasks independently; external and internal control.
As a result, students should:
be able to divide natural numbers;
be able to solve equations and word problems;
be able to draw conclusions;
be able to develop an algorithm of actions;
use mathematically literate language;
display the content of the actions performed in speech;
evaluate yourself and your comrades.
Forms of student work: frontal, steam room, individual.
Necessary Technical equipment: computer, multimedia projector, mathematics textbooks, handouts (for mental calculation, for work in class, for homework), electronic presentation made in Power Point.
Routing lesson.
| Lesson stage | Tasks | Time | Task performance indicators |
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| teachers | student |
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| Stage 1. Organizational. | Checking class readiness. | The short duration of the moment. |
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| Stage 2. Checking homework. | The teacher collects notebooks with homework. | Students hand in their notebooks. | Before the lesson. | Homework will be checked for each student. |
| Stage 3. Updating knowledge. | Teacher's opening speech. Verbal counting. Game "Mathematical Lotto". Historical reference. | Solve examples of mental calculation. Answer the question posed by the teacher. They work in pairs. | Development of group work skills. Students' basic knowledge was tested. |
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| Stage 4. | Together with the students, he determines the purpose of the lesson. | Determine the purpose of the lesson. | The goal of the lesson has been set. |
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| Stage 5. | Directs students' work. | Solve tasks involving calculating the values of numerical expressions, equations, and problems. Perform self-checks and draw conclusions. | Establishing the correctness and awareness of studying the topic. Identification of comprehension and correction of identified gaps. |
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| Stage 6. Physical exercise. | Manages the presentation. | The change of activity provided emotional relief for students. |
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| Stage 7. | Directs students' work. | Perform test tasks independently. | The correctness and awareness of the studied topic is established. |
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| Stage 8. | Self-assessment of activity. |
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| Stage 9. | Students write down the assignment in their diary. | Students understood the purpose, content and methods of completing homework. |
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Description of the procedural part of the lesson.
| Lesson stage | Teacher activities | Student activity |
| Stage 1. Organizational. | The teacher welcomes students and checks their readiness for the lesson. | Greet the teacher and sit down. |
| Stage 2. Checking homework. | The teacher checks the availability of homework notebooks. | All students handed in their notebooks for checking. |
| Stage 3. Updating knowledge. | It is difficult to master any topic in mathematics without the ability to count quickly and accurately, therefore, as always, we begin the lesson with mental calculation. (Work in pairs). Hold hands and show that you are a couple. There are envelopes on your tables for mental calculations. Solve the examples orally and cover them with a card with the answer. Using the key (slide No. 1), replace the resulting numbers corresponding letters. Read the given word. | Solve one of 3 tasks. 42-d; 22nd; 10-l; 15th; 37th; 19-o; 39th; 9-t; 700 l; 20-hour; 16-a; 1-s; 36-n; 110o; 22nd. Received the words: dividend, divisor, quotient. |
| Stage 4. Setting goals, lesson objectives, motivational activities of students. | What action do all these concepts refer to? Yes, today we will continue to work on dividing natural numbers. This is not the first lesson on the topic. What goal can you set for yourself for this lesson? In the meantime, a little additional information. Students prepared reports on the topic. (Slides No. 2, No. 3, No. 4). №2 . Vladimir Ivanovich Dal - author “Explanatory Dictionary of the Living Great Russian Language” in his dictionary he writes: Divide - break into parts, crush, fragment, make a section. Divide one number by another - find out how much times one is contained in a different. №3. At first there was no sign for this action. They wrote with a word, Indian mathematicians - with the first letter of the name of the action. The colon sign to indicate division came into use at the end of the 17th century. (in 1684) thanks to the famous German mathematician Gottfried Wilhelm Leibniz. №4. What other sign represents division? /(slash). This sign was first used by the 13th century Italian scientist Fibonacci. . | Answer: to division. Answer: Strengthen your knowledge on the topic. Listen to student messages. |
| Stage 5. Understanding the content and sequence of application of practical actions when performing upcoming tasks. | Open your notebooks, write down the date and topic of the lesson. (Slide No. 5) Guides students' work at this stage. Task No. 1 . Open the textbook on page 76, No. 481 (a,b,). Solve independently, 2 students complete the task on individual boards. There is an additional task on the card. Task No. 2 . Solve the equation and choose correct solution out of 2 proposed. Explain the correct solution and indicate the error in another .(slide No. 7) | Write down the date and topic of the lesson. a) 7585: 37 + 95 = 300 1) 7585:37=205 2) 205+95=300 b)(6738 – 834) : 123= 48 1) 6738-834=5904 2) 5904:123=48 Self-check, draw conclusions. Individual reflection. Additionally: 1440:12:24=5 1)1440:12=120 2) 120:24=5 Solve the equation (x-15)*7=70 1 solution. x-15=70:7 x=25 Answer: 25 2nd solution. x-15=70:7 |
| Stage 6. Physical exercise. | Slide number 8. | Do exercises for the hands and eyes. |
| Continuation of stage 5. | Task No. 3 . Solve a problem: One team of the plant produced 636 parts, which is 3 times more than the 2nd team and 4 times more than the 3rd team. How many parts did all the teams produce together? The student solves on the board, the rest in the notebook. Additional task: The train traveled 450 km in x hours. Find the speed of the train. Write an expression and calculate if x = 9; x=15. Task No. 4 (Slide number 10). They brought 100 kg of apples, x kg in each box, and 120 kg of pears, y kg in each box. What does the expression mean: a) 100:x b) 120:y c) 100:x+120:y d) 120:y-100:x | №3. Read the problem and make up short note, solution algorithm, draw up the solution to the problem in a notebook. Solution. 1) 636:3=212(d) was manufactured by the 2nd brigade 2) 636:4=159(d) was manufactured by the 3rd brigade 3) 636+212+159=1007(d) were produced by 3 brigades together Answer: 1007 parts. Additional task. 450:x (km/h) - train speed. If x=9, then 450:9=50 (km/h) If x=15, then 450:15=30 (km/h) Answer : 50 (km/h), 30 (km/h) Give oral answers. a) the number of boxes of apples c) total number of boxes d) how many more boxes are there with pears than with apples? |
| Stage 7. Self-execution students assignments. | Directs students' work. | Perform test tasks independently. The leaves are submitted for verification. A1. What are the components of division called? 1) factors 2) quotient 3) dividend and divisor 4) terms A2. In one building there are 240 apartments, and in the second there are 2 times fewer apartments. How many apartments are there in the second building? 480 2) 138 3) 120 4) 242 A3. On day 1, tourists walked 15 km, which is 3 times more than on day 2. How many kilometers did the tourists walk on day 2? 1) 5km 2) 45km 3)12km 4)18km A4. Enter a number that is not divisible by 7. 1) 56 2) 48 3) 35 4) 21 IN 1. What number is 2 times greater than 36? Write this number down. AT 2. How many times is 890 greater than 178? Write this number down. C1. How many even three-digit numbers can be made from the numbers 4, 5, 6? (Numbers may be repeated) |
| Stage 8. Summing up the lesson. Reflection. | Summarizes students' work and gives grades. | Analyze their work in class. They answer the questions asked. |
| Stage 9. Information about homework, instructions for its implementation. | Sets differentiated homework. | Students write down the assignment in their diary. They take the task cards home. Required task: №1. Calculate: 2001:69 + 58884:84 №2. Solve the equation: a) x:17=34 b) (x – 8) *12=132 Additional task: On Sunday m people visited the museum, on Monday 4 times less than on Sunday, and on Tuesday - 33 people less than on Sunday. How many people visited the museum during these three days? Make up an expression and calculate for m =48, m = 100. |
Literature:
Mathematics. 5th grade: textbook for educational institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Shvartsburd. – 25th edition, erased. – M.: Mnemosyne, 2009;
Testing and measuring materials. Mathematics: 5th grade / Compiled by L.V. Popova.-M.: VAKO, 2011;
Chesnokov A.S., Neshkov K.I. Didactic materials in mathematics for grade 5. M.: Classic Style, 2007.
In this article we will look at the rules and algorithms for dividing natural numbers. Let us immediately note that here we are looking only at division as a whole, that is, without a remainder. Read about dividing natural numbers with a remainder in our separate material.
Before formulating the rule for dividing natural numbers, you need to understand the connection between division and multiplication. After we establish this connection, we will sequentially consider the simplest cases: dividing a natural number by itself and by one. Next, we will analyze division using the multiplication table, division by sequential subtraction, division by numbers that are multiples of 10, various powers of 10.
For each case, we will give and consider examples in detail. At the end of the article we will show how to check the division result.
Relationship between division and multiplication
To trace the connection between division and multiplication, remember that division is represented as partitioning the original divisible set into several identical sets. Multiplication involves combining several identical sets into one.
Division is the inverse action of multiplication. What does it mean? Let's give an analogy. Let's imagine that we have b sets, each of which contains c objects. The total number of objects in all sets is a. Multiplication is the union of all sets into one. Mathematically it will be written like this:
The reverse process of partitioning the resulting general set into b sets of objects each corresponds to division:
Based on what has been said, we can move on to the following statement:
If the product of natural numbers c and b is equal to a, then the quotient of a and b is equal to c. Let's rewrite it in letter form.
If b c = a, then a ÷ b = c
Using the commutative property of multiplication, we can write:
It also follows that a ÷ c = b.
Based on what has been said, a general conclusion can be formulated. If the product of numbers c and b is equal to a, then the quotients a ÷ b and a ÷ c are equal to c and b, respectively.
Let us summarize all of the above and give a definition of division of natural numbers.
Division of natural numbers
Division - finding an unknown factor from a known product and another known factor.
This definition will become the basis on which we will build rules and methods for dividing natural numbers.
Division by sequential subtraction
We just talked about division in the context of multiplication. Based on this knowledge, the division operation can be carried out. However, there is another approach that is quite simple and worthy of attention - division by sequential subtraction. This method is intuitive, so let’s look at it using an example, without giving theoretical calculations.
Heading
What is 12 divided by 4?
In other words, this problem can be formulated as follows: there are 12 objects (for example, oranges), and they need to be divided into equal groups of 4 objects (put into boxes of 4 pieces). How many such groups or boxes of four oranges each will there be?
Step by step we will subtract 4 oranges from the original quantity and form groups of 4 until the oranges run out. The number of steps we have to take will be the answer to the original question.
Of the 12 oranges, put the first four in a box. After this, 12 - 4 = 8 citrus fruits remain in the original pile of oranges. Of these eight, we take 4 more into another box. Now there are 8 - 4 = 4 oranges left in the original pile of oranges. From these four pieces you can just form one more, separate third box, after which 4 - 4 = 0 oranges will remain in the original pile.
So, we received 3 boxes, 4 items each. In other words, we divided 12 by 4, and the result was 3.
When working with numbers, you don’t need to draw an analogy with objects every time. What did we do with the dividend and divisor? We successively subtracted the divisor from the dividend until we got a zero remainder.
Important!
When dividing by the sequential subtraction method, the number of subtraction operations until a zero remainder is obtained is the quotient of the division.
To reinforce this, let's look at another, more complex example.
Example 1: Division by sequential subtraction
Let's calculate the result of dividing the number 108 by 27 using the sequential subtraction method.
First action: 108 - 27 = 81.
Second action: 81 - 27 = 54.
Third action: 54 - 27 = 27.
Fourth action: 27 - 27 = 0.
No further action is required. We received the answer:
Note that this method is convenient only in cases when required amount successive subtractions are small. In other cases, it is advisable to apply the division rules, which we will consider below.
Division of equal natural numbers
According to the properties of natural numbers, we formulate a rule on how to divide equal natural numbers.
Division of equal natural numbers
The quotient of a natural number divided by its equal natural number is equal to one!
For example:
1 ÷ 1 = 1 ; 141 ÷ 141 = 1; 2589 ÷ 2589 = 1; 100000000 ÷ 100000000 = 1.
Division by one
Based on the properties of natural numbers, we can also formulate a rule for dividing a natural number by one.
Dividing a natural number by one
The quotient of any natural number divided by one is equal to the number itself being divided.
For example:
1 ÷ 1 = 1 ; 141 ÷ 1 = 141 ; 2589 ÷ 1 = 2589 ; 100000000 ÷ 1 = 100000000.
Multiplication table - handy tool, which allows you to find products of single-valued natural numbers. However, it can also be used for division.
The multiplication table allows you to find not only the result of a product of factors, but also a factor from a known product and another factor. As we found out earlier, division is precisely finding an unknown factor from a known product and another factor.
Using the multiplication table, you can divide any number on a yellow background by any single-digit natural number. We'll show you how to do it. There are two methods, the use of which we will consider with examples.
Divide 48 by 6.
Method one.
In the column whose top cell contains the divisor 6, we find the dividend 48. The result of the division is in the leftmost cell of the row containing the dividend. It is circled in blue.
Method two.
First, in the line with divisor 6, we find dividend 48. The result of the division is in the uppermost cell of the column containing the dividend. It is circled in blue.
So we divided 48 by 6 and got 8. The result was found using the multiplication table in two ways. Both methods are absolutely identical.
To reinforce this, let's look at another example. Divide 7 by 1. Here are some pictures illustrating the division process.
As a result of dividing the number 7 by 1, you guessed it, the number 7 is obtained. In division using the multiplication table, it is very important to know this table by heart, since it is not always possible to have it at hand.
Division by 10, 100, 1000, etc.
Let us immediately formulate the rule for dividing natural numbers by 10, 100, 1000, etc. Let us immediately assume that division without a remainder is possible.
Division by 10, 100, 1000, etc.
The result of dividing a natural number by 10, 100, 1000, etc. is a natural number whose notation is obtained from the notation of the dividend if 1, 2, 3, etc. are discarded to the right of it. zeros.
As many zeros are discarded as there are in the divisor entry!
For example, 30 ÷ 10 = 3. We removed one zero from the number 30.
The quotient of 120000 ÷ 1000 is equal to 120 - from the number 120000 we discard three zeros on the right, that is how many are contained in the divisor.
The justification for the rule is based on the rule for multiplying a natural number by 10, 100, 1000, etc. Let's give an example. Let's say we need to divide 10200 by 100.
10200 = 102 100
10200 ÷ 100 = 102 100 100 = 102.
Representation of the dividend as a product
When dividing natural numbers, do not forget about the property of dividing the product of two numbers by a natural number. Sometimes the dividend can be represented as a product, one of the factors in which is divided by the divisor.
Let's look at typical cases.
Example 2. Representing the dividend as a product
Divide 30 by 3.
The dividend 30 can be represented as the product 30 = 3 10.
We have: 30 ÷ 3 = 3 10 ÷ 3
Using the property of dividing the product of two numbers, we get:
3 10 ÷ 3 = 3 ÷ 3 10 = 1 10 = 10
Let's give a few more similar examples.
Example 3. Representing the dividend as a product
Let's calculate the quotient 7200 ÷ 72.
We represent the dividend as 7200 = 72 100. In this case, the result of division will be as follows:
7200 ÷ 72 = 72 100 ÷ 72 = 72 ÷ 72 ÷ 100 = 100
Example 4. Representing the dividend as a product
Let's calculate the quotient: 1600000 ÷ 160.
1600000 = 160 10000
1600000 ÷ 160 = 160 10000 ÷ 160 = 160 ÷ 160 10000 = 10000
In more complex examples It is convenient to use the multiplication table. Let's illustrate this.
Example 5. Representing the dividend as a product
Divide 5400 by 9.
The multiplication table tells us that 54 is divisible by 9, so it is advisable to represent the dividend as a product:
5400 = 54 100.
Now let's finish the division:
5400 ÷ 9 = 54 100 ÷ 9 = 54 ÷ 9 100 = 6 100 = 600
To secure of this material Let's look at another example, without detailed verbal explanations.
Example 6. Representation of the dividend as a product
Let's calculate how much 120 divided by 4 is.
120 ÷ 4 = 12 10 ÷ 4 = 12 ÷ 4 10 = 3 10 = 30
Dividing natural numbers ending in zero
When dividing numbers that end in 0, it is useful to remember the property of dividing a natural number by the product of two numbers. In this case, the divisor is represented as a product of two factors, after which this property is used in conjunction with the multiplication table.
As always, we will explain this with examples.
Example 7. Dividing natural numbers ending in 0
Divide 490 by 70.
Let's write 70 as:
Using the property of dividing a natural number by a product, we can write:
490 ÷ 70 = 490 ÷ 7 10 = 490 ÷ 10 ÷ 7.
We have already discussed division by 10 in the previous paragraph.
490 ÷ 10 ÷ 7 = 49 ÷ 7 = 7
To reinforce this, let’s look at another, more complex example.
Example 8: Dividing natural numbers ending in 0
Let's take the numbers 54000 and 5400 and divide them.
54000 ÷ 5400 = ?
Let's represent 5400 as 54 100 and write:
54000 ÷ 5400 = 54000 ÷ 54 100 = 54000 ÷ 100 ÷ 54 = 540 ÷ 54.
Now we represent the dividend 540 as 54 10 and write:
540 ÷ 54 = 54 10 ÷ 54 = 54 ÷ 54 10 = 10
54000 ÷ 5400 = 10.
Let us summarize what is stated in this paragraph.
Important!
If the entries for the dividend and divisor contain zeros on the right, then you need to get rid of the same number of zeros in both the dividend and the divisor. After this, divide the resulting numbers.
For example, dividing the numbers 64000 and 8000 will be reduced to dividing the numbers 64 and 8.
Private selection method
Before considering this method of division, we introduce some conditions.
Let the numbers a and b be divisible by each other, and the product b · 10 gives a number greater than a. In this case, the quotient a ÷ b is a single-valued natural number. In other words, it is a number from 1 to 9. This is a typical situation when the quotient selection method is convenient and applicable. Sequentially multiplying the divisor by 1, 2, 3, . . , 9 and comparing the result with the dividend, you can find the quotient.
Let's look at an example.
Example 9. Selection of private
Divide 108 by 27.
It is easy to see that 27 · 10 = 270 ; 270 > 108 .
Let's start selecting a private one.
27 1 = 27 27 2 = 54 27 3 = 81 27 4 = 108
Bingo! The quotient was found using the selection method:
Note that in cases where b · 10 > a it is also convenient to find the quotient by the method of sequential subtraction.
Representing the dividend as a sum
Another way that can help find the quotient is to represent the dividend as the sum of several natural numbers, each of which is easily divisible by the divisor. After this, we will need the property of dividing the sum of natural numbers by a number. Together with an example, we will consider the algorithm and answer the question: in the form of what terms should we represent the dividend?
Let the dividend be 8551 and the divisor be 17.
- Let's calculate how many more digits there are in the notation of the dividend than in the notation of the divisor. In our case, the divisor contains two signs, and the dividend contains four. This means that the dividend has two more decimal places. Remember the number 2.
- Add two zeros to the right of the divisor. Why two? In the previous paragraph, we just determined this number. However, if the resulting number turns out to be greater than the divisor, you need to subtract 1 from the number obtained in the previous paragraph. In our example, by adding zeros to the divisor, we got the number 1700< 8551 . Таким образом, отнимать единицу из двойки, полученной в первом пункте, не нужно. В памяти так же оставляем число 2 .
- To the number 1 on the right we assign zeros in the amount a certain number from the previous paragraph. Thus we get work unit category, with which we will operate further. In our case, two zeros are assigned to one. Working category - hundreds.
- We sequentially multiply the divisor by 1, 2, 3, etc. units of the working digit until we get a number greater than the dividend. 17 100 = 1700; 17 · 200 = 3400 ; 17 · 300 = 5100 ; 17 · 400 = 6400 ; 17 · 500 = 8500 ; 17 · 600 = 10200 We are interested in the penultimate result, since the next result of the product after it is greater than the dividend. The number 8500, which was obtained in the penultimate step of multiplication, is the first addend. Remember the equality that we will use further: 8500 = 17 500.
- We calculate the difference between the dividend and the found term. If it is not equal to zero, we return to the first point and begin the search for the second term, using the already obtained difference instead of the dividend. We repeat the steps until the result is zero. In our example, the difference is 8551 - 8500 = 51. 51 ≠ 0, therefore, go to point 1.
We repeat the algorithm:
- We compare the number of digits in the new dividend 51 and the divisor 17. Both entries have two digits, the difference in the number of characters is zero. Remember the number 0.
- Since we remember the number 0, there is no need to add additional zeros to the divisor.
- We will also not add zeros to one. Again, because in the first paragraph we remembered the number 0. Thus, our working digit is units
- We successively multiply 17 by 1, 2, 3, . . etc. We get: 17 · 1 = 17 ; 17 · 2 = 34 ; 17 3 = 51.
- Obviously, in the third step we got a number equal to the divisor. This is the second term. Since 51 - 51 = 0, at this stage we stop the search for terms - it is completed.
Now all that remains is to find the quotient. We presented the dividend 8551 as the sum 8500 + 51. Let's write down:
8500 + 51 ÷ 17 = 8500 ÷ 17 + 51 ÷ 17.
The results of divisions in brackets are known to us from previous actions.
8500 + 51 ÷ 17 = 8500 ÷ 17 + 51 ÷ 17 = 500 + 3 = 503.
Result of division: 8551 ÷ 17 = 503.
Let's look at a few more examples, without commenting on each action in such detail.
Example 10. Division of natural numbers
Let's find the quotient: 64 ÷ 2.
1. The dividend has one more sign than the divisor. Remember the number 1.
2. We assign one zero to the right of the divisor.
3. We add one zero to the number 1 and get the unit of the working digit - 10. The working category is thus tens.
4. We begin sequential multiplication of the divider by units of the working digit. 2 · 10 = 20 ; 2 20 = 40 ; 2 · 30 = 60 ; 2 · 40 = 80 ; 80 > 64 .
The first term found is the number 60.
The equality 60 ÷ 2 = 30 will be useful to us in the future.
5. We are looking for the second term. To do this, calculate the difference 64 - 60 = 4. The number 4 is divisible by 2 without a remainder, obviously this is the second term.
Now we find the quotient:
64 ÷ 2 = 60 + 4 ÷ 2 = 60 ÷ 2 + 4 ÷ 2 = 30 + 2 = 32.
Example 11. Division of natural numbers
Let's solve: 1178 ÷ 31 = ?
1. We see that the dividend has two more digits than the divisor. Remember the number 2.
2. Add two zeros to the divisor on the right. We get the number 3100.
3100 > 1178, so the memorized number 2 from the first point needs to be reduced by one.
3. We add one zero to the one on the right and get the working digit - tens.
4. Multiply 31 by 10, 20, 30, . . etc.
31 · 10 = 310 ; 31 · 20 = 620 ; 31 · 30 = 930 ; 31 40 = 1240
1240 > 1178, therefore, the first term is the number 930.
5. Calculate the difference 1178 - 930 = 248. With the number 248 in place of the dividend, we begin to look for the second term.
1. The number 248 has one more digit than the number 31. Remember the number 1.
2. To 31 we add one zero to the right. Since 310 > 248, we reduce the unit obtained in the previous paragraph, and as a result we have the number 0.
3. Since we remember the number 0, there is no need to add additional zeros to the unit, and the ones digit is the working digit.
4. Consistently multiply 31 by 1, 2, 3, . . etc., comparing the result with the dividend.
31 · 1 = 31 ; 31 · 2 = 62 ; 31 · 3 = 93 ; 31 · 4 = 124 ; 31 · 5 = 155 ; 31 · 6 = 186 ; 31 · 7 = 217 ; 31 8 = 248
Thus, it is the number 248 that is the second term, which is divisible by 31.
5. The difference 248 - 248 is zero. We finish searching for terms, remember the ratio 248 ÷ 31 = 8 and find the quotient.
1178 ÷ 31 = 930 + 248 ÷ 31 = 930 ÷ 31 + 248 ÷ 31 = 30 + 8 = 38.
We gradually increase the complexity of the examples.
Example 12. Division of natural numbers
Divide 13984 by 32.
In this case, the algorithm described above will need to be applied three times. We will not give all the calculations, we will simply indicate in the form of which terms the divisor will be represented. You can test yourself and do the calculations yourself.
The first term is equal to 12800.
12800 ÷ 32 = 400.
The second term is equal to 960.
960 ÷ 32 = 30.
The third term is equal to 224.
Result:
13984 ÷ 32 = 12800 + 960 + 224 ÷ 32 = 12800 ÷ 32 + 960 ÷ 32 + 224 ÷ 32 = 400 + 30 + 7 = 437.
It would seem that we have considered almost everything possible ways division of natural numbers. At this point, the topic can be considered closed. However, there is a method that in some cases allows division to be carried out faster and more rationally.
Let's look at it one last time.
Representation of the dividend as the difference of natural numbers
Sometimes it is easier and more convenient to represent the dividend as a difference rather than a sum. This can greatly speed up and facilitate the division process. How exactly? Let's show it with an example.
Example 13. Division of natural numbers
Divide 594 by 6.
If we use the algorithm from the previous paragraph, we will get the result:
594 ÷ 6 = 540 + 54 ÷ 6 = 540 ÷ 6 + 54 ÷ 6 = 90 + 9 = 99.
However, if the number 594 is represented as the difference 600 - 6, everything becomes much more obvious. Both numbers 600 and 6) are divisible by 6. By the property of dividing the difference of natural numbers, we get:
594 ÷ 6 = 600 - 6 ÷ 6 = 600 ÷ 6 - 6 ÷ 6 = 100 - 1 = 99
The result is the same, but the actions are objectively easier and simpler.
Let's solve another example using the same method. Note that it is important to be able to correctly notice what manipulation to do with numbers in order to carry out the division easily. Let's even say that there is some element of art in this.
Example 14. Division of natural numbers
Let's remember the multiplication table and understand: the number 483 can be conveniently represented as 483 = 490 - 7.
490 ÷ 7 = 70 7 ÷ 7 = 1
We carry out the division:
483 ÷ 7 = (490 - 7) ÷ 7 = 490 ÷ 7 - 7 ÷ 7 = 70 - 1 = 69.
Checking the division result
Checking is never superfluous, especially if we divided large numbers. How to check if natural numbers are divided correctly? Using multiplication!
To check whether the division was performed correctly, you need to multiply the quotient by the divisor. The result should be the dividend.
The meaning of this action is very simple. For example, we had a objects, and we divided these a objects into b piles. Each pile contained items. Mathematically it looks like this:
Now let's combine back all b piles of c items. The result should be the same collection of objects a.
Let's look at the test using two examples.
Example 15. Checking the result of dividing natural numbers
The number 475 is divided by 19. The result was 25. Is the division done correctly?
Let's multiply the quotient of 25 by the divisor of 19 and find out whether the numbers were divided correctly.
25 19 = 475.
The number 475 is equal to the dividend, which means the division was done correctly.
Example 16. Checking the result of dividing natural numbers
Divide and check the result:
We will represent the dividend as a sum of terms and carry out the division.
1024 ÷ 32 = 960 + 64 ÷ 32 = 960 ÷ 32 + 64 ÷ 32 = 30 + 2 = 32.
Let's check the result:
32 32 = 1024.
Conclusion: the division was performed correctly.
Checking the result of dividing numbers by division
The verification method discussed above is based on multiplication. There is also a division test. How to carry it out?
Checking the division result
To check whether the quotient was found correctly, you need to divide the dividend by the resulting quotient. The result should be a divisor.
If it turns out differently, we can conclude that an error has crept in somewhere.
The rule is based on the same connection between dividend, divisor and quotient as the rule from the previous paragraph.
Let's look at examples.
Example 17. Checking the result of dividing natural numbers
Is the equality true:
Let's divide the dividend by the quotient:
104 ÷ 8 = 80 + 24 ÷ 8 = 80 ÷ 8 + 24 ÷ 8 = 10 + 3 = 13.
The result is a divisor, which means the division was done correctly.
Example 18. Checking the result of dividing natural numbers
Let's calculate and check: 240 ÷ 15 = ?
Representing the dividend as a sum, we get:
240 ÷ 15 = 150 + 90 ÷ 15 = 150 ÷ 15 + 90 ÷ 15 = 10 + 6 = 16.
Let's check the result:
240 ÷ 16 = 160 + 80 ÷ 16 = 160 ÷ 16 + 80 ÷ 16 = 10 + 5 = 15.
The division is done correctly.
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Although mathematics seems difficult to most people, it is far from true. Many mathematical operations are quite easy to understand, especially if you know the rules and formulas. So, knowing the multiplication table, you can quickly multiply in your head. The main thing is to constantly train and not forget the rules of multiplication. The same can be said about division.
Let's look at the division of integers, fractions and negatives. Let's remember the basic rules, techniques and methods.
Division operation
Let's start, perhaps, with the very definition and name of the numbers that participate in this operation. This will greatly facilitate further presentation and perception of information.
Division is one of the four basic mathematical operations. Its study begins in primary school. It is then that the children are shown the first example of dividing a number by a number and the rules are explained.
The operation involves two numbers: the dividend and the divisor. The first is the number that is being divided, the second is the number that is being divided by. The result of division is the quotient.
There are several notations for writing this operation: “:”, “/” and a horizontal bar - writing in the form of a fraction, when the dividend is at the top, and the divisor is below, below the line.
Rules
When studying one or another mathematical operation The teacher is obliged to introduce students to the basic rules that they should know. True, they are not always remembered as well as we would like. That's why we decided to refresh your memory a little on the four fundamental rules.
Basic rules for dividing numbers that you should always remember:
1. You cannot divide by zero. This rule should be remembered first.
2. You can divide zero by any number, but the result will always be zero.
3. If a number is divided by one, we get the same number.
4. If a number is divided by itself, we get one.
As you can see, the rules are quite simple and easy to remember. Although some may forget such a simple rule as impossibility or confuse the division of zero by a number with it.
per number
One of the most useful rules- a sign by which the possibility of dividing a natural number by another without a remainder is determined. Thus, the signs of divisibility by 2, 3, 5, 6, 9, 10 are distinguished. Let us consider them in more detail. They make it much easier to perform operations on numbers. We also give an example for each rule of dividing a number by a number.
These rules-signs are quite widely used by mathematicians.
Test for divisibility by 2
The easiest sign to remember. A number that ends in an even digit (2, 4, 6, 8) or 0 is always divisible by two. Quite easy to remember and use. So, the number 236 ends in an even digit, which means it is divisible by two.
Let's check: 236:2 = 118. Indeed, 236 is divisible by 2 without a remainder.
This rule is best known not only to adults, but also to children.
Test for divisibility by 3
How to correctly divide numbers by 3? Remember the following rule.
A number is divisible by 3 if the sum of its digits is a multiple of three. For example, let's take the number 381. The sum of all digits will be 12. This is three, which means it is divisible by 3 without a remainder.
Let's also check this example. 381: 3 = 127, then everything is correct.
Divisibility test for numbers by 5
Everything is simple here too. You can divide by 5 without a remainder only those numbers that end in 5 or 0. For example, let’s take numbers such as 705 or 800. The first ends in 5, the second in zero, therefore they are both divisible by 5. This is one one of the simplest rules that allows you to quickly divide by a single-digit number 5.
Let's check this sign using the following examples: 405:5 = 81; 600:5 = 120. As you can see, the sign works.
Divisibility by 6
If you want to find out whether a number is divisible by 6, then you first need to find out whether it is divisible by 2, and then by 3. If so, then the number can be divided by 6 without a remainder. For example, the number 216 is divisible by 2 , since it ends with an even digit, and with 3, since the sum of the digits is 9.
Let's check: 216:6 = 36. The example shows that this sign is valid.
Divisibility by 9
Let's also talk about how to divide numbers by 9. The sum of digits whose divisible by 9 is divided by this number. Similar to the rule of dividing by 3. For example, the number 918. Let's add all the digits and get 18 - a number that is a multiple of 9. So, it divisible by 9 without a remainder.
Let's solve this example to check: 918:9 = 102.
Divisibility by 10
One last sign to know. Only those numbers that end in 0 are divisible by 10. This pattern is quite simple and easy to remember. So, 500:10 = 50.
That's all the main signs. By remembering them, you can make your life easier. Of course, there are other numbers for which there are signs of divisibility, but we have highlighted only the main ones.
Division table
In mathematics, there is not only a multiplication table, but also a division table. Once you learn it, you can easily perform operations. Essentially, a division table is a reverse multiplication table. Compiling it yourself is not difficult. To do this, you should rewrite each line from the multiplication table in this way:
1. Put the product of the number in first place.
2. Put a division sign and write down the second factor from the table.
3. After the equal sign, write down the first factor.
For example, take the following line from the multiplication table: 2*3= 6. Now we rewrite it according to the algorithm and get: 6 ÷ 3 = 2.
Quite often, children are asked to create a table on their own, thus developing their memory and attention.
If you don’t have time to write it, you can use the one presented in the article.
Types of division
Let's talk a little about the types of division.
Let's start with the fact that we can distinguish between division of integers and fractions. Moreover, in the first case we can talk about operations with integers and decimals, and in the second - only about fractional numbers. In this case, a fraction can be either the dividend or the divisor, or both at the same time. This is due to the fact that operations on fractions are different from operations on integers.
Based on the numbers that participate in the operation, two types of division can be distinguished: into single-digit numbers and into multi-digit ones. The simplest is division by a single digit number. Here you will not need to carry out cumbersome calculations. In addition, a division table can be a good help. Dividing by other - two-, three-digit numbers - is harder.
Let's look at examples for these types of division:
14:7 = 2 (division by a single digit number).
240:12 = 20 (division by a two-digit number).
45387: 123 = 369 (division by a three-digit number).
The last one can be distinguished by division, which involves positive and negative numbers. When working with the latter, you should know the rules by which a result is assigned a positive or negative value.
When dividing numbers with different signs(the dividend is a positive number, the divisor is negative, or vice versa) we get a negative number. When dividing numbers with the same sign (both the dividend and the divisor are positive or vice versa), we get a positive number.
For clarity, consider the following examples:
Division of fractions
So, we have looked at the basic rules, given an example of dividing a number by a number, now let’s talk about how to correctly perform the same operations with fractions.
Although dividing fractions may seem like a lot of work at first, working with them is actually not that difficult. Dividing a fraction is done in much the same way as multiplying, but with one difference.
In order to divide a fraction, you must first multiply the numerator of the dividend by the denominator of the divisor and record the resulting result as the numerator of the quotient. Then multiply the denominator of the dividend by the numerator of the divisor and write the result as the denominator of the quotient.
It can be done simpler. Rewrite the divisor fraction by swapping the numerator with the denominator, and then multiply the resulting numbers.
For example, let's divide two fractions: 4/5:3/9. First, let's turn the divisor over and get 9/3. Now let's multiply the fractions: 4/5 * 9/3 = 36/15.
As you can see, everything is quite easy and no more difficult than dividing by a single-digit number. The examples are not easy to solve if you do not forget this rule.
conclusions
Division is one of the mathematical operations that every child learns in elementary school. Eat certain rules, which you should know, techniques that make this operation easier. Division can be with or without a remainder; there can be division of negative and fractional numbers.
It is quite easy to remember the features of this mathematical operation. We have sorted out the most important points, we looked at more than one example of dividing a number by a number, we even talked about how to work with fractional numbers.
If you want to improve your knowledge of mathematics, we advise you to remember these simple rules. In addition, we can advise you to develop memory and mental arithmetic skills by doing mathematical dictations or simply trying to verbally calculate the quotient of two random numbers. Believe me, these skills will never be superfluous.