In mental counting, as elsewhere, there are tricks, and in order to learn how to count faster, you need to know these tricks and be able to put them into practice.

Today we will do this!

1. How to quickly add and subtract numbers

Consider three random examples:

1. 25 – 7 =
2. 34 – 8 =
3. 77 – 9 =

Type 25 - 7 = (20 + 5) - (5- 2) = 20 - 2 = (10 + 10) - 2 = 10 + 8 = 18

Agree that such operations are difficult to turn in your head.

But there is an easier way:

25 - 7 \u003d 25 - 10 + 3, since -7 \u003d -10 + 3

It's much easier to subtract 10 from 10 and add 3 than it is to do complex calculations.

Let's go back to our examples:

1. 25 – 7 =
2. 34 – 8 =
3. 77 – 9 =

Optimizing subtracted numbers:

1. Subtract 7 = subtract 10 add 3
2. Subtract 8 = subtract 10 add 2
3. Subtract 9 = subtract 10 add 1

In total we get:

1. 25 – 10 + 3 =
2. 34 – 10 + 2 =
3. 77 – 10 + 1 =

Now it's much more interesting and easier!

Now count the examples below in this way:

1. 91 – 7 =
2. 23 – 6 =
3. 24 – 5 =
4. 46 – 8 =
5. 13 – 7 =
6. 64 – 6 =
7. 72 – 19 =
8. 83 – 56 =
9. 47 – 29 =

2. How to quickly multiply by 4, 8 and 16

In the case of multiplication, we also break numbers into simpler ones, for example:

If you remember the multiplication table, then everything is simple. And if not?

Then you need to simplify the operation:

We put the largest number first, and decompose the second into simpler ones:

8 * 4 = 8 * 2 * 2 = ?

It is much easier to double numbers than to quadruple or eight them.

We get:

8 * 4 = 8 * 2 * 2 = 16 * 2 = 32

Examples of decomposing numbers into simpler ones:

1. 4 = 2*2
2. 8 = 2*2 *2
3. 16 = 22 * 2 2

Practice this with the following examples:

1. 3 * 8 =
2. 6 * 4 =
3. 5 * 16 =
4. 7 * 8 =
5. 9 * 4 =
6. 8 * 16 =

3. Divide a number by 5

Let's take the following examples:

1. 780 / 5 = ?
2. 565 / 5 = ?
3. 235 / 5 = ?

Division and multiplication with the number 5 is always very simple and pleasant, because five is half of ten.

And how to solve them quickly?

1. 780 / 10 * 2 = 78 * 2 = 156
2. 565 /10 * 2 = 56,5 * 2 = 113
3. 235 / 10 * 2 = 23,5 *2 = 47

In order to work out this method, solve the following examples:

1. 300 / 5 =
2. 120 / 5 =
3. 495 / 5 =
4. 145 / 5 =
5. 990 / 5 =
6. 555 / 5 =
7. 350 / 5 =
8. 760 / 5 =
9. 865 / 5 =
10. 1270 / 5 =
11. 2425 / 5 =
12. 9425 / 5 =

4. Multiplication by single digits

Multiplication is a little more difficult, but not much, how would you solve the following examples?

1. 56 * 3 = ?
2. 122 * 7 = ?
3. 523 * 6 = ?

Without special counters, solving them is not very pleasant, but thanks to the Divide and Conquer method, we can count them much faster:

1. 56 * 3 = (50 + 6)3 = 50 3 + 6*3 = ?
2. 122 * 7 = (100 + 20 + 2)7 = 100 7 + 207 + 2 7 = ?
3. 523 * 6 = (500 + 20 + 3)6 = 500 6 + 206 + 3 6 =?

We just have to multiply single-digit numbers, some of them with zeros, and add the results.

To work through this technique, solve the following examples:

1. 123 * 4 =
2. 236 * 3 =
3. 154 * 4 =
4. 490 * 2 =
5. 145 * 5 =
6. 990 * 3 =
7. 555 * 5 =
8. 433 * 7 =
9. 132 * 9 =
10. 766 * 2 =
11. 865 * 5 =
12. 1270 * 4 =
13. 2425 * 3 =

Divisibility of a number by 2, 3, 4, 5, 6 and 9

Check the numbers: 523, 221, 232

A number is divisible by 3 if the sum of its digits is divisible by 3.

For example, let's take the number 732 and represent it as 7 + 3 + 2 = 12. 12 is divisible by 3, which means that the number 372 is divisible by 3.

Check which of the following numbers are divisible by 3:

12, 24, 71, 63, 234, 124, 123, 444, 2422, 4243, 53253, 4234, 657, 9754

A number is divisible by 4 if the number consisting of its last two digits is divisible by 4.

For example, 1729. The last two digits form 20, which is divisible by 4.

Check which of the following numbers are divisible by 4:

20, 24, 16, 34, 54, 45, 64, 124, 2024, 3056, 5432, 6872, 9865, 1242, 2354

A number is divisible by 5 if its last digit is 0 or 5.

Check which of the following numbers are divisible by 5 (the easiest exercise):

3, 5, 10, 15, 21, 23, 56, 25, 40, 655, 720, 4032, 14340, 42343, 2340, 243240

A number is divisible by 6 if it is divisible by both 2 and 3.

Check which of the following numbers are divisible by 6:

22, 36, 72, 12, 34, 24, 16, 26, 122, 76, 86, 56, 46, 126, 124

A number is divisible by 9 if the sum of its digits is divisible by 9.

For example, let's take the number 6732 and represent it as 6 + 7 + 3 + 2 = 18. 18 is divisible by 9, which means that the number 6732 is divisible by 9.

Check which of the following numbers are divisible by 9:

9, 16, 18, 21, 26, 29, 81, 63, 45, 27, 127, 99, 399, 699, 299, 49

Game "Fast Addition"

1. Speeds up mental counting
2. Trains attention
3. Develops creative thinking

An excellent simulator for the development of fast counting. A 4x4 table is given on the screen, and numbers are shown above it. The largest number you need to collect in the table. To do this, click on two numbers with the mouse, the sum of which is equal to this number. For example, 15+10 = 25.

Game "Quick Score"

The game "quick count" will help you improve your thinking. The essence of the game is that in the picture presented to you, you will need to choose the answer "yes" or "no" to the question "are there 5 identical fruits?". Follow your goal, and this game will help you with this.

Game "Guess the operation"

The game "Guess the operation" develops thinking and memory. The main essence of the game is to choose a mathematical sign so that the equality is true. Examples are given on the screen, look carefully and put the desired “+” or “-” sign so that the equality is true. The sign "+" and "-" are located at the bottom of the picture, select the desired sign and click on the desired button. If you answer correctly, you score points and continue playing.

Game "Simplify"

The game "Simplify" 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 action is given, the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need with the mouse. If you answer correctly, you score points and continue playing.

Task for today

Solve all the examples and practice for at least 10 minutes in the Quick Addition game.

It is very important to work out all the tasks of this lesson. The better you perform the tasks, the more you will benefit. If you feel that there are not enough tasks for you, you can make up examples for yourself and solve them and train in mathematical educational games.

The lesson is taken from the course "Oral counting in 30 days"

Learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even take roots. I will teach you how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.

Other developmental courses

Money and the mindset of a millionaire

Why are there money problems? In this course, we will answer this question in detail, look deep into the problem, consider our relationship with money from a psychological, economic and emotional point of view. From the course, you will learn what you need to do to solve all your financial problems, start saving money and invest it in the future.

Knowing the psychology of money and how to work with them makes a person a millionaire. 80% of people with an increase in income take out more loans, becoming even poorer. Self-made millionaires, on the other hand, will make millions again in 3-5 years if they start from scratch. This course teaches the proper distribution of income and cost reduction, motivates you to learn and achieve goals, teaches you to invest money and recognize a scam.

Speed ​​reading in 30 days

Increase your reading speed by 2-3 times in 30 days. From 150-200 to 300-600 wpm or from 400 to 800-1200 wpm. The course uses traditional exercises for the development of speed reading, techniques that speed up the work of the brain, a method for progressively increasing the speed of reading, understands the psychology of speed reading and the questions of course participants. Suitable for children and adults reading up to 5,000 words per minute.

Development of memory and attention in a child 5-10 years old

The purpose of the course is to develop the child's memory and attention so that it is easier for him to study at school, so that he can remember better.

After completing the course, the child will be able to:

1. 2-5 times better to remember texts, faces, numbers, words
2. Learn to remember for longer
3. The speed of remembering the necessary information will increase

Super memory in 30 days

Memorize the information you need quickly and permanently. Wondering how to open the door or wash your hair? I am sure not, because it is part of our life. Easy and simple memory training exercises can be made part of life and done little by little during the day. If you eat the daily norm of food at a time, or you can eat in portions throughout the day.

Why do we need a mental account, if it is the 21st century in the yard, and all kinds of gadgets are capable of almost instantly performing any arithmetic operations? You can even not poke your finger at the smartphone, but give a voice command - and immediately get the right answer. Now even elementary school students who are too lazy to divide, multiply, add and subtract on their own are doing this successfully.

But this medal also has a downside: scientists warn that if you don’t train, don’t load it with work and make tasks easier for him, he starts to be lazy, he is reduced. In the same way, without physical training, our muscles also weaken.

Mikhail Vasilyevich Lomonosov spoke about the benefits of mathematics, calling it the most beautiful of sciences: “Mathematics is already worth loving because it puts the mind in order.”

The oral account develops attention, speed of reaction. No wonder there are more and more new methods of quick oral counting, designed for both children and adults. One of them is the Japanese oral counting system, which uses the ancient Japanese soroban abacus. The technique itself was developed in Japan 25 years ago, and now it is successfully used in some of our schools of oral counting. It uses visual images, each of which corresponds to a certain number. Such training develops the right hemisphere of the brain, which is responsible for spatial thinking, building analogies, etc.

It is curious that in just two years, students of such schools (children aged 4–11 years old are accepted here) learn to perform arithmetic operations with 2-digit, or even 3-digit numbers. Kids who do not know multiplication tables here know how to multiply. They add and subtract large numbers without writing down their column. But, of course, the goal of training is the balanced development of the right and.

You can also master mental arithmetic with the help of the problem book “1001 tasks for mental arithmetic at school”, compiled back in the 19th century by a village teacher and well-known educator Sergei Alexandrovich Rachinsky. This problem book is supported by the fact that it went through several editions. This book can be found and downloaded online.

People who practice quick counting recommend Yakov Trakhtenberg's book "Quick Counting System". The history of this system is very unusual. In order to survive in the concentration camp where he was sent by the Nazis in 1941, and not to lose his mental clarity, the Zurich professor of mathematics began to develop algorithms for mathematical operations that allow him to quickly calculate in his head. And after the war, he wrote a book in which the quick counting system is presented in such a clear and accessible way that it is still in demand.

Good reviews about the book by Yakov Perelman “Quick Count. Thirty Simple Examples of Oral Counting. The chapters in this book are devoted to multiplication by single and double digits, in particular, multiplication by 4 and 8, 5 and 25, by 11/2, 11/4, *, division by 15, squaring, calculating by formula.

The simplest ways of oral counting

People with certain abilities will quickly master this skill, namely: the ability to think logically, the ability to concentrate and store several images in short-term memory at the same time.

Equally important is the knowledge of special algorithms of actions and some mathematical laws that allow, as well as the ability to choose the most effective for a given situation.

And, of course, you can not do without regular training!

The most common quick counting methods are as follows:

1. Multiplying a two-digit number by a one-digit number

Multiplying a two-digit number by a one-digit number is easiest by decomposing it into two components. For example, 45 - by 40 and 5. Next, we multiply each component by the desired number, for example, by 7, separately. We get: 40 × 7 = 280; 5 × 7 = 35. Then add the results: 280 + 35 = 315.

2. Multiply a three-digit number

Multiplying a three-digit number in your mind is also much easier if you decompose it into its components, but presenting the multiplicand in such a way that it is easier to perform mathematical operations with it. For example, we need to multiply 137 by 5.

We represent 137 as 140 - 3. That is, it turns out that now we must multiply by 5 not 137, but 140 - 3. Or (140 - 3) x 5.

Knowing the multiplication table within 19 x 9, you can count even faster. We decompose the number 137 into 130 and 7. Then we multiply by 5, first 130, and then 7, and add the results. So 137 x 5 = 130 x 5 + 7 x 5 = 650 + 35 = 685.

You can decompose not only the multiplicand, but also the multiplier. For example, we need to multiply 235 by 6. We get six by multiplying 2 by 3. Thus, we first multiply 235 by 2 and get 470, and then we multiply 470 by 3. Total 1410.

The same operation can be performed differently by representing 235 as 200 and 35. It turns out 235 × 6 = (200 + 35) × 6 = 200 × 6 + 35 × 6 = 1200 + 210 = 1410.

In the same way, decomposing numbers into components, you can perform addition, subtraction and division.

3. Multiply by 10

Everyone knows how to multiply by 10: just add zero to the multiplicand. For example, 15 × 10 = 150. Based on this, it is no less easy to multiply by 9. First, add 0 to the multiplicand, that is, multiply it by 10, and then subtract the multiplier from the resulting number: 150 × 9 = 150 × 10 = 1500 − 150 = 1350.

4. Multiply by 5

It is easy to multiply by 5. You just need to multiply the number by 10, and divide the resulting result by 2.

5. Multiply by 11

It is interesting to multiply two-digit numbers by 11. Let's take, for example, 18. Let's mentally expand 1 and 8, and write the sum of these numbers between them: 1 + 8. We get 1 (1 + 8) 8. Or 198.

6. Multiply by 1.5

If you need to multiply some number by 1.5, divide it by two and add the resulting half to the whole: 24 × 1.5 = 24 / 2 + 24 = 36.

These are just the simplest ways of mental counting, with the help of which we can train our brain in everyday life. For example, counting the cost of purchases while standing in line at the checkout. Or perform mathematical operations with the numbers on the numbers of cars passing by. Those who like to "play" with numbers and want to develop their mental abilities can refer to the books of the above-mentioned authors.

Why count in the mind, if you can solve any arithmetic problem on a calculator. Modern medicine and psychology prove that mental counting is an exercise for gray cells. Performing such gymnastics is necessary for the development of memory and mathematical abilities.

There are many tricks to simplify mental calculations. Everyone who has seen the famous painting by Bogdanov-Belsky "Mental Account" is always surprised - how do peasant children solve such a difficult task as dividing the sum of five numbers that must first be squared?

It turns out that these children are students of the famous teacher-mathematician Sergei Alexandrovich Rachitsky (he is also depicted in the picture). These are not child prodigies - elementary school students of a nineteenth-century village school. But they all already know how to simplify arithmetic calculations and have learned the multiplication table! Therefore, it is quite possible for these kids to solve such a problem!

Secrets of mental counting

There are methods of oral counting - simple algorithms that it is desirable to bring to automatism. After mastering simple techniques, you can move on to mastering more complex ones.

We add the numbers 7,8,9

To simplify the calculations, the numbers 7,8,9 must first be rounded up to 10, and then subtract the increase. For example, to add 9 to a two-digit number, you must first add 10 and then subtract 1, and so on.

Examples :

Add two digit numbers quickly

If the last digit of a two-digit number is greater than five, round it up. We perform the addition, subtract the “additive” from the resulting amount.

Examples :

54+39=54+40-1=93

26+38=26+40-2=64

If the last digit of a two-digit number is less than five, then add up by digits: first add tens, then ones.

Example :

57+32=57+30+2=89

If the terms are reversed, then you can first round the number 57 to 60, and then subtract 3 from the total:

32+57=32+60-3=89

Adding three-digit numbers in your mind

Quick counting and addition of three-digit numbers - is it possible? Yes. To do this, you need to parse three-digit numbers into hundreds, tens, units and add them one by one.

Example :

249+533=(200+500)+(40+30)+(9+3)=782

Subtraction features: reduction to round numbers

Subtracted are rounded up to 10, up to 100. If you need to subtract a two-digit number, you need to round it up to 100, subtract, and then add an amendment to the remainder. This is true if the correction is small.

Examples :

576-88=576-100+12=488

Mind subtracting three-digit numbers

If at one time the composition of numbers from 1 to 10 was well mastered, then subtraction can be done in parts and in the indicated order: hundreds, tens, units.

Example :

843-596=843-500-90-6=343-90-6=253-6=247

Multiply and Divide

Instantly multiply and divide in your mind? It is possible, but one cannot do without knowledge of the multiplication table. is the golden key to quick mental counting! It applies to both multiplication and division. Recall that in the elementary grades of a village school in the pre-revolutionary Smolensk province (the painting "Mental Counting"), children knew the continuation of the multiplication table - from 11 to 19!

Although in my opinion it is enough to know the table from 1 to 10 in order to be able to multiply larger numbers. for example:

15*16=15*10+(10*6+5*6)=150+60+30=240

Multiply and divide by 4, 6, 8, 9

Having mastered the multiplication table for 2 and 3 to automatism, making the rest of the calculations will be as easy as shelling pears.

For multiplication and division of two- and three-digit numbers, we use simple tricks:

multiplying by 4 is twice multiplying by 2;

to multiply by 6 means to multiply by 2 and then by 3;

multiplying by 8 is three times multiplying by 2;

multiplying by 9 is twice multiplying by 3.

for example :

37*4=(37*2)*2=74*2=148;

412*6=(412*2) 3=824 3=2472

Similarly:

divided by 4 is twice divided by 2;

divide by 6 is first divide by 2 and then by 3;

divided by 8 is three times divided by 2;

Divide by 9 is twice divided by 3.

for example :

412:4=(412:2):2=206:2=103

312:6=(312:2):3=156:3=52

How to multiply and divide by 5

The number 5 is half of 10 (10:2). Therefore, we first multiply by 10, then we divide the result in half.

Example :

326*5=(326*10):2=3260:2=1630

The rule of division by 5 is even simpler. First, we multiply by 2, and then we divide the result by 10.

326:5=(326 2):10=652:10=65.2.

Multiply by 9

To multiply a number by 9, it is not necessary to multiply it twice by 3. It is enough to multiply it by 10 and subtract the multiplied number from the resulting number. Compare which is faster:

37*9=(37*3)*3=111*3=333

37*9=37*10 - 37=370-37=333

Also, particular patterns have long been noticed that greatly simplify the multiplication of two-digit numbers by 11 or by 101. So, when multiplied by 11, a two-digit number seems to move apart. The numbers that make it up remain at the edges, and their sum is in the center. For example: 24*11=264. When multiplying by 101, it is enough to attribute the same to a two-digit number. 24*101= 2424. The simplicity and logic of such examples is admirable. Such tasks are very rare - these are entertaining examples, the so-called little tricks.

Counting on fingers

Today you can still meet many defenders of "finger gymnastics" and the method of mental counting on the fingers. We are convinced that learning to add and subtract by bending and unbending fingers is very visual and convenient. The range of such calculations is very limited. As soon as the calculations go beyond one operation, difficulties arise: it is necessary to master the next technique. Yes, and bending your fingers in the era of iPhones is somehow undignified.

For example, in defense of the "finger" technique, the technique of multiplying by 9 is given. The trick of the technique is as follows:

• To multiply any number within the first ten by 9, you need to turn your palms towards you.
• Counting from left to right, bend the finger corresponding to the number being multiplied. For example, to multiply 5 by 9, you need to bend the little finger on your left hand.
• The remaining number of fingers on the left will correspond to tens, on the right - units. In our example - 4 fingers on the left and 5 on the right. Answer: 45.

Yes, indeed, the solution is quick and visual! But this is from the field of tricks. The rule only works when multiplying by 9. Isn't it easier to learn the multiplication table to multiply 5 by 9? This trick will be forgotten, and a well-learned multiplication table will remain forever.

There are also many more similar tricks using fingers for some single mathematical operations, but this is relevant while you use it and is immediately forgotten when you stop using it. Therefore, it is better to learn standard algorithms that will remain for life.

Oral account on the machine

First, you need to know the composition of the number and the multiplication table well.

Secondly, you need to remember the methods of simplifying calculations. As it turned out, there are not so many such mathematical algorithms.

Thirdly, in order for the technique to turn into a convenient skill, it is necessary to constantly conduct brief “brainstorming sessions” - to practice oral calculations using one or another algorithm.

Workouts should be short: mentally solve 3-4 examples using the same technique, then move on to the next one. We must strive to use every free minute - and useful, and not boring. Thanks to simple training, all calculations over time will be done at lightning speed and without errors. This is very useful in life and will help out in difficult situations.

Send

cool

How long have you been counting in your head, and not in a column, and even more so not with a calculator? By the way, counting in the mind is not only fashionable, but also useful: this is how you develop short-term memory, concentration and attention. And also, what a thrill you feel when you can calculate how much you should be given change while standing in line, mmm ...

Just a few months of daily training for 5-10 minutes, and you will feel how your brain has accelerated.

Addition

Let's start with a simple one - addition of single-digit numbers. Having learned to instantly add single-digit numbers, you can easily add multi-digit numbers, because all calculations come down to performing typical actions. You will see this soon.

Single digit addition

There are no problems with examples whose results are within 10. These combinations of numbers just need to be remembered as the basis of the basics.

But for examples "with the transition through 10" there is already a technique - "reliance on a dozen." The bottom line is to bring one term to 10, and then subtract from the second term as much as we added to the first.

For example, we need to add 5 and 8:

1. The number 5 is not enough to 10, the same number is 5.
2. Now imagine 8 as the sum of 5 and some other number (that's 3).
3. And add to 5 that part of the number 8, which is missing to 10, and then the remainder. It will turn out 10 and 3, that is, 13.

Multi-digit addition

The principle of adding multi-digit numbers is to add the same digits to each other: thousands with thousands, hundreds with hundreds, tens with tens, ones with ones.

For example, we need to add 245 and 917:

1. 245 consists of three digits - 200, 40 and 5. And 917 from 900, 10 and 7.

Let's add bit parts to each other:

200 + 900 = 1100, 40 + 10 = 50, 5 + 7 = 12.

And now we add the resulting numbers in reverse order, “closing” the zeros:

62 + 1100 = 1162.

Subtraction

As with addition, there is nothing complicated with subtracting single-digit numbers from single-digit numbers. And when subtracting a single-digit number from a two-digit number, it is convenient to use the same rule of "reliance on a dozen."

Single digit subtraction

For example, subtract 13 − 7:

1. We remove enough from 13 to get 10 - that is, 3.
2. We remove the same amount from 7 - it turns out 4.
3. Now just subtract 4 from 10.

Multi-digit subtraction

Here everything is even simpler than with the addition of multi-digit numbers, because only the number that is being subtracted needs to be decomposed into bit parts.

For example, subtract 734 − 427:

1. We decompose 427 into digits: 400, 20 and 7. Now we subtract them sequentially from 734.
2. Subtracting 734 − 400 is very easy because it only works on hundreds. Roughly speaking, we subtract 4 from 7 - we get 3, or rather, 334.
3. With tens, everything is the same: subtract 30 - 20, we get 10 - 314.

Now we subtract units through ten: 314 - 7.

We remove 4 from 314 and 7, we get 310 - 3. Well, here it’s already quite simple - the answer is 307.

Little tricks

When subtracting 9 from a number, first subtract 10, and then add 1:

321 − 9 = 321 − 10 + 1 = 312

When subtracting 8 from a number, first subtract 10, and then add 2:

321 − 8 = 321 − 10 + 2 = 313

When subtracting 7 from a number, first subtract 10, and then add 3:

321 − 7 = 321 − 10 + 3 = 314

Multiplication

This is when you add the same thing over and over again. For example, 7 × 3 = 7 + 7 + 7 = 21.

To learn how to quickly multiply any numbers in your mind (except for the very cosmic ones), you need to ideally multiply single-digit numbers, that is, know the multiplication table.

Moreover, it is not necessary to know it perfectly, it is enough to remember the reference numbers for yourself, which will help in the calculations. Multiply 6 × 7. Mnemotechnically, we know that 6 × 6 = 36. That is, 6 more must be added to 36 to get the answer - 42.

It is believed that of all the examples in the multiplication table 7 × 8 is the most difficult. To remember the answer, there is an excellent five-six-seven-eight rule: 56 = 7 × 8.

Multiplication of a single-digit number by a two-digit number

Multiply 387 × 8:

1. First of all, we decompose 387 into digits - 300, 80 and 7 - and multiply each of them by 8.

We start with hundreds: 300 × 8 is the same as multiplying 3 × 8, and then adding two zeros to the result. I.e:

3 x 8 x 100 = 24 x 100 = 2400.

By analogy, 80 × 8 = 640, 7 × 8 = 56.

And now we add the resulting numbers, combining them by digits:

2400 + 640 + 56 = 2000 + 400 + 600 + 40 + 50 + 6 = 2000 + (400 + 600) + (40 + 50) + 6 = 2000 + 1000 + 90 + 6 = 3000 + 90 + 6 = 3096

Little tricks

Any number can be easily multiplied by 9: you just need to multiply by 10 (or add zero at the end), and then subtract the original number.

47 x 9 = (47 x 10) - 47 = 470 - 47 = 423

A non-round number can easily be multiplied by 2 by first rounding it to the nearest convenient value.

For example, 237 × 2. First, it is easier to multiply 240 × 2 = 480. And then subtract 6 from the result (3 × 2 = 6 - after all, 3 was not enough for us to reach 240). Total:

237 x 2 = 240 x 2 − (3 x 2) = 476

To multiply any two-digit number by 11, you need to add two digits of this two-digit number to each other, and then enter it between the digits of the original number:

True, if the sum of the two digits of the original number is greater than 10, you need to put a unit digit between the digits of the original number, and add ten to the left digit:

Multiplication of two-digit numbers

Although it seems that multiplying two-digit numbers is the pinnacle of mental calculations, solving such examples is not much more difficult than in the previous paragraph. Let's take a look at an example.

Multiply 83 × 34:

1. Let's break 34 into 30 and 4 to make it easier, and then multiply each by 83.

Multiplying 83 by 30 is easy - it's like multiplying 83 × 3, and then multiplying the result by another 10. We figured out how to multiply single-digit and double-digit numbers. We believe:

83 × 3 = 80 × 3 + 3 × 3 = 240 + 9 = 249. So 84 × 30 = 2490.

Now multiply

83 x 4 = 80 x 4 + 3 x 4 = 320 + 12 = 332.

Let's sum up the results:

2490 + 332 = 2000 + 400 + 300 + 90 + 30 + 2 = 2000 + 700 + 120 + 2 = 2822.

Division

This is the inverse of multiplication. Let's start again with the simplest.

Dividing a two-digit number by a one-digit number

Divide 48: 3. The main task is to choose a number that can be multiplied by 3 and get 48. From the multiplication table, we remember that the only number whose result of multiplication by 3 has the number 8 at the end is 6. And 3 × 6 \u003d 18 That is, we have 30: 3 = 10. In total, it turns out 48: 3 = 16.

Division of a multi-digit number by a one-digit number

Divide 6475: 7. In such examples, the main task is to “take” the maximum “round” parts that can be divided into 6 without a remainder.

1. Let's select from 6475 the largest part that can be divided by 7 without a remainder. 6475 is close to 7000 (i.e. 7 × 1000), so we can try to take 900 × 7 = 6300. Great!
2. It remains 175. In the same way, we select from 175 the largest number that can be divided by 7 according to the multiplication table - this is 140. And 140: 7 \u003d 20. Let's remember this number and subtract 175 - 140. Hundreds result in zero, and 7 − 4 = 3. That is, the balance at the moment is 35.
3. We recall that according to the multiplication table 7 × 5 = 35, and add up all the resulting numbers: 900 + 20 + 5 = 925.

Division by two digits

With division by a two-digit number, everything is much more interesting. The task is to find the limits within which the result lies.

For example, let's divide 6351:73:

1. First, let's try to guess which ten the result is in. Remember that according to the multiplication table 7 × 8 = 56, so we try to multiply 73 × 80 = 5840. This is the closest ten, because if you add another 730 (that is, 73 × 10), you get 6570 already - more than necessary. Therefore, our number lies between 80 and 90.
2. Now let's look at the last digits of our numbers - 1 and 3. From the multiplication table, we remember that only one number, when multiplied by 3 at the end, gives 1 - this is 7. We try to multiply 73 × 7 = 511. We add 5840 + 511 = 6351. Hooray, the answer is 87!

Little tricks

Non-round numbers can be easily divided by 2 by rounding them up. For example, we divide 358 by 2. We round 358 to 360, and then we divide it by 2 - we get 130. And then we subtract this number 1 (obtained as a result of dividing by 2 added 2).

358: 2 = 360: 2 − 2: 2 = 130 − 1 = 129

1. There is a pattern according to which multiplying by 5 can almost be equated to dividing by For example, if you multiply 47 × 5 = 235, and if you divide 47: 2 = 23.5. Magic, right? That is, to multiply any number by 5, it must first be divided by 2, and then multiplied by 10.

To multiply a number by 25, it's sometimes easier to divide it by 4 and then multiply by 100 (or add two zeros):

12 x 25 = 12: 4 x 100 = 3 x 100 = 300

These methods are enough to train yourself to count confidently in your head. Remember that you need to do this regularly, devoting only 5-10 minutes every day. Try to catch your rhythm so that solving such problems brings pleasure. And rest on the correctness of the answers, not the speed - it will come with time. And don't quit.

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Quick Counting Techniques: Magic Available to All

In order to understand the role that numbers play in our lives, set up a simple experiment. Try to do without them for a while. No numbers, no calculations, no measurements... You will find yourself in a strange world where you will feel absolutely helpless, bound hand and foot. How to get to a meeting on time? Distinguish one bus from another? Make a phone call? Buy bread, sausage, tea? Cook soup or potatoes? Without numbers, and therefore, without counting, life is impossible. But how hard this science is sometimes given! Try to quickly multiply 65 by 23? Does not work? The hand itself reaches for a mobile phone with a calculator. Meanwhile, semi-literate Russian peasants 200 years ago calmly did this, using only the first column of the multiplication table - multiplication by two. Don't believe? But in vain. This is reality.

stone age computer

Even without knowing the numbers, people have already tried to count. If our ancestors, who lived in caves and wore skins, needed to exchange something with a neighboring tribe, they acted simply: they cleared the site and laid out, for example, an arrowhead. Near lay a fish or a handful of nuts. And so on until one of the exchanged goods ran out, or the head of the "trading mission" decided that enough was enough. Primitive, but in its own way very convenient: you won’t get confused, and you won’t be deceived.

With the development of cattle breeding, the tasks became more complicated. A large herd had to be counted somehow in order to know whether all the goats or cows were in place. The "calculating machine" of illiterate but smart shepherds was a dugout pumpkin with pebbles. As soon as the animal left the pen, the shepherd put a pebble in the gourd. In the evening, the herd returned, and the shepherd took out a stone with each animal that entered the pen. If the gourd was empty, he knew the flock was all right. If there were pebbles, he went to look for the loss.

When the numbers appeared, things got more fun. Although for a long time our ancestors used only three numerals: "one", "pair" and "many".

Can you count faster than a computer?

Outrun a device that performs hundreds of millions of operations per second? Impossible... But the one who says this is cruelly disingenuous, or simply deliberately overlooks something. A computer is just a set of chips in plastic; it doesn't count by itself.

Let's put the question in another way: can a person, calculating in his mind, overtake someone who performs calculations on a computer? And here the answer is yes. Indeed, in order to receive an answer from the "black suitcase", the data must first be entered into it. This will be done by a person with the help of fingers or voice. And all these actions have time limits. Insurmountable restrictions. Nature itself supplied them to the human body. Everything except one organ. Brain!

The calculator can only perform two operations: addition and subtraction. Multiplication for him is multiple addition and division is multiple subtraction.

Our brain behaves differently.

The class where the future king of mathematics, Carl Gauss, studied, somehow received the task: add up all the numbers from 1 to 100. Carl wrote the absolutely correct answer on his board as soon as the teacher finished explaining the task. He did not diligently add numbers in order, as any self-respecting computer would do. He applied the formula he discovered himself: 101 x 50 = 5050. And this is far from the only trick that speeds up mental calculations.

The simplest tricks for quick counting

They are taught at school. The simplest: if you need to add 9 to any number, add 10 and subtract 1 if 8 (+ 10 - 2), 7 (+ 10 - 3), etc.

54 + 9 = 54 + 10 - 1 = 63. Fast and convenient.

Two-digit numbers add up just as easily. If the last digit in the second term is greater than five, the number is rounded up to the next ten, and then the "excess" is subtracted. 22 + 47 = 22 + 50 – 3 = 69

With three-digit numbers, there are no difficulties in the same way. We add them, as we read, from left to right: 321 + 543 \u003d 300 + 500 + 20 + 40 + 1 + 3 \u003d 864. Much easier than in a column. And much faster.

What about subtraction? The principle is the same: we round the subtracted to the nearest integer and add the missing one: 57 - 8 = 57 - 10 + 2 = 49; 43 - 27 \u003d 43 - 30 + 3 \u003d 16. Faster than on a calculator - and no complaints from the teacher even during the test!

Do I need to learn the multiplication table?

Children usually hate this. And they do it right. No need to teach her! But do not rush to be outraged. No one claims that the table does not need to be known.

Its invention is attributed to Pythagoras, but, most likely, the great mathematician only gave a complete, concise form to what was already known. At the excavations of ancient Mesopotamia, archaeologists found clay tablets with the sacramental: "2 x 2". People have been using this highly convenient system of calculations for a long time and have discovered many ways that help to comprehend the internal logic and beauty of the table, to understand - and not stupidly, mechanically memorize.

In ancient China, the table began to be taught by multiplying by 9. It’s easier this way, and not least because you can multiply by 9 “on your fingers”.

Place both hands on the table, palms down. The first finger from the left is 1, the second is 2, and so on. Let's say you need to solve a 6 x 9 problem. Raise your sixth finger. Fingers on the left will show tens, on the right - ones. Answer 54.

Example: 8 x 7. The left hand is the first multiplier, the right hand is the second. There are five fingers on the hand, and we need 8 and 7. We bend three fingers on the left hand (5 + 3 = 8), on the right 2 (5 + 2 = 7). We have five bent fingers, which means five dozen. Now multiply the rest: 2 x 3 = 6. These are units. Total 56.

This is just one of the simplest methods of "finger" multiplication. There are many of them. "On the fingers" you can operate with numbers up to 10,000!

The "finger" system has a bonus: the child perceives it as a fun game. He engages willingly, experiences a lot of positive emotions, and as a result, very soon begins to perform all operations in his mind, without the help of his fingers.

You can also divide with your fingers, but it's a little more complicated. Programmers still use their hands to convert numbers from decimal to binary - it's more convenient and much faster than on a computer. But within the framework of the school curriculum, you can learn to quickly divide even without fingers, in your mind.

Let's say you need to solve example 91: 13. Column? No need to mess up paper. The dividend ends with one. And the divisor is three. What is the very first thing in the multiplication table where the triple is involved, and ends with one? 3 x 7 = 21. Seven! That's it, we got her. Need 84: 14. Remember the table: 6 x 4 = 24. The answer is 6. Simple? Still would!

number magic

Most of the quick counting tricks are similar to magic tricks. Take at least the most famous example of multiplying by 11. To, for example, 32 x 11, you need to write 3 and 2 along the edges, and put their sum in the middle: 352.

To multiply a two-digit number by 101, simply write the number twice. 34 x 101 = 3434.

To multiply a number by 4, multiply it by 2 twice. To divide, divide by 2 twice.

Many witty and, most importantly, quick tricks help to raise a number to a power, to extract the square root. The famous "Perelman's 30 tricks" for mathematically minded people will be cooler than the Copperfield show, because they also UNDERSTAND what is happening and how it is happening. Well, the rest can just enjoy the beautiful focus. For example, you need to multiply 45 by 37. Let's write the numbers on a sheet and separate them with a vertical line. We divide the left number by 2, discarding the remainder, until we get one. Right - multiply until the number of lines in the column is equal. Then we cross out from the RIGHT column all those numbers opposite which an even result is obtained in the LEFT column. We add the remaining numbers from the right column. It turns out 1665. Multiply the numbers in the usual way. The answer will fit.

"Charging" for the mind

Quick counting techniques can make life easier for a child at school, for mom in a store or kitchen, and for dad at work or in the office. But we prefer the calculator. Why? We don't like to stress. It's hard for us to keep numbers, even two-digit ones, in our heads. For some reason they don't hold up.

Try to go to the middle of the room and sit on the twine. For some reason "does not sit down", right? And the gymnast does it quite calmly, without straining. Need to train!

The easiest way to train and, at the same time, warm up the brain: verbal counting aloud (required!) through the number to one hundred and back. In the morning, standing in the shower, or preparing breakfast, count: 2.. 4.. 6.. 100... 98.. 96. You can count in three, in eight - the main thing is to do it out loud. After just a couple of weeks of regular practice, you will be surprised how EASIER it becomes to deal with numbers.