How to train mental counting. How to learn to count quickly in your head
IN Lately In Russia, a new method for developing intelligence is beginning to gain popularity in our country. Instead of the usual chess sections, parents send their children to mental arithmetic schools. How kids are taught to count in their heads, how much such classes cost and what experts say about them - in the material "AiF-Volgograd".
What is mental arithmetic?
Mental arithmetic is a Japanese technique for developing a child’s intellectual abilities through calculations on special soroban abacus, which is sometimes called an abacus.
“When performing actions with numbers in their minds, children imagine these abacus and in a split second they mentally add, subtract, multiply and divide any numbers - even three-digit, even six-digit,” says Natalya Chaplieva, teacher of the Volga club, where children are taught using this method.
According to her, when children are just learning all these actions, they count the numbers directly on the soroban, fingering the bones. Then they gradually move from counting to a “mental map” - a picture depicting them. At this stage of learning, they stop touching the abacus and begin to imagine in their minds how they move the bones on it. Then, the children stop using the mental map and begin to completely visualize the soroban for themselves.
Abacus soroban. Photo: AiF/ Evgeniy Strokan
“We recruit children from 4 to 12 years old into groups. At this age, the brain is most plastic; the child absorbs information like a sponge, and therefore easily masters learning methods. It’s much more difficult for an adult to learn mental arithmetic,” says Ekaterina Grigorieva, teacher of the mental arithmetic club.
How much does it cost?
The abacus has a rectangular frame that contains 23-31 spokes, each of which has 5 bones strung on them, separated by a transverse crossbar. Above it there is one domino, which denotes “five”, and below it there are 4 dominoes, denoting ones.
You need to move the bones with only two fingers - the thumb and forefinger. The counting on the soroban starts from the very first knitting needle on the right. It stands for units. The knitting needle to the left of it is tens, the next one is hundreds, etc.
Soroban is not sold in regular stores. You can buy such accounts on the Internet. Depending on the number of knitting needles and material, the price of soroban can range from 170 to 1,000 rubles.
At the first stage, children work with abacus. Photo: AiF/ Evgeniy Strokan
If you don’t want to spend money on bills at all, you can download a free application for your phone - an online simulator that simulates an abacus.
Mental arithmetic classes for children in Volgograd cost about 500-600 rubles per hour. You can buy a subscription for 8 classes for 4,000 rubles and 16 classes for 7,200 rubles. Classes are held 2 times a week. The Volga school gives out abacus, mental maps and notebooks to children free of charge, and students can take them home. At the end of the course, the child can keep the soroban as a souvenir.
Children have to learn mental arithmetic for about 1-2 years, depending on their abilities.
Assignments for students. Photo: AiF/ Evgeniy Strokan
If you don’t have money for classes at a special school, then you can try to look for video lessons on YouTube. True, some of them are posted on the website by organizations providing lessons for money for the purpose of self-promotion. Their videos are very short - 3 minutes long. With their help you can learn the basics of mental arithmetic, but nothing more.
What do experts say about this?
Teachers who conduct mental arithmetic classes are confident that the training is worth the money spent on it.
“Mental arithmetic develops well the child’s imagination, creativity, thinking, memory, fine motor skills, attentiveness, perseverance. Classes are aimed at ensuring that the child develops both hemispheres at the same time, which is very important, because the traditional preparation of a child for school only develops right hemisphere brain," says teacher Natalya Chaplieva.
Psychologist Natalya Oreshkina believes that in the case of children 4-5 years old, mental arithmetic classes will be effective only if they take place in a playful way.
“Children of this age generally have difficulty concentrating for such a time, unless we are talking about watching a cartoon,” says the expert. - But if the lesson is structured in a playful way, if children practice abacus and color something, then they will learn knowledge while being in their natural environment - in a game. In addition, it should not be difficult for children; they should not exceed the permissible load level. For example, for 4-year-olds, classes should last no more than 30 minutes. I can say that mental arithmetic for children is very interesting. But if a child lags behind his peers in some way, then such activities will be too difficult for him. If a child does not have an internal resource for activities, then it will be a waste of time, effort and money.”
No matter how ashamed I was, by the age of 30 I realized that I was very bad at counting elementary numbers in my head and wasted a lot of time on it. I decided to correct this shortcoming and found tools on the Internet that helped me learn to count in my head.
There are key patterns in arithmetic that need to be brought to automaticity.
Subtraction 7,8,9 To subtract 9 from any number, you need to subtract 10 from it and add 1. To subtract 8 from any number, you need to subtract 10 from it and add 2. To subtract 7 from any number, you need to subtract 10 from it and add 3. If usually you think differently, then for best result you need to get used to this new way.
Multiply by 9. A quick way to multiply any number by 9 is by first multiplying the number by 10 (just add a 0 at the end) and then subtracting the number itself from the result. For example 89*9=890-89=801. This operation must be brought to automation.
Multiply by 2. For mental arithmetic, it is very important to be able to quickly multiply any number by 2. To multiply by 2 non-round numbers, try rounding them to the nearest more convenient number. So it’s easier to calculate 139*2 if you first multiply 140*2 (140*2=280). and then subtract 1*2=2 (exactly 1 needs to be added to 139 to get 140) Total: 140*2-1*2=278
Divide by 2. For mental counting, it is also important to be able to quickly divide any number by 2. Despite the fact that multiplication and division by 2 is quite simple for many, in difficult cases also try to round numbers. For example, to divide 198 by 2, you must first divide 200 (this is 198+2) by 2 and subtract 1 (we got 1 by dividing the added 2 by 2) Total: 198/2=200/2-2/2=100- 1=99.
Dividing and multiplying by 4 and 8. Division (or multiplication) by 4 and 8 are double or triple division (or multiplication) by 2. It is convenient to perform these operations sequentially. For example, 46*4=46*2*2=922*2=184
Multiply by 5. Multiplying by 5 is very simple. Multiplying by 5 and dividing by 2 are practically the same thing. So 88*5=440, and 88/2=44, so always multiply a number by 5 by dividing the number by 2 and multiplying it by 10.
Multiplying by single digit numbers. To quickly count in your head, it is useful to be able to multiply two- and three-digit numbers by single-digit numbers. To do this, you need to multiply a two- or three-digit number bit by bit. For example, let's multiply 83*7. To do this, first multiply 8 by 7 (and add 0, since 8 is the tens place) and add to this number the product of 3 and 7. Thus, 83*7=80*7+3*7=560+21=581. Let's take a more complex example 236*3. So, we multiply the complex number by 3 bitwise: 200*3+30*3+6*3=600+90+18=708.
Definition of ranges. In order not to get confused in the algorithms and mistakenly give a completely wrong answer, it is important to be able to construct an approximate range of answers. So multiplying single-digit numbers by each other can give the result no more than 90 (9*9=81), two-digit numbers - no more than 10,000 (99*99 =9801), three-digit numbers no more - 1,000,000 (999*999=998001)
Dividing 1000 by 2,4,8,16. And finally, it is useful to know the division of numbers that are multiples of 10 by numbers that are multiples of two: 100=2*500=4*250=8*125=16*62.5
In the century cash registers and calculators, people count less and less in their heads. They have almost completely switched to computer technology, but it often fails, or it simply won’t be there when it’s needed. Imperceptibly, we lose the skills of accurate and quick counting and sometimes we belatedly realize that we are no longer so good at this matter. But, quickly counting in your head is an undeniable advantage and advantage. A person who easily operates with numbers will almost never be deceived in calculations. But the important thing is that it will develop and maintain mental abilities, which is important for children and young people.
How to learn to count quickly in your child's head
All skills are best developed and reinforced in childhood. You can learn to count, just like reading, from the age of 1.5-2 years. The peculiarities of this age are that the child will first accumulate passive knowledge - he will understand, know, but due to little vocabulary, there will be little talking. Until the age of five, a child can learn to mentally perform simple operations - subtraction and addition within twenty. If at two to three and a half years old you use visual methods in teaching, then later the baby will be able to operate only with numbers, without reinforcement with visual material.
If you want your child to have a better chance that the process of handling large values and mathematical operations will be easier and faster, then you need to teach him to count as early as possible.
It is better to teach children under four years of age with visual materials. You can count whatever you want. Fire trucks rushing to fight a fire, motorcyclists roaring past you, cats basking in the sun, flocks of birds - everything you can count around you. With numeracy skills, observation and attention will simultaneously develop. Gradually increase the load. In the morning you saw 2 cats, and when you returned home, 3 more. Ask your child: “Did he notice that there are so many cats today! How much did he notice? Praise him for his accuracy and observation, because these qualities will be useful to him in life.
IN primary school The child needs to quickly and freely perform any calculations within the limits determined by the school curriculum. To learn to count quickly, constant training is necessary. Therefore, the task of parents is to encourage the baby to count and make it interesting. The more often your child practices, the easier it will be for him to make accurate and quick mental calculations.
How to learn to count quickly as an adult
If a child has learned to count quickly since childhood, then over time he will be able to operate with large values. But if a person of a more mature age or a student decides to master quick counting, then it is necessary to apply a simple technique that will undoubtedly bring positive results.
Any learning starts small. If you know the multiplication tables, that's great. If you forgot, or never knew, you should use this method of counting. For example, you need to find out how much 8x6 is. Let's write the example this way:
2 4
--=48
8x6
Answer 48. We got it by writing down the example 8x6, drawing a straight line over it and above each number we wrote down how much is missing to 10. Above 8 we write 2, on 6 we write 4. The first digit of the answer is the difference between the numbers in the bottom and top lines diagonally. 8-4=4, 6-2=4 – you can take any pair to calculate – the answer will always be the same. So we realized that the first digit is 4. Now let's find the second. To do this, multiply the numbers on the top line by 2x4=8. Our example is solved: 8x6=48.
Larger numbers are calculated a little differently. For example, you need to count 11x13.
1 3
--=140+3=143
11x13
On the bottom line we write the example 11x13. At the top we write how much these numbers exceed 10. We get 1 and 3. Let's add the numbers diagonally. We get 11+3=14, 13+1=14. We got 14 tens, since the original numbers exceed 10. Therefore, we multiply 14 by 10. 14x10 = 140. All that remains is to multiply the top numbers 1x3=3 and add the resulting figure to the answer.
Such calculation methods are difficult to carry out only at first. So start with simple examples and gradually make it more difficult. But in order to learn to count in your head, you need to completely get rid of notes and do everything in your head.
Children can also be taught using these methods, but only when they fully know the school curriculum. Otherwise you won't achieve positive results, but will only harm the acquisition of school knowledge.
Once you have mastered the manipulation of two-digit numbers, you can move on to calculating multi-digit numbers - hundreds and even thousands.
Video lessons
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Many people ask how to learn to quickly count in their heads so that it looks unnoticeable and not stupid. After all modern technologies allow you to use your memory and mental abilities less. But sometimes these technologies are not at hand and sometimes it is easier and faster to calculate something in your head. Many people have started counting even basic things on a calculator or phone, which is also not very good. The ability to do mental math remains a useful skill for modern man, despite the fact that he owns all sorts of devices that can count for him. The ability to do without special devices and quickly solve an arithmetic problem at the right time is not the only use of this skill. In addition to the utilitarian purpose, mental calculation techniques will allow you to learn how to organize yourself in various life situations. In addition, the ability to count in your head will undoubtedly have a positive impact on the image of your intellectual abilities and will distinguish you from the surrounding “humanists.”
Quick counting methods
There is a certain set of simple arithmetic rules and patterns that you not only need to know for mental calculation, but also constantly keep in mind in order to quickly apply the most effective algorithm at the right time. To do this, it is necessary to bring their use to automaticity, consolidate it in mechanical memory, so that from solving the simplest examples you can successfully move on to more complex arithmetic operations. Here are the basic algorithms that you need to know, remember and apply instantly, automatically:
Subtraction 7, 8, 9
To subtract 9 from any number, you need to subtract 10 from it and add 1. To subtract 8 from any number, you need to subtract 10 from it and add 2. To subtract 7 from any number, you need to subtract 10 from it and add 3. If usually If you think differently, then for a better result you need to get used to this new method.
Multiply by 9
You can quickly multiply any number by 9 using your fingers.
Division and multiplication by 4 and 8
Division (or multiplication) by 4 and 8 are double or triple division (or multiplication) by 2. It is convenient to perform these operations sequentially.
For example, 46*4=46*2*2 =92*2= 184.
Multiply by 5
Multiplying by 5 is very simple. Multiplying by 5 and dividing by 2 are practically the same thing. So 88*5=440, and 88/2=44, so always multiply by 5 by dividing the number by 2 and multiplying it by 10.
Multiply by 25
Multiplying by 25 is the same as dividing by 4 (followed by multiplying by 100). So 120*25 = 120/4*100=30*100=3000.
Multiplying by single digits
For example, let's multiply 83*7.
To do this, first multiply 8 by 7 (and add zero, since 8 is the tens place), and add to this number the product of 3 and 7. Thus, 83*7=80*7 +3*7= 560+21=581 .
Let's take a more complex example: 236*3.
So, we multiply the complex number by 3 bitwise: 200*3+30*3+6*3=600+90+18=708.
Defining ranges
In order not to get confused in the algorithms and mistakenly give a completely wrong answer, it is important to be able to construct an approximate range of answers. Thus, multiplying single-digit numbers by each other can give a result of no more than 90 (9*9=81), two-digit numbers - no more than 10,000 (99*99=9801), three-digit numbers no more than 1,000,000 (999*999=998001).
Layout in tens and units
The method consists of dividing both factors into tens and ones and then multiplying the resulting four numbers. This method is quite simple, but requires the ability to hold up to three numbers in memory simultaneously and at the same time perform arithmetic operations in parallel.
For example:
63*85 = (60+3)*(80+5) = 60*80 + 60*5 +3*80 +3*5=4800+300+240+15=5355
Such examples can be easily solved in 3 steps:
1. First, tens are multiplied by each other.
2. Then add 2 products of units and tens.
3. Then the product of units is added.
This can be schematically described as follows:
First action: 60*80 = 4800 - remember
- Second action: 60*5+3*80 = 540 - remember
- Third action: (4800+540)+3*5= 5355 - answer
For the fastest possible effect, you will need a good knowledge of the multiplication table for numbers up to 10, the ability to add numbers (up to three digits), as well as the ability to quickly switch attention from one action to another, keeping the previous result in mind. It is convenient to train the last skill by visualizing the arithmetic operations being performed, when you should imagine a picture of your solution, as well as intermediate results.
Mental visualization of columnar multiplication
56*67 - count in a column. Probably the count in a column contains maximum amount actions and requires constantly keeping auxiliary numbers in mind.
But it can be simplified:
First action: 56*7 = 350+42=392
Second action: 56*6=300+36=336 (or 392-56)
Third action: 336*10+392=3360+392=3,752
Private techniques for multiplying two-digit numbers up to 30
The advantage of the three methods of multiplying two-digit numbers for mental calculation is that they are universal for any numbers and, with good mental calculation skills, they can allow you to quickly come to the correct answer. However, the multiplication efficiency of some double digit numbers in the mind can be higher due to fewer actions when using special algorithms.
Multiplying by 11
To multiply any two-digit number by 11, you need to enter the sum of the first and second digits between the first and second digits of the number being multiplied.
For example: 23*11, write 2 and 3, and between them put the sum (2+3). Or in short, that 23*11= 2 (2+3) 3 = 253.
If the sum of the numbers in the center gives a result greater than 10, then add one to the first digit, and instead of the second digit we write the sum of the digits of the number being multiplied minus 10.
For example: 29*11 = 2 (2+9) 9 = 2 (11) 9 = 319.
You can quickly multiply by 11 orally not only two-digit numbers, but also any other numbers.
For example: 324 * 11=3(3+2)(2+4)4=3564
Squared sum, squared difference
To square a two-digit number, you can use the squared sum or squared difference formulas. For example:
23²= (20+3)2 = 202 + 2*3*20 + 32 = 400+120+9 = 529
69² = (70-1)2 = 702 - 70*2*1 + 12 = 4,900-140+1 = 4,761
Squaring numbers ending in 5. To square numbers ending in 5. The algorithm is simple. The number up to the last five, multiply by the same number plus one. Add 25 to the remaining number.
25² = (2*(2+1)) 25 = 625
85² = (8*(8+1)) 25 = 7,225
This is also true for more complex examples:
155² = (15*(15+1)) 25 = (15*16)25 = 24,025
The technique for multiplying numbers up to 20 is very simple:
16*18 = (16+8)*10+6*8 = 288
Proving the correctness of this method is simple: 16*18 = (10+6)*(10+8) = 10*10+10*6+10*8+6*8 = 10*(10+6+8) +6*8. The last expression is a demonstration of the method described above. Essentially, this method is a special way of using reference numbers. In this case, the reference number is 10. In the last expression of the proof, we can see that it is by 10 that we multiply the bracket. But any other numbers can be used as a reference number, the most convenient of which are 20, 25, 50, 100...
Reference number
Look at the essence of this method using the example of multiplying 15 and 18. Here it is convenient to use the reference number 10. 15 is greater than ten by 5, and 18 is greater than ten by 8.
In order to find out their product, you need to perform the following operations:
1. To any of the factors add the number by which the second factor is greater than the reference one. That is, add 8 to 15, or 5 to 18. In the first and second cases, the result is the same: 23.
2. Then we multiply 23 by the reference number, that is, by 10. Answer: 230
3. To 230 we add the product 5*8. Answer: 270.
The reference number when multiplying numbers up to 100. The most popular multiplication technique large numbers in the mind is the technique of using the so-called reference number
Reference number for multiplication- this is the number to which both factors are close and by which it is convenient to multiply. When multiplying numbers up to 100 with reference numbers, it is convenient to use all numbers that are multiples of 10, and especially 10, 20, 50 and 100.
The technique for using the reference number depends on whether the factors are greater than or less than the reference number. There are three possible cases here. We will show all 3 methods with examples.
Both numbers are less than the reference (below the reference). Let's say we want to multiply 48 by 47.
These numbers are close enough to the number 50, and therefore it is convenient to use 50 as a reference number.
To multiply 48 by 47 using the reference number 50:
1. From 47, subtract as much as 48 is missing to 50, that is, 2. It turns out 45 (or
subtract 3 from 48 - it's always the same)
2. Next we multiply 45 by 50 = 2250
3. Then add 2*3 to this result - 2,256
50 (reference number)
3(50-47) 2(50-48)
(47-2)*50+2*3=2250+6=2256
If the numbers are less than the reference number, then from the first factor we subtract the difference between the reference number and the second factor. If the numbers are greater than the reference number, then to the first factor we add the difference between the reference number and the second factor.
50(reference number)
(51+13)*50+(13*1)=3200+13=3213
One number is below the reference, and the other is above. The third case of using a reference number is when one number is greater than the reference number and the other is less. Such examples are no more difficult to solve than the previous ones. We increase the smaller factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors. Or we reduce the larger factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors.
50(reference number)
5(50-45) 2(52-50)
(52-5)*50-5*2=47*50-10=2340 or (45+2)*50-5*2=47*50-10=2340
When multiplying two-digit numbers from different tens, it is more convenient to use a reference number
take a round number that is greater than the larger factor.
90(reference number)
63 (90-27) 1 (90-89)
(89-63)*90+63*1=2340+63=2403
Thus, by using a single reference number, it is possible to multiply a large combination of two-digit numbers. The methods described above can be divided into universal (suitable for any numbers) and specific (convenient for specific cases).
As a last resort, you can use a “peasant” account. To multiply one number by another, say 21*75, we need to write the numbers in two columns. The first number in the left column is 21, the first number in the right column is 75. Then divide the numbers in the left column by 2 and discard the remainder until we get one, and multiply the numbers in the right column by 2. Cross out all lines with even numbers in the left column, and we add up the remaining numbers in the right column, we get the exact result.
Conclusion
Like all calculation methods, these fast calculation methods have their advantages and disadvantages:
PROS:
1.Using various methods even the least educated person can do fast calculations.
2. Quick counting methods can help get rid of a complex action by replacing it with several simpler ones.
3.Quick counting methods are useful in situations where columnar multiplication cannot be used.
4. Fast counting methods can reduce calculation time.
5. Mental arithmetic develops mental activity, which helps to quickly navigate difficult life situations.
6. The mental calculation technique makes the calculation process more fun and interesting.
MINUSES:
1. Often, solving an example using quick calculation methods turns out to be longer than simply multiplying by column, since you have to perform a larger number of actions, each of which is simpler than the original one.
2. There are situations when a person, out of excitement or something else, forgets the methods of quick counting or even gets confused in them; in such cases, the answer is incorrect, and the methods are actually useless.
3.Quick counting methods have not been developed for all cases.
4. When calculating using the quick counting technique, you need to keep many answers in your head, which can cause you to get confused and come to an erroneous result.
Surely practice makes a difference vital role in the development of any abilities. But the skill of mental calculation does not rely on experience alone. This is proven by people who can count in their heads complex examples. For example, such people can multiply and divide three-digit numbers, perform arithmetic operations that not every person can count in a column. What you need to know and be able to do to an ordinary person to master such a phenomenal ability? Today there are various techniques, helping to learn how to quickly count in your head.
Having studied many approaches to teaching the skill of counting orally, we can highlight 3 main components of this skill:
1. Abilities. The ability to concentrate and the ability to hold several things in short-term memory at the same time. Predisposition to mathematics and logical thinking.
2. Algorithms. Knowledge of special algorithms and the ability to quickly select the necessary, most effective algorithm in each specific situation.
3. Training and experience, the importance of which for any skill has not been canceled. Constant training and gradual complication of solved problems and exercises will allow you to improve the speed and quality of mental calculation. It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others with a quick score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, since having in your arsenal the abilities and a set of the necessary algorithms, you can surprise even the most experienced “accountant”, provided that you have trained for the same amount of time.
IN modern world With many ultra-progressive devices, mental arithmetic has not lost its relevance.
Sometimes we come across people who can add, multiply and divide complex numbers with lightning speed. Such people do not have supernatural abilities, they simply know the simplified counting formulas and regularly train their skills.
Three components of successful learning
- Capabilities. In order to learn to count in your head, you should be able to concentrate on the task at hand and retain complex numbers in memory.
- Formulas. To easily and simply perform calculations in your head, you should remember the basic mathematical formulas.
- Practice. Frequent training will allow you to develop and improve the skill.
There are several simple ways multiplying a number by 11.
Method 1
When multiplying a 2-digit number by 11, we expand the digits of the multiplier.
For example (54 * 11):
5 _ 4 * 11=…
Now we sum up the units and tens, and write the resulting result in the answer:
5 (5+4) 4 * 11 = 5 (9) 4 = 594
If, when summing tens and ones, you get a 2-digit number, leave only the ones, and add “1” to the tens.
For example (89 * 11):
8 _ (8+9) _9 = 8 _ (17) _ 9 = _ (8+1) _ 79 = 979
Method 2
When multiplying by 11, we decompose the number 11 into the sum: 10+1, and multiply the parts.
For example:
12 * 11 = 12 * (10+1) = 120 + 12 = 132
The same goes for 3-digit numbers:
114 * 11 = 114 * (10+1) = 1140 + 114 = 1254
Multiply by 9 and 11
When multiplying by "9", we simply multiply the number by 10 and then subtract the same original number. If we multiply by “11”, the number should be multiplied by “10” and add the original number.
Examples:
15 * 9 = 15 * 10 – 15 = 150 - 15 = 135
57 * 11 = 57 * 10 + 57 = 570 + 57 = 627
Squaring a number ending in 5
Enough simple technique. Multiply ten by itself +1, and add “25” at the end.
For example (35 * 35):
35 * 35 = 3 * (3+1)_25 = 1225
Verbal multiplication by 5, 25, 50, 125
Multiplying numbers up to 10 by 5 is no problem
Let's learn how to multiply two-digit and three-digit numbers just as easily.
Method 1
Let's divide our multiplier by "2". Did you get a whole number? This means we add “0” to it at the end; if the number is not equally divisible, we discard the remainder and add “5” at the end.
For example (1482 * 5):
1482 * 5 = (1482/2) _ (+0 or +5) = 741 _(+0) = 7410 – the number is divisible by 2 without a remainder
2269 * 5 = (2269/2) _ (+0 or +5) = 1134.5 _ (+5) = 11345 – the number is divisible by 2 with a remainder
Method 2
When multiplying a number by 5, 25, 50, 125, you can use the following formulas:
A * 5 = A * 10 / 2
A * 50 = A * 100 / 2
A * 25 = A * 100 / 4
A * 125 = A * 1000 / 8
Examples:
44 * 5 = 44 * 10 / 2 = 440 / 2 = 220
24 * 50 = 24 * 100 / 2 = 2400 / 2 = 1200
26 * 25 = 26 * 100 / 4 = 2600 / 4 = 650
54 * 125 = 54 * 1000 / 8 = 54000 / 8 = 6750
Learning to multiply by 4 orally
A fairly simple method that does not require much effort.
We multiply the number by “2”, and then multiply the resulting result again by “2”.
For example:
27 * 4 = 27 * 2 * 2 = 54 * 2 = 108
Calculate 15% of the number in your head
Find 10% of the number and add ½ of 10%.
For example:
15% of 664 = (10%) + (10% / 2) = 66.4 + 33.2 = 99.6
Multiply large numbers in your head, one of which is even
When multiplying large numbers, one of which is even, we will use the method of simplifying factors. An even number is halved, and an odd number is increased by the same amount.
For example:
48 * 125 = 24 * 250 = 12 * 500 = 6 * 1000 = 6000
Learning to divide by 5, 50, 25
One simple trick will help you quickly divide in your head: multiply our number by “2” and move the decimal point back one digit.
145 / 5 = 145 * 2 = 290 (shift the comma) = 29
1200 / 5 = 1200 * 2 = 2,400 (offset the decimal point) = 240
When dividing by 50, 25, it is convenient to use the formulas:
A / 50 = A * 2 / 100
A / 25 – A * 4 / 100
Examples:
2350 / 50 = 2350 * 2 / 100 = 4700 / 100 = 47
2600 / 25 = 2600 * 4 / 100 = 10400 / 100 = 104
Subtract from 1000
To subtract a number from 1000, subtract each digit of the number from “9” and subtract the last digit from 10.
For example:
1000 – 248 = (9-2) _ (9-4) _ (10-8) = 752
Multiplying prime numbers
This method is often called diagonal. Above the numbers we add how much they lack to “10”, subtract diagonally and get the 1st digit of the number, then multiply the upper numbers and write down the 2nd digit.
Example, multiply 7 by 8: 3 __ 2
7 8
8 – 3 = 5 _
3 * 2 = 6
Total: 56
Multiply numbers from 10 to 20
In order to quickly multiply numbers from 10 to 20 in your head, you should know one trick: add the units of another to one number, multiply the sum by 10, and add the product of the units to the resulting result.
Example:
13 * 15 = (13 + 5) * 10 + 3 * 5 = 180 + 15 = 195
Adding and subtracting natural numbers
1. If a term is increased by a certain number, then the same number should be subtracted from the resulting amount.
For example:
650 + 346 = (650 + 346 + 4) – 4 = (650 + 350) – 2 = 1000 – 2 = 998
2. If one term is reduced by a certain number and the same number is added to the second term, the sum will not change.
For example:
335 + 765 = (335 + 5) + (765 - 5) = 340 + 760 = 1100
3. If you add the same number to the minuend and subtrahend, the result will not change.
For example:
225 - 339 = (225 + 5) - (339 + 5) = 230 - 344 = 114
We multiply numbers with the same number of tens, the sum of whose units = 10
The arithmetic is quite simple: we multiply the tens of one of the factors by the number greater than “1”, multiply the units, and write down the result one by one.
For example:
302 * 308 = ..
1). 30 * (30 + 1) = 900 + 30 = 930
2). 2 * 8 = 16
Multiply by a number consisting of digits 9
How to multiply by the number 9, 99, 999?
To do this, simply add the missing units and perform the calculation.
Example:
154 * 99 = 154 * (100 - 1) = 15400 - 154 = 15246
Add numbers that are close in size
We calculate a series of numbers that are close in value
They can be expanded and folded in parts.
For example:
19 + 22 + 23 + 21+ 24 + 17=…
Let's expand the terms:
19 = 20 - 1
22 = 20 + 2
23 = 20 + 3
21 = 20 + 1
24 = 20 + 4
17 = 20 -3
Total: 20 * 6 + (2-1+3+1+4-3) = 120 + 6 = 126
We hope that our tips will help you master the techniques of quick mental counting. It should be remembered that theory is only 20% of success. The remaining 80% is your desire and practice.