Let's look at how we can multiply two digit numbers using the traditional methods we are taught in school. Some of these methods can allow you to quickly multiply two-digit numbers in your head with enough practice. Knowing these methods is helpful. However, it is important to understand that this is just the tip of the iceberg. AT this lesson the most popular methods of multiplying two-digit numbers are considered.

The first way is the layout into tens and ones

The easiest way to understand how to multiply two-digit numbers is the one we were taught in school. It consists in splitting both factors into tens and ones, followed by multiplying the resulting four numbers. This method is quite simple, but requires the ability to keep up to three numbers in memory at the same time 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

It is easier to solve such examples in 3 steps. First, tens are multiplied by each other. Then add 2 products of units by tens. Then the product of units is added. Schematically, this can be 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 effect, you need good knowledge multiplication tables 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. The last skill is convenient to train by visualizing the performed arithmetic operations when you have to imagine a picture of your solution, as well as intermediate results.

Conclusion. It is not difficult to make sure that this method is not the most efficient, i.e., one that allows one to obtain correct result. Other methods should be taken into account.

The second way is arithmetic fittings

Bringing an example to a convenient form is a fairly common way of counting in the mind. Customizing an example is useful when you need to quickly find an approximate or exact answer. The desire to adjust examples to certain mathematical patterns is often brought up in mathematics departments at universities or in schools in classes with a mathematical bias. People are taught to find simple and convenient solution algorithms various tasks. Here are some fitting examples:

Example 49*49 can be solved like this: (49*100)/2-49. First, 49 is counted by one hundred - 4900. Then 4900 is divided by 2, which equals 2450, then 49 is subtracted. Total 2401.

The product 56*92 is solved like this: 56*100-56*2*2*2. It turns out: 56*2= 112*2=224*2=448. We subtract 448 from 5600, we get 5152.

This method may be more effective than the previous one only if you own verbal account on the basis of multiplying two-digit numbers by single-digit ones and you can keep several results in mind at the same time. In addition, one has to spend time searching for a solution algorithm, and also takes a lot of attention for the correct observance of this algorithm.

Conclusion. The method when you try to multiply 2 numbers by decomposing them into simpler arithmetic procedures perfectly trains your brain, but is associated with great mental costs, and the risk of getting an incorrect result is higher than with the first method.

The third way is mental visualization of multiplication in a column

56 * 67 - count in a column.

Probably the column contains maximum amount actions and requires constantly keeping auxiliary numbers in mind. But it can be simplified. In the second lesson, it was told that it is important to be able to quickly multiply single digits to double digits. If you already know how to do this automatically, then counting in a column in your mind will not be so difficult for you. The algorithm is

First action: 56*7 = 350+42=392 - remember and don't forget until the third step.

Second action: 56*6=300+36=336 (or 392-56)

Third action: 336 * 10 + 392 = 3360 + 392 = 3 752 - it’s more complicated here, but you can start calling the first number that you are sure of - “three thousand ...”, but for now, add 360 and 392.

Conclusion: counting in a column is directly difficult, but you can, if you have the skill of quickly multiplying two-digit numbers by single-digit ones, simplify it. Add this method to your arsenal. In a simplified form, the column count is some modification of the first method. Which is better is an amateur question.

As you can see, none of the methods described above allows you to count in your mind fast enough and accurately all examples of multiplication of two-digit numbers. It must be understood that the use of traditional methods of multiplication for counting in the mind is not always rational, that is, allowing you to achieve the maximum result with the least effort.

There are three general methods: direct multiplication, the reference number method, and the Trachtenberg method.

Master them all, as each may be more preferable in a given situation.

You can practice the acquired skills using the training table.

Direct multiplication

This method is useful when one of the factors is in the range of 12-18 or ends in 1, and the other is significantly different from it.

One of the multipliers is mentally divided into tens and ones. Then multiply another factor by tens, then by units and add.

For example, 62x13 = 62x10 + 62x3 = 620 + 186 = 806.

Sometimes it is convenient to break a larger multiplier into tens and ones: 42x17 = 17x40 + 17x2 = 714.

Reference number method

It takes a little practice to get used to, but it's very handy when the two factors are close numbers. In particular, this is the main way to square two-digit numbers.

The reference number is a round number close to both factors. It can be less than both factors, greater than both factors, or lies between them.

As a reference number, you should choose numbers that are easy to multiply by. For example, 50 or 100 if they are close to two factors.

Depending on how the reference number and factors are related, the multiplication technique is slightly different.

a. The reference number is less than two factors. For example, you need to multiply 32 by 36.

  • The reference number is 30. The multipliers are greater than the reference number by 2 and 6.
  • Add 6 to the first multiplier and multiply by the reference number: 38 × 30 = 1140.
  • Add the product of 2 and 6: 1140 + 2x6 = 1152.

b. The reference number is greater than two factors. For example, you need to multiply 43 by 48.

  • The reference number is 50. The factors are less than the reference number by 7 and 2.
  • Subtract 2 from the first factor and multiply by the reference number: 41 × 50 = 2050.
  • Add the product of 7 and 2: 2050 + 7x2 = 2064.

in. The reference number is between the factors. For example, you need to multiply 37 by 42.

  • The reference number is 40. The first factor is less by 3, the second is greater by 2.
  • Add 2 to the smaller factor and multiply by the reference number: 39 × 40 = 1560.
  • Subtract the product of 3 and 2: 1440 − 3×2 = 1554.
Trachtenberg method

The Trachtenberg method is the most general. It is convenient to use it whenever special tricks do not work. It also extends to multiplication multi-digit numbers.

Since the Trachtenberg method is not quite familiar, it is better to have multipliers in front of your eyes when mastering it. In the future, practice without writing down the original numbers.

Let's analyze the method using the example of multiplying 87 by 32.

  • Present the numbers in sequence: 8732. Multiply the two internal numbers (7 and 3), the two external numbers (8 and 2) and add. It turns out 37.
  • Multiply the tens: 80x30 = 2400. Add 37x10. It turns out 2770.
  • Add the product of units (7 and 2). Total 2784.

Verbal counting- an occupation that in our time bothers less and less people. It is much easier to get a calculator on your phone and calculate any example.

But is it really so? In this article, we will present math hacks that will help you learn how to quickly add, subtract, multiply and divide numbers in your mind. Moreover, operating not in units and tens, but at least two-digit and three-digit numbers.

After mastering the methods in this article, the idea of ​​​​reaching the phone for a calculator no longer seems so good. After all, you can not waste time and calculate everything in your mind much faster, but at the same time stretch your brains and impress others (of the opposite sex).

We warn you! If you a common person, and not a child prodigy, then it will take training and practice, concentration and patience to develop mental numeracy. At first, everything can turn out slowly, but then things will go smoothly, and you can quickly count any numbers in your head.

Gauss and mental arithmetic

One of the mathematicians with a phenomenal rate of mental calculation was the famous Carl Friedrich Gauss (1777-1855). Yes, yes, the same Gauss who came up with the normal distribution.

In his own words, he learned to count before he could speak. When Gauss was 3 years old, the boy looked at his father's payroll and declared, "The calculations are wrong." After the adults checked everything, it turned out that little Gauss was right.

In the future, this mathematician reached considerable heights, and his works are still actively used in theoretical and applied sciences. Until his death, Gauss did most of his calculations in his head.

Here we will not deal with complex calculations, but start with the simplest.

Adding numbers in your mind

To learn how to add large numbers in your mind, you need to be able to accurately add numbers up to 10 . Ultimately, any complex task comes down to performing a few trivial actions.

Most often, problems and errors occur when adding numbers with a "pass through 10 ". When adding (and even when subtracting), it is convenient to use the technique of “reliance on a dozen”. What is it? First, we mentally ask ourselves how much one of the terms is missing before 10 , and then add to 10 the difference remaining up to the second term.

For example, let's add the numbers 8 and 6 . To out 8 get 10 , lacks 2 . Then to 10 it remains to add 4=6-2 . As a result, we get: 8+6=(8+2)+4=10+4=14

The main trick with adding large numbers is to break them into bit parts, and then add these parts together.

Suppose we need to add two numbers: 356 and 728 . Number 356 can be imagined as 300+50+6 . Likewise, 728 will look like 700+20+8 . Now we add up:

356+728=(300+700)+(50+20)+(8+6)=1000+70+14=1084

Subtracting numbers in your mind

Subtracting numbers will also be easy. But unlike addition, where each number is divided into bit parts, when subtracting, you only need to “break” the number that we subtract.

For example, how much will 528-321 ? Breaking down the number 321 into bit parts and we get: 321=300+20+1 .

Now we consider: 528-300-20-1=228-20-1=208-1=207

Try to visualize the process of addition and subtraction. At school, everyone was taught to count in a column, that is, from top to bottom. One way to restructure thinking and speed up counting is not to count from top to bottom, but from left to right, breaking numbers into place parts.

Multiplying numbers in your mind

Multiplication is the repeated repetition of a number. If you need to multiply 8 on the 4 , which means that the number 8 need to repeat 4 times.

8*4=8+8+8+8=32

Since everything challenging tasks are reduced to simpler ones, you need to be able to multiply all single-digit numbers. There is a great tool for this - multiplication table . If you do not know this table by heart, then we strongly recommend that you first learn it and only then take up the practice of mental counting. In addition, there is, in fact, nothing to learn there.

Multiplication of multi-digit numbers by single-digit

First, practice multiplying multi-digit numbers by single-digit numbers. Let's multiply 528 on the 6 . Breaking down the number 528 into ranks and go from oldest to youngest. We multiply first and then add the results.

528=500+20+8

528*6=500*6+20*6+8*6=3000+120+48=3168

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Multiplication of two-digit numbers

There is nothing complicated here either, only the load on short-term memory is a little more.

Multiply 28 and 32 . To do this, we reduce the whole operation to multiplication by single-digit numbers. Imagine 32 as 30+2

28*32=28*30+28*2=20*30+8*30+20*2+8*2=600+240+40+16=896

One more example. Let's multiply 79 on the 57 . This means that you need to take the number " 79 » 57 once. Let's break the whole operation into stages. Let's multiply first 79 on the 50 , and then - 79 on the 7 .

  • 79*50=(70+9)*50=3500+450=3950
  • 79*7=(70+9)*7=490+63=553
  • 3950+553=4503

Multiply by 11

Here is a quick mental counting trick that will help you multiply any two-digit number by 11 at phenomenal speed.

To multiply a two-digit number by 11 , we add two digits of the number with each other, and enter the resulting amount between the digits of the original number. The resulting three-digit number is the result of multiplying the original number by 11 .

Check and multiply 54 on the 11 .

  • 5+4=9
  • 54*11=594

Take any two digit number, multiply it by 11 and see for yourself - this trick works!

Squaring

With the help of another interesting method of mental counting, you can easily and quickly square two-digit numbers. It is especially easy to do this with numbers that end in 5 .

The result begins with the product of the first digit of the number by the one following it in the hierarchy. That is, if this figure is denoted by n , then the next digit in the hierarchy will be n+1 . The result ends with the square of the last digit, i.e. the square 5 .

Let's check! Let's square the number 75 .

  • 7*8=56
  • 5*5=25
  • 75*75=5625

Division of numbers in the mind

It remains to deal with the division. In fact, this is the inverse operation of multiplication. With division up to 100 no problems should arise at all - after all, there is a multiplication table that you know by heart.

Division by a single number

When dividing multi-digit numbers by a single-digit one, it is necessary to select the largest possible part, which can be divided using the multiplication table.

For example, there is a number 6144 , to be divided by 8 . Remember the multiplication table and understand that on 8 will divide the number 5600 . Let's imagine an example in the form:

6144:8=(5600+544):8=700+544:8

544:8=(480+64):8=60+64:8

Left to divide 64 on the 8 and get the result by adding all the results of the division

64:8=8

6144:8=700+60+8=768

Division by two digits

When dividing by a two-digit number, you need to use the rule for the last digit of the result when multiplying two numbers.

When multiplying two multi-digit numbers, the last digit of the multiplication result always coincides with the last digit of the result of multiplying the last digits of these numbers.

For example, let's multiply 1325 on the 656 . As a rule, the last digit in the resulting number will be 0 , as 5*6=30 . Really, 1325*656=869200 .

Now, armed with this valuable information, consider dividing by a two-digit number.

How much will 4424:56 ?

Initially, we will use the “fitting” method and find the limits within which the result lies. We need to find the number that, when multiplied by 56 will give 4424 . Intuitively, let's try the number 80.

56*80=4480

So the required number is less than 80 and obviously more 70 . Let's determine its last digit. Her work on 6 must end with a number 4 . According to the multiplication table, the results are suitable for us 4 and 9 . It is logical to assume that the result of division can be either a number 74 , or 79 . We check:

79*56=4424

Done, solution found! If the number didn't fit 79 , the second option would certainly be correct.

In conclusion, we present a few useful tips, which will help you quickly learn oral counting:

  • Don't forget to exercise every day;
  • do not quit training if the result does not come as quickly as you would like;
  • download mobile app for oral counting: so you don’t have to come up with examples for yourself;
  • Read books on quick mental counting techniques. There are different mental counting techniques, and you will be able to master the one that suits you best.

The benefits of mental arithmetic are undeniable. Practice, and every day you will count faster and faster. And if you need help in solving more complex and multi-level tasks, contact the student service specialists for fast and qualified help!

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Learn multiplication table - game

Try our educational e-game. Using it, you will be able to decide tomorrow math problems in class at the blackboard with no answers, without resorting to the tablet to multiply the numbers. One has only to start playing, and after 40 minutes there will be an excellent result. And to consolidate the result, train several times, not forgetting the breaks. Ideally, every day (save the page so you don't lose it). game form The simulator is suitable for both boys and girls.

Result: 0 points

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Multiplication directly on the site (online)

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Multiplication table (numbers 1 to 20)
× 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320
17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340
18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360
19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380
20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

How to multiply numbers by a column (mathematics video)

To practice and learn quickly, you can also try to multiply numbers by a column.

Those who treated mathematics lessons with disdain at school must have been in an awkward situation at least a few times in their lives. How to calculate how much to leave for a tip or amount utility bill? If you know a couple of simple tricks, it will take you literally a second. And during the exam, knowing the rules for multiplying large numbers can help save critically missing time. Mel shares with Creu simple secrets computing.

For those who are preparing for the main school exam

1. Multiply by 11

We all know that when multiplying by ten, zero is added to the number, but did you know that there is an equally simple way to multiply a two-digit number by 11? There he is:

Take the original number and imagine the gap between two digits (in this example, we use the number 52): 5_2

Now add the two numbers and write them in the middle: 5_(5+2)_2.

Thus, your answer is: 572. If adding the numbers in brackets results in a two-digit number, just remember the second digit, and add one to the first number: 9_(9+9)_9 (9+1)_8_9 10_8_9 1089. This always works.

2. Fast squaring

This technique will help you quickly square a two-digit number that ends in five. Multiply the first number by itself +1 and add 25 at the end. That's it! 252 = (2x(2+1)) & 25

3. Multiply by five

The multiplication table for five is very easy for most, but when you have to deal with large numbers, it becomes more difficult to do this.

This trick is incredibly simple. Take any number and divide it in half. If the result is an integer, add a zero at the end. If not, ignore the comma and add five at the end. This always works:

2682x5 = (2682 / 2) & 5 or 0

2682 / 2 = 1341 (whole number so add 0)

Let's try another example:

2943.5 (fractional, omit comma, add 5)

4. Multiply by nine

It's simple. To multiply any number from one to nine by nine, look at the hands. Bend the finger that corresponds to the multiplied number (for example, 9x3 - bend the third finger), count the fingers up to the crooked finger (in the case of 9x3 it is two), then count after the crooked finger (in our case seven). The answer is 27.

5. Multiply by four

This is a very simple technique, although obvious only to some. The trick is to simply multiply by two and then multiply by two again: 58x4 = (58x2) + (58x2) = (116) + (116) = 232.

6. Tip counting

If you need to leave a 15% tip, there is an easy way to do so. Calculate 10% (divide the number by ten), and then add the resulting number to half of it and get the answer:

15% of $25 = (10% of 25) + ((10% of 25) / 2)

$2.50 + $1.25 = $3.75

7. Complex multiplication

If you need to multiply large numbers, and one of them is even, you can simply rearrange them to get the answer:

32x125 is the same as:

16x250 is the same as:

8×500 is the same as:

8. Divide by five

In fact, dividing large numbers by five is very simple. You just need to multiply by two and move the comma:

1 . 195 * 2 = 390

2 . Move the comma: 39.0 or just 39.

1 . 2978 * 2 = 5956

2 . 595,6

9. Subtraction from 1000

To subtract from 1000, you can use this simple rule. Subtract all digits from nine except the last one. And subtract the last digit from ten:

1 . Subtract 6 = 3 from 9

2 . Subtract 4 from 9 = 5

3 . Subtract 8 = 2 from 10

10. Systematized multiplication rules

Multiply by 5: Multiply by 10 and divide by 2.

Multiply by 6: Sometimes it's easier to multiply by 3 and then by 2.

Multiply by 9: Multiply by 10 and subtract the original number.

Multiply by 12: Multiply by 10 and add the original number twice.

Multiply by 13: Multiply by 3 and add the original number 10 times.

Multiply by 14: Multiply by 7 and then by 2.

Multiply by 15: Multiply by 10 and add 5 times the original number, as in the previous example.

Multiply by 16: If you want, multiply 4 times by 2. Or multiply by 8, and then by 2.

Multiply by 17: Multiply by 7 and 10 times add the original number.

Multiply by 18: Multiply by 20 and subtract the original number twice.

Multiply by 19: Multiply by 20 and subtract the original number.

Multiply by 24: Multiply by 8 and then by 3.

Multiply by 27: Multiply by 30 and subtract the original number 3 times.

Multiply by 45: Multiply by 50 and subtract the original number by 5 times.

Multiply by 90: Multiply by 9 and assign 0.

Multiply by 98: Multiply by 100 and subtract the original number twice.

Multiply by 99: Multiply by 100 and subtract the original number.

BONUS: interest

Calculate 7% of 300.

First you need to understand the meaning of the word "percent" (percent). The first part of the word is about (per). Per = for each. The second part is a cent, which is like 100. For example, a century = 100 years. 100 cents in one dollar and so on. So percentage = for every hundred.

So, it turns out that 7% of 100 will be seven. (Seven for every hundred, only one hundred).

8% of 100 = 8.

35.73% of 100 = 35.73

But how can this be useful? Let's return to the problem of 7% of 300.

7% of the first hundred is 7. 7% of the second hundred is the same as 7, and 7% of the third hundred is still the same 7. So, 7 + 7 + 7 = 21. If 8% of 100 = 8, then 8 % of 50 = 4 (half of 8).

Split each number if you need to calculate percentages of 100, if the number is less than 100, just move the comma to the left.

Examples:

8%200 =? 8 + 8 = 16.

8%250 =? 8 + 8 + 4 = 20,

8%25 = 2.0 (Move decimal point to the left).

15%300 = 15+15+15 =45

15%350 = 15+15+15+7,5 = 52,5

It's also good to know that you can always swap the numbers: 3% of 100 is the same as 100% of 3. And 35% of 8 is the same as 8% of 35.