Short entry: Short notation: 98*41 = 100*39 + 118 = 4,018

Thus, by using a single reference number, it is possible to multiply a large combination of two-digit numbers. If you are good at multiplying by 30, 40, 60, 70 or 80, then you can use this technique to multiply any numbers (up to 100 and even more).

Using Multiple Reference Numbers

The multiplication technique using reference numbers allows you to use 2 reference numbers. This is convenient when the reference number of one factor can be expressed in terms of the reference number of another. For example, in the product "23 * 88" it is convenient to use the reference number 20 for 23 and 80 for 88. Multiplying these numbers using two references is convenient because 20 = 80:4.

The technique of 2 reference numbers is that we first divide 88 by 4 and get 22, multiply 23 by 22 and multiply the product again by 4. That is, we first divide the product by 4, and then multiply by 4. It turns out: 23*22 = 250*2+6= 506, and 506*4 = 2024 - this is the answer!

For visualization, you can use the already familiar diagram. The product 23*88 is calculated as follows:

We write down a convenient reference number “20” and add a factor of 4 next to it, with which we can express 80 in terms of 20.
Then we do, as before, write how much 23 exceeds 20 (3), and 88 exceeds 80 (8).
Above the triple we write the product 3 by 4 (that is, 3 by the reference multiplier).
To 88 we add the product of 3 by 4 and multiply by the reference (20), we get 100*20 = 2000
We add to 2000 the product of 3 and 8. Result: 2024

Examples of 3 digit by 2 digit multiplication. Ways to quickly multiply numbers verbally. Visual Geometry Game

With the best free game learns very quickly. Check it out for yourself!

Learn multiplication tables - game

Try our educational e-game. Using it, you will be able to decide tomorrow math problems in class at the blackboard without answers, without resorting to a tablet to multiply numbers. You just have to start playing, and in about 40 minutes it will be excellent result. And to consolidate the result, train several times, not forgetting about breaks. Ideally, every day (save the page so as not to lose it). Game form The exercise machine is suitable for both boys and girls.

See cheat sheets below full form.

Multiplication directly on the site (online)

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Multiplication table (numbers from 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 in a column (mathematics video)

To practice and learn quickly, you can also try multiplying numbers by column.

Let's look at how you can multiply two-digit numbers using traditional methods, which we are taught in school. Some of these methods may allow you to quickly multiply two-digit numbers in your head with enough practice. It is useful to know these methods. However, it is important to understand that this is just the tip of the iceberg. This lesson covers the most popular multiplication techniques. double digit numbers.

The first method is the layout into tens and units

The easiest way to understand multiplying two-digit numbers is the one we were taught at school. It 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

It’s easier to solve such examples in 3 steps. First, the tens are multiplied by each other. Then the 2 products of ones and tens are added. 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 maximum quick 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 performed, when you should imagine a picture of your solution, as well as intermediate results.

Conclusion. It is not difficult to see that this method is not the most effective, that is, allowing least actions get correct result. Other methods should be taken into account.

The second method is arithmetic adjustments

Bringing an example to convenient view is a fairly common way of counting in your head. Fitting an example is useful when you need to quickly find an approximate or exact answer. The desire to fit examples to certain mathematical patterns is often cultivated in mathematics departments at universities or in schools in classes with a mathematical bias. People are taught to find simple and convenient algorithms for solving various problems. Here are some examples of fitting:

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

The product 56*92 is solved as follows: 56*100-56*2*2*2. It turns out: 56*2= 112*2=224*2=448. From 5600 we subtract 448, we get 5152.

This method may be more effective than the previous one only if you own verbal counting based on multiplying two-digit numbers by single-digit numbers and you can keep several results in mind at the same time. In addition, you have to spend time searching for a solution algorithm, and a lot of attention is also spent on correctly following this algorithm.

Conclusion. The method where you try to multiply 2 numbers by breaking them down into simpler arithmetic procedures is a great way to train your brain, but it involves a lot of mental effort, and the risk of getting the wrong result is higher than with the first method.

The third method is mental visualization of multiplication in a column

56*67 - count in a column.

Probably the count in a column contains maximum quantity actions and requires constantly keeping auxiliary numbers in mind. But it can be simplified. The second lesson taught that it is important to be able to quickly multiply single-digit numbers by double-digit numbers. If you already know how to do this automatically, then counting in a column in your head will not be so difficult for you. The algorithm is as follows

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 saying the first number you’re sure of - “three thousand...”, and while you’re talking, add 360 and 392.

Conclusion: Counting in a column is directly complicated, but if you have the skill of quickly multiplying two-digit numbers by single-digit numbers, you can simplify it. Add this method to your arsenal. In a simplified form, counting in a column is some modification of the first method. Which is better is not a question for everyone.

As you can see, none of the methods described above allows you to count all examples of multiplication of two-digit numbers in your head quickly and accurately enough. It is necessary to understand that the use traditional ways multiplication for mental calculation is not always rational, that is, allowing you to achieve the maximum result with the least effort.

Don't like math? You just don't know how to use it! In fact, this is fascinating science. And our selection of unusual multiplication methods confirms this.

Multiply on your fingers like a merchant

This method allows you to multiply numbers from 6 to 9. To begin, bend both hands into fists. Then on your left hand, bend as many fingers as the first factor is greater than the number 5. On your right hand, do the same for the second factor. Count the number of extended fingers and multiply the sum by ten. Now multiply the sum of the bent fingers of the left and right hand. By adding both sums, you get the result.

Example. Let's multiply 6 by 7. Six is more than five by one, which means we bend one finger on our left hand. And seven is two, which means there are two fingers on the right. The total is three, and after multiplying by 10 it is 30. Now let’s multiply the four bent fingers of the left hand and three of the right. We get 12. The sum of 30 and 12 gives 42.

Actually, here we are talking about a simple multiplication table, which it would be good to know by heart. But this method is good for self-testing, and it’s also useful to stretch your fingers.

Multiply like Ferrol

This method was named after the German engineer who used it. Method allows you to quickly multiply numbers from 10 to 20. If you practice, you can do it even in your head.

The point is simple. The result will always be a three-digit number. So first we count units, then tens, then hundreds.

Example. Let's multiply 17 by 16. To get units, multiply 7 by 6, tens - add the product of 1 and 6 with the product of 7 and 1, hundreds - multiply 1 by 1. As a result, we get 42, 13 and 1. For convenience, write them in a column and let's add it up That's the result!

Multiply like a Japanese

This graphic method, which is used by Japanese schoolchildren, makes it easy to multiply two- and even three-digit numbers. To try it out, have some paper and pen ready.

Example. Let's multiply 32 by 143. To do this, draw a grid: reflect the first number with three and two lines with a horizontal indent, and the second with one, four and three lines vertically. Place dots where the lines intersect. As a result, we should get a four-digit number, so we will conditionally divide the table into 4 sectors. And let's count the points that fall into each of them. We get 3, 14, 17 and 6. To get the answer, add the extra ones from 14 and 17 to the previous number. We get 4, 5 and 76 - 4576.

Multiply like an Italian

Another interesting graphic method is used in Italy. Perhaps it is simpler than the Japanese one: you definitely won’t get confused when transferring tens. To multiply large numbers using it, you need to draw a grid. We write down the first factor horizontally from above, and the second factor vertically to the right. In this case, there should be one cell for each number.

Now let's multiply the numbers in each row by the numbers in each column. We write the result in a cell (divided in two) at their intersection. If you get a single digit number, then top part We write 0 in the cells, and in the lower one - the result obtained.

All that remains is to add up all the numbers in the diagonal stripes. We start from the bottom right cell. In this case, we add tens to the units in the adjacent column.

This is how we multiplied 639 by 12.

Fun, right? Have fun with mathematics! And remember that humanities specialists are also needed in IT!

The most popular technique for multiplying large numbers in the mind is the technique of using the so-called reference number. In the last lesson, when we showed how to multiply numbers up to 20, we essentially used the reference number 10. It is also worth noting that you can learn more about the method of using the reference number in the book "" by Bill Handley.

General rules for using a reference number

The reference number is useful when multiplying numbers that are close together and when squaring them. You already understood how you can use the reference number method from the last lesson, now let's summarize everything that has been said.

The reference number for multiplication 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:

47*48

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)
Next we multiply 45 by 50 = 2250
Then we add 2*3 to this result and voila - 2,256!

It is convenient to visualize the table below schematically in your mind.

(reference number)

(48-3)*50 = 45*50 = 2 250

(or (47-2)*50= 45*50 remember that multiplying by 5 is the same as dividing by 2)

Answer:

2 250 + 6 = 2 256

We write the reference number to the left of the product. If the numbers are less than the reference number, then the difference between them and the reference is written below these numbers. To the right of 48*47 we write the calculation with the reference number, to the right of remainders 2 and 3 we write their product.

If we use a simplified scheme, the solution looks like this: 47*48=45*50 + 6= 2,256

Let's look at other examples:

Multiply 18*19

(reference number)	18	*	19	(18-1)20 = 340*
	2	*	1	+2
Answer:				342

Short entry: 18*19 = 20*17+2 = 342

Multiply 8*7

(reference number)	8	*	7	(8-3)*10 = 50
	2	*	3	+6
Answer:				56

Short entry: 8*7 = 10*5+6 = 56

Multiply 98*95

(reference number)	98	*	95	(95-2)100 = 9300*
	2	*	5	+10
Answer:				9310

Short entry: 98*95 = 100*93 + 10 = 9 310

Multiply 98*71

(reference number)	98	*	71	(71-2)100 = 6900*
	2	*	29	+58
Answer:				6958

Short entry: 98*71 = 100*69 + 58 = 6 958

Both numbers are greater than the reference (above the reference)

Let's say we want to multiply 54 by 53. These numbers are close enough to the number 50, and therefore it is convenient to use 50 as a reference number. But unlike previous examples, these numbers are larger than the reference one. In fact, the model of their multiplication does not change, but now you need to add, rather than subtract, remainders.

To 54 add as much as 53 exceeds 50, that is, 3. It turns out 57 (or add 4 to 53 - it’s always the same)
Next we multiply 57 by 50 = 2,850 (multiplying by 50 is similar to dividing by 2)
Then add 4*3 to this result. Answer: 2862

+12

(reference number)

(54+3)*50 = 2 850

or (53+4)*50= 57*50 (remember that multiplying by 5 is the same as dividing by 2)

Answer:

2 862

The short solution looks like this: 50*57+12 = 2,862

For clarity, below are examples:

Multiply 23*27

				+21
(reference number)	23	*	27	(23+7)20 = 600*
Answer:				621

Short entry: Short notation: 23*27 = 20*30 + 21 = 621

Multiply 51*63

				+13
(reference number)	51	*	63	(63+1)50 = 3 200*
Answer:				3 213

Short entry: Short notation: 51*63 = 64*50 + 13 = 3,213

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.

Multiply 45*52

The product 45*52 is calculated as follows:

We subtract 5 from 52 or add 2 to 45. In either case we get: 47
Next we multiply 47 by 50 = 2,350 (multiplying by 50 is similar to dividing by 2)
Then we subtract (and not add, as before!) 2*5. Answer: 2 340

			2
(reference number)	45	*	52	(45+2)50 = 2 350*
	5			-10
Answer:				2 340

Short notation: 45*52 = 47*50-10 = 2,340

We also do the same with similar examples:

Multiply 91*103

			3
(reference number)	91	*	103	(91+3)100 = 9400*
	9			-27
Answer:				9 373

Only one number is close to the reference number, and the other is not

As you have already seen from the examples, the reference number is convenient to use if even only one number is close to the reference number. It is desirable that the difference between this number and the reference number is no more than 2-x or 3-x or equal to a number that is convenient to multiply by (for example, 5, 10, 25 - see the second lesson)

Multiply 48*73

			23
(reference number)	48	*	73	(73-2)50 = 3 550*
	2			-46
Answer:				3 504

Short solution: 48*73 = 71*50 - 23*2 = 3 504

Multiply 23*69

	3		49	147
(reference number)	23	*	69	(3+69)20 = 1440*
Answer:				1 587

Short entry: Short solution: 23*69 = 72*20 + 147 = 1,587 - a little more complicated

+118

Answer:

4018

	3*4=12
	3	*	8	+24
(reference number)	23	*	88	(88+12)20 = 2 000*
Answer:				2 024 88	(23-3)100 = 2 000*
	2		12	+24
			12:4=3
Answer:				2 024

Short entry: 23*88 = (23-12:4)*100 + 24 = 2024

As you can see, the answer is the same.

The method using two reference numbers is somewhat more complicated and requires additional steps. First, you must understand which 2 reference numbers you are comfortable using. Secondly, you need to perform an additional action to find the number that needs to be multiplied by the reference.

It is better to use this technique when you have already mastered multiplication with one reference number quite well.

Training

If you want to improve your skills on the topic this lesson, you can use next game. The points you receive are affected by the correctness of your answers and the time spent on completion. Please note that the numbers are different each time.

Some quick ways oral multiplication We’ve already figured it out, now let’s take a closer look at how to quickly multiply numbers in your head using various auxiliary methods. You may already know, and some of them are quite exotic, such as the ancient Chinese way multiplying numbers.

Layout by ranks

Is the most simple trick fast multiplication of two-digit numbers. Both factors need to be divided into tens and ones, and then all these new numbers must be multiplied by each other.

This method requires the ability to hold up to four numbers in memory at the same time, and to do calculations with these numbers.

For example, you need to multiply numbers 38 And 56 . We do it this way:

38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + 8 * 50 + 30 * 6 + 8 * 6 = 1500 + 400 + 180 + 48 = 2128 It will be even easier to do oral multiplication of two-digit numbers in three operations. First you need to multiply the tens, then add two products of ones by tens, and then add the product of ones by ones. It looks like this: 38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + (8 * 50 + 30 * 6) + 8 * 6 = 1500 + 580 + 48 = 2128 In order to successfully use this method, you need to know the multiplication table well, be able to quickly add two-digit and three-digit numbers, and switch between mathematical operations without forgetting intermediate results. The last skill is achieved through help and visualization.

This method is not the fastest and most effective, so it is worth exploring other methods of oral multiplication.

Fitting the numbers

You can try to bring the arithmetic calculation to a more convenient form. For example, the product of numbers 35 And 49 can be imagined this way: 35 * 49 = (35 * 100) / 2 — 35 = 1715
This method may be more effective than the previous one, but it is not universal and is not suitable for all cases. It is not always possible to find a suitable algorithm to simplify the problem.

On this topic, I remembered an anecdote about how a mathematician sailed along the river past a farm and told his interlocutors that he was able to quickly count the number of sheep in the pen, 1358 sheep. When asked how he did it, he said it was simple - you need to count the number of legs and divide by 4.

Visualization of columnar multiplication

This one is one of the most universal methods oral multiplication of numbers, developing spatial imagination and memory. First, you should learn to multiply two-digit numbers by single-digit numbers in a column in your head. After this, you can easily multiply two-digit numbers in three steps. First, a two-digit number must be multiplied by the tens of another number, then multiplied by the units of another number, and then sum the resulting numbers.

It looks like this: 38 * 56 = (38 * 5) * 10 + 38 * 6 = 1900 + 228 = 2128

Visualization with number arrangement

Very interesting way Multiplying two-digit numbers is as follows. You need to sequentially multiply the digits in numbers to get hundreds, ones and tens.

Let's say you need to multiply 35 on 49 .

First you multiply 3 on 4 , you get 12 , then 5 And 9 , you get 45 . Recording 12 And 5 , with a space between them, and 4 remember.

You receive: 12 __ 5 (remember 4 ).

Now you multiply 3 on 9 , And 5 on 4 , and sum up: 3 * 9 + 5 * 4 = 27 + 20 = 47 .

Now we need to 47 add 4 which we remember. We get 51 .

We write 1 in the middle and 5 add to 12 , we get 17 .

In total, the number we were looking for is 1715 , it is the answer:

35 * 49 = 1715
Try multiplying in your head in the same way: 18 * 34, 45 * 91, 31 * 52 .

Chinese or Japanese multiplication

In Asian countries, it is customary to multiply numbers not in a column, but by drawing lines. For Eastern cultures, the desire for contemplation and visualization is important, which is probably why they came up with such a beautiful method that allows you to multiply any numbers. This method is complicated only at first glance. In fact, greater clarity allows you to use this method much more effectively than column multiplication.

In addition, knowledge of this ancient oriental method increases your erudition. Agree, not everyone can boast that they know the ancient multiplication system that the Chinese used 3000 years ago.

Video about how the Chinese multiply numbers

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