Multiplying two-digit numbers. Multiplication Multiplying two-digit numbers

How to quickly multiply large numbers, how to master such useful skills? Most people find it difficult to verbally multiply two-digit numbers by single-digit numbers. And there is nothing to say about complex arithmetic calculations. But if desired, the abilities inherent in every person can be developed. Regular training, a little effort and application developed by scientists, effective techniques will allow you to achieve amazing results.

Choosing traditional methods

Methods of multiplying two-digit numbers that have been proven for decades do not lose their relevance. The simplest techniques help millions of ordinary schoolchildren, students of specialized universities and lyceums, as well as people engaged in self-development, improve their computing skills.

Multiplication using number expansion

Most the easy way How to quickly learn to multiply large numbers in your head is to multiply tens and units. First, the tens of two numbers are multiplied, then the ones and tens alternately. The four numbers received are summed up. To use this method, it is important to be able to remember the results of multiplication and add them in your head.

For example, to multiply 38 by 57 you need:

• factor the number into (30+8)*(50+7) ;
• 30*50 = 1500 – remember the result;
• 30*7 + 50*8 = 210 + 400 = 610 – remember;
• (1500 + 610) + 8*7 = 2110 + 56 = 2166

Naturally, it is necessary to have excellent knowledge of the multiplication table, since it will not be possible to quickly multiply in your head in this way without the appropriate skills.

Multiplication by column in the mind

Many people use a visual representation of the usual columnar multiplication in calculations. This method is suitable for those who can memorize auxiliary numbers for a long time and perform arithmetic operations with them. But the process becomes much easier if you learn how to quickly multiply two-digit numbers by single-digit numbers. To multiply, for example, 47*81 you need:

• 47*1 = 47 – remember;
• 47*8 = 376 – remember;
• 376*10 + 47 = 3807.

Speaking them out loud while simultaneously summing them up in your head will help you remember intermediate results. Despite the difficulty of mental calculations, after some training this method will become your favorite.

The above multiplication methods are universal. But knowing more efficient algorithms for some numbers will greatly reduce the number of calculations.

Multiplying by 11

This is perhaps the simplest method that is used to multiply any two-digit numbers by 11.

It is enough to insert their sum between the digits of the multiplier:
13*11 = 1(1+3)3 = 143

If the number in brackets is greater than 10, then one is added to the first digit, and 10 is subtracted from the amount in brackets.
28*11 = 2 (2+8) 8 = 308

Multiplying large numbers

It is very convenient to multiply numbers close to 100 by decomposing them into their components. For example, you need to multiply 87 by 91.

• Each number must be represented as the difference between 100 and one more number:
  (100 - 13)*(100 - 9)
  The answer will consist of four digits, the first two of which are the difference between the first factor and the subtracted from the second bracket, or vice versa - the difference between the second factor and the subtracted from the first bracket.
  87 – 9 = 78
  91 – 13 = 78
• The second two digits of the answer are the result of multiplying those subtracted from two parentheses. 13*9 = 144
• As a result, the numbers 78 and 144 are obtained. If, when writing down the final result, a number of 5 digits is obtained, the second and third digits are summed. Result: 87*91 = 7944 .

These are the most simple ways multiplication. After using them repeatedly, bringing the calculations to automation, you can master more complex techniques. And after a while, the problem of how to quickly multiply two-digit numbers will no longer worry you, and your memory and logic will improve significantly.

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

Learn multiplication tables - game

Try our educational e-game. Using it, tomorrow you will be able to solve mathematical problems in class at the blackboard without answers, without resorting to a tablet to multiply numbers. You just have to start playing, and within 40 minutes you will have an excellent result. And to consolidate the results, 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 the full cheat sheet below.

Multiplication directly on the site (online)

*
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.

Multiplying two-digit numbers | Online trainer

The exercise is considered completed after 7 correct answers.

The norm for performing the exercise is 3 minutes

To successfully complete the exercise, familiarize yourself with the theory and work through the previous lessons

Multiplying two-digit numbers | Theory

In general, it is convenient to multiply two-digit numbers in your head in the following order:

1. For the base (first or left) number, take the number with the largest second digit;
2. multiply the base (first) two-digit number by the tens of another (second) two-digit number;
3. multiply the base (first) two-digit number by the units of another (second) two-digit number;
4. add the two results.

Challenge: 42 x 36

1) 36 x 42 (the number 36 is taken as the base (first) number, since 6>1)

2) 36 x 40 = (30+6) x 4 x 10

30 x 4 = 120; 6 x 4 = 24; 120 + 24 = 144; 144 x 10 = 1440*

3) 36 x 2 = (30+6) x 2

30 x 2 = 60; 6 x 2 = 12; 60 + 12 = 72

4) 1440 + 72 = 1752

Challenge: 47 x 52

1) 47 x 52 (the number 47 is taken as the base (first) number, since 7>2)

2) 47 x 50 = 2350

4) 2350 + 94 = 2444

If one of the numbers ends in 9, then it is more convenient to solve the problem in the following order:

1. for the second (located on the right) number take the number ending in 9;
2. round the second number up to tens by adding 1 to it;
3. multiply the first number by the rounded second number;
4. subtract the first number from the result of step 3.

Challenge: 39 x 56

1) 56 x 39 (the number 39 is taken as the second (on the right) number, since it ends in 9)

2) 56 x 39(40-1)

3) 56 x 40 = (50+6) x 4 x 10

50 x 4 = 200; 6 x 4 = 24; 200 + 24 = 224; 224 x 10 = 2240

4) 2240 - 56 = 2184

If one of the two-digit numbers is 11, then solving such a problem will be much easier if you use the technique outlined in Lesson 1.

In many cases, solving the problem of multiplying two-digit numbers in your head is much easier if you use the factorization method.

Factorization is the transformation of a number into a product of simpler numbers. For example, the number 24 can be converted into the product of 8 and 3 (24 = 8 x 3) or 6 and 4 (24 = 6 x 4). The number 24 can also be represented as the product of 12 and 2 (24 = 12 x 2), but when doing mental arithmetic it is more convenient to deal with single-digit numbers.

Individual two-digit numbers can also be represented as the product of three single-digit numbers. For example, 84 = 7 x 6 x 2 = 7 x 4 x 3.

Let's solve the multiplication problem using factorization.

Problem: 34 x 42

Factoring the number 24 gives 8 and 3 or 6 and 4. To solve the problem, we will represent the number 24 as the product of 6 and 4, but if you prefer, you can choose the product of 8 and 3.

Multiply the first number by 6, then multiply the result by 4:

34 x 6 = 204

204 x 4 = 816

To know which two-digit numbers can be factorized, you need to carefully study the multiplication table. You can write down all two-digit numbers that can be factorized, indicating possible ways their factorization.

If both of the two-digit numbers being multiplied can be factorized, then in most cases it is more convenient to factor the smaller number.

Challenge: 36 x 72

The number 36 can be represented as the product of 6 and 6, and the number 72 as the product of 9 and 8.

Since 36

72 x 6 = 432

432 x 6 = 2592

Example with factorization by three numbers.

Challenge: 57 x 75

If one of the two-digit numbers being multiplied consists of identical digits (22, 33, 44, etc.), then it is more convenient to factor it by 11 and 2, 3, 4, etc.), since multiplication by 11 is not difficult, as was shown in lesson 11.

Problem: 81 x 44

If the numbers are close in value to a round number, then when multiplying them in your mind it is convenient to use the following formulas: (C+a)(C+b) = (C+a+b)C+ab; (C-a)(C-b) = (C-a-b)C+ab; (C+a)(C-b) = (C+a-b)C-ab**, where “C” is a round number close to the two numbers being multiplied, and “a” and “b” are the differences between the numbers being multiplied and the round number .

Challenge: 67 x 64

(60 + 7) x (60 + 4) = (60 + 7 + 4) x 60 + 7 x 4 = 71 x 60 + 28 = 4260 + 28 = 4288

Problem: 39 x 38

(40 - 1) x (40 - 2) = (40 - 1 - 2) x 40 + 1 x 2 = 37 x 40 + 2 = 1480 + 2 = 1482

Challenge: 41 x 38

(40 + 1) x (40 – 2) = (40 + 1 – 2) x 40 + 1 x 2 = 39 x 40 – 2 = 1558

It is more convenient to multiply two-digit numbers, the first digits (tens) of which are equal, and the second digits (units) add up to 10, in the following order:

1. multiply the first digit of two-digit numbers by the same digit increased by one;
2. multiply the second digits of two-digit numbers;
3. place the results of point 1 and point 2 one after the other.

Challenge: 76 x 74

Don't be discouraged or give up if you have trouble multiplying two-digit numbers at first. To confidently perform such an operation mentally requires practice, as well as creativity.

* To memorize intermediate results of calculations in your mind, you can use mnemonics based on the association of numbers with images.

** Proof of formulas by transformation: (C+a)(C+b) = (C+a)C+(C+a)b = C 2 +Ca+Cb+ab = (C+a+b)C+ab ; (C-a)(C-b) = (C-a)C-(C-a)b = C 2 -Ca-Cb+ab = (C-a-b)C+ab; (C+a)(C-b) = (C+a)C-(C+a)b = C 2 +Ca-Cb-ab = (C+a-b)C-ab.

***Proof of the method: according to the formula used in the previous method (C+a)(C+b) = (C+a+b)C+ab; since a+b=10, then (C+a)(C+b) = (C+10)C+ab; since the product of two-digit round numbers C and C+10 gives a number with two zeros at the end, and the product of a and b gives a two-digit number, then to find the sum of these two expressions it is enough to put the product of a and b instead of the last two zeros of the first expression.

Page 1 of 4

Exact products of two-digit numbers 11 -- 50 (Bradis Table 1)

Bradys table products of two-digit numbers consists of 89 tablets of the products of each of the natural numbers from 11 to 99, indicated in bold numbers on the right, by all the integers from 0 to 99. To get, for example, the product of 57-49, you need to take the tablet with the number 57 and find the intersection of the line with the heading (left) 40 and Column with heading (top) 9. The same product 2793 can be obtained from plate 49 at the intersection of row 50 and column 7.

Using the distributive property, you can use the Bradis table to simplify the product of any multi-digit number by a two-digit number, as well as the multiplication of any multi-digit by multi-digit number. To avoid errors, it is better to write three-digit products, such as 35-17 = 595, as four-digit products by adding a zero on the left: 35-17 = 0595. If the factor contains an odd number of digits, it is useful to add a zero on the right, discarding it in the final result.

Bradys table 1 also simplifies the division of any multi-digit number by a two-digit number: while ordinary written division gives the digits of the quotient one at a time, using the table gives them two at once. A plate with a number equal to the divisor is used; two digits of the dividend must be taken down at once. If, when dividing with a remainder, only one (rightmost) digit of the dividend is added, then in the quotient only one (last) digit is obtained. But if the quotient must be found in the form of a decimal fraction, then the last digit of the dividend is taken together with zero tenths.

	0	1	2	3	4	5	6	7	8	9
0	0	11	22	33	44	55	66	77	88	99	11
10	110	121	132	143	154	165	176	187	198	209
20	220	231	242	253	264	275	286	297	308	319
30	330	341	352	363	374	385	396	407	418	429
40	440	451	462	473	484	495	506	517	528	539
50	550	561	572	583	594	605	616	627	638	649
60	660	671	682	693	704	715	726	737	748	759
70	770	781	792	803	814	825	836	847	858	869
80	880	891	902	913	924	935	946	957	968	979
90	990	1001	1012	1023	1034	1045	1056	1067	1078	1089
0	0	12	24	36	48	60	72	84	96	108	12
10	120	132	144	156	168	180	192	204	216	228
20	240	252	264	276	288	300	312	324	336	348
30	360	372	384	396	408	420	432	444	456	468
40	480	492	504	516	528	540	552	564	576	588
50	600	612	624	636	648	660	672	684	696	708
60	720	732	744	756	768	780	792	804	816	828
70	840	852	864	876	888	900	912	924	936	948
80	960	972	984	996	1008	1020	1032	1044	1056	1068
90	1080	1092	1104	1116	1128	1140	1152	1164	1176	1188
0	0	13	26	39	52	65	78	91	104	117	13
10	130	143	156	169	182	195	208	221	234	247
20	260	273	286	299	312	325	338	351	364	377
30	390	403	416	429	442	455	468	481	494	507
40	520	533	546	559	572	585	598	611	624	637
50	650	663	676	689	702	715	728	741	754	767
60	780	793	806	819	832	845	858	871	884	897
70	910	923	936	949	962	975	988	1001	1014	1027
80	1040	1053	1066	1079	1092	1105	1118	1131	1144	1157
90	1170	1183	1196	1209	1222	1235	1248	1261	1274	1287
0	0	14	28	42	56	70	84	98	112	126	14
10	140	154	168	182	196	210	224	238	252	266
20	280	294	308	322	336	350	364	378	392	406
30	420	434	448	462	476	490	504	518	532	546
40	560	574	588	602	616	630	644	658	672	686
50	700	714	728	742	756	770	784	798	812	826
60	840	854	868	882	896	910	924	938	952	966
70	980	994	1008	1022	1036	1050	1064	1078	1092	1106
80	1120	1134	1148	1162	1176	1190	1204	1218	1232	1246
90	1260	1274	1288	1302	1316	1330	1344	1358	1372	1386
	0	1	2	3	4	5	6	7	8	9

	0	1	2	3	4	5	6	7	8	9
0	0	15	30	45	60	75	90	105	120	135	15
10	150	165	180	195	210	225	240	255	270	285
20	300	315	330	345	360	375	390	405	420	435
30	450	465	480	495	510	525	540	555	570	585
40	600	615	630	645	660	675	690	705	720	735
50	750	765	780	795	810	825	840	855	870	885
60	900	915	930	945	960	975	990	1005	1020	1035
70	1050	1065	1080	1095	1110	1125	1140	1155	1170	1185
80	1200	1215	1230	1245	1260	1275	1290	1305	1320	1335
90	1350	1365	1380	1395	1410	1425	1440	1455	1470	1485
0	0	16	32	48	64	80	96	112	128	144	16
10	160	176	192	208	224	240	256	272	288	304
20	320	336	352	368	384	400	416	432	448	464
30	480	496	512	528	544	560	576	592	608	624
40	640	656	672	688	704	720	736	752	768	784
50	800	816	832	848	864	880	896	912	928	944
60	960	976	992	1008	1024	1040	1056	1072	1088	1104
70	1120	1136	1152	1168	1184	1200	1216	1232	1248	1264
80	1280	1296	1312	1328	1344	1360	1376	1392	1408	1424
90	1440	1456	1472	1488	1504	1520	1536	1552	1568	1584
0	0	17	34	51	68	85	102	119	136	153	17
10	170	187	204	221	238	255	272	289	306	323
20	340	357	374	391	408	425	442	459	476	493
30	510	527	544	561	578	595	612	629	646	663
40	680	697	714	731	748	765	782	799	816	833
50	850	867	884	901	918	935	952	969	986	1003
60	1020	1037	1054	1071	1088	1105	1122	1139	1156	1173
70	1190	1207	1224	1241	1258	1275	1292	1309	1326	1343
80	1360	1377	1394	1411	1428	1445	1462	1479	1496	1513
90	1530	1547	1564	1581	1598	1615	1632	1649	1666	1683
0	0	18	36	54	72	90	108	126	144	162	18
10	180	198	216	234	252	270	288	306	324	342
20	360	378	396	414	432	450	468	486	504	522
30	540	558	576	594	612	630	648	666	684	702
40	720	738	756	774	792	810	828	846	864	882
50	900	918	936	954	972	990	1008	1026	1044	1062
60	1080	1098	1116	1134	1152	1170	1188	1206	1224	1242
70	1260	1278	1296	1314	1332	1350	1368	1386	1404	1422
80	1440	1458	1476	1494	1512	1530	1548	1566	1584	1602
90	1620	1638	1656	1674	1692	1710	1728	1746	1764	1782
	0	1	2	3	4	5	6	7	8	9