Multiplication of Consecutive Numbers

Multiplication of Consecutive Numbers

Dr Himanshu Shekhar

Kids! This time, I am exploring another aspect of Multiplication through examples, where method to find out product of consecutive integers will be illustrated.

In fact, in one of the post entitled “Multiplication of Numbers with Complimentary Digits”, such requirements are spelt out, which is reproduced below:

122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

145x145 = 14x15 | 5x5 = 21025

The left partition invariably require, multiplication of two consecutive numbers. Rather than partitioning, as discussed in previous posts, the current post explores addition and some other properties based on last digits of the products. For ease of calculation, explanation using product of 2-digit consecutive numbers is discussed in the post, which can be extended further by practice.

The product of consecutive numbers, with unit place digit of one of the numbers as 0 or 5 can be obtained by direct multiplication, easily, as given below, with slight modification (extraction of 2&5 to give 2x5 = 10 from numbers).

89x90 = 8010; 
90x91 = 8190; 
94x95 = 47x19x10 = 8930; 
95x96 = 10x19x48 = 9120;

79x80 = 6320; 
80x81 = 6480; 
84x85 = 42x17x10 = 7140; 
85x86 = 10x17x43 = 7310;

The product of consecutive integers, where unit place digits of smaller number are 1 and 2, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

91x92 = 90x93+2 = 8370+2 = 8372;

81x82 = 80x83+2 = 6640+2 = 6642;

71x72 = 70x73+2 = 5110+2 = 5112;

61x62 = 60x63+2 = 3780+2 = 3782;

Examples with 2 less than smaller and 2 more than larger number is taken:

92x93 = 90x95+6 = 8550+6 = 8556;

82x83 = 80x85+6 = 6800+6 = 6806;

72x73 = 70x75+6 = 5250+6 = 5256;

62x63 = 60x65+6 = 3900+6 = 3906;

The product of consecutive integers, where unit place digits of the smaller number are 6 and 7, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of 5 less than each unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

96x97 = 95x98+1x2 = 10x19x49+2 = 9310+2 = 9312;

86x87 = 85x88+1x2 = 10x17x44+2 = 7480+2 = 7482;

76x77 = 75x78+1x2 = 10x15x39+2 = 5850+2 = 5852;

66x67 = 65x68+1x2 = 10x13x34+2 = 4420+2 = 4422;

Examples with 2 less than smaller and 2 more than larger number is taken:

97x98 = 95x100+6 = 9500+6 = 9506;

87x88 = 85x90+6 = 7650+6 = 7656;

77x78 = 75x80+6 = 6000+6 = 6006;

67x68 = 65x70+6 = 4550+6 = 4556;

Now two type of consecutive number multiplication is left, where unit place digit of larger number is 4 or 9.

Main part of addition will be product of one less than smaller and one more than larger number and 2 is added to it. Examples are given below.

93x94 = 92x95+2 = 10x46x19+2 = 8740+2 = 8742;

98x99 = 97x100+2 = 9700+2 = 9702;

83x84 = 82x85+2 = 10x41x17+2 = 6970+2 = 6972;

88x89 = 87x90+2 = 7830+2 = 7832;

73x74 = 72x75+2 = 10x36x15+2 = 5400+2 = 5402;

78x79 = 77x80+2 = 6160+2 = 6162;

Now any set of multiplication of consecutive numbers can be carried out easily. It is just practice, which makes this method perfect. It is mental mathematics, which needs quick action by mind to get product of consecutive numbers.

Kids! Explore this method and compare with known conventional method. Enjoy.

Dr Himanshu Shekhar

Multiplication with Reference: Part 4

Multiplication with Reference: Part 4

Dr Himanshu Shekhar

Kids! Multiplication is such a beautiful mathematical operation that it can be explored in several different ways.

The previous few posts are exploring multiplication of numbers using a reference, which are invariably some power of 10. However, nearness to a power of 10 is one pre-requisite of such calculation. In the current post, reference number is taken as some multiple of power of 10, thus expanding the domain of this mental mathematical technique.

In this post reference can be 20 (2x10), 50 (5x10 or ½ x 100), 60 (6x10), 300 (3x100), 700 (7x100) and so on. The product calculation is again carried out under two heads – left partition and right partition.

Ø  Left partition is decided by multiplication of (i) multiplier of power of 10 and (ii) sum of multipliers minus reference.

Ø  Right partition has number of digits as number of zeros in the base of reference and is given by product of differences of multipliers from reference.

In nutshell, only difference will come in the left partition, where it will be multiplied by a number given by multiple of power of 10. Let us solve examples.

54x56 = 5x6|4x6 = 30|24 = 3024. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

54x56 = 5x(54+56-50)|4x6 = 5x60|24 = 300|24 = 300+2|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only).

54x56 = ½ x (54+56-50)|4x6 = ½ x60|24 = 30|24 = 3024. (Reference is ½ x100:, Multiplier ½ : Right partition has 2-digit only).

48x63 = 4x(48+63-40)|8x23 = 4x71|184 = 284|184 = 284+18|4 = 302|4 = 3024. (Reference is 4x10:, Multiplier 4: Right partition has 1-digit only).

48x63 = 5x(48+63-50)|-2x13 = 5x61|-26 = 305|-26 = 305|4-30 = 305|34 = 305-3|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only: -26 = 34).

48x63 = ½ x(48+63-50)|-2x13 = ½ x61|-26 = 30.5|-26 = 30|50-26 = 30|24 = 3024.  (Reference is ½ x100:, Multiplier ½: Right partition has 2-digit only: 0.5 on left partition = 0.5x100 = 50 on right partition).

48x63 = 6x(48+63-60)|-12x3 = 6x51|-36 = 306|-36 = 306|4-40 = 306|44 = 306-4|4 = 302|4 = 3024. (Reference is 6x10:, Multiplier 6: Right partition has 1-digit only).

312x318 = 31x32|2x8 = 992|16 = 99216. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

312x318 = 3x(312+318-300)|12x18 = 3x330|216 = 990+2|16 = 992|16 = 99216. (Reference is 3 x 100:, Multiplier 3 : Right partition has 2-digit only).

798x805 = 8x(798+805-800)|-2x5 = 8x803|-10 = 6424|90-100 = 6424|190 = 6424-1|90 = 6423|90 = 642390. (Reference is 8 x 100: Multiplier 8 : Right partition has 2-digit only: -10 = 190).

798x805 = 4/5 x(798+805-800)|-2x5 = 4/5 x803|-10 = 3212/5 |-10 = 642.4|-10 = 642|400-10 = 642|390 = 642390. (Reference is 4/5 x 1000:, Multiplier 4/5 : Right partition has 3-digit only: 0.4 on left partition = 0.4x1000 = 400 on right partition).

237x245 = ¼ x (237+245-250)|-13x-5 = ¼ x232|065 = 58|065 = 58065. (Reference is ¼ x 1000:, Multiplier ¼ : Right partition has 3-digit only).

758x753 = ¾ x (758+753-750)|8x3 = ¾ x 761 | 024 = 2283/4 | 024 = 570.75 | 024 = 570| 750+024 = 570|774 = 570774. (Reference is ¾  x 1000:, Multiplier ¾ : Right partition has 3-digit only: 0.75 on left partition is 750 on right partition).

Kids can explore themselves with the help of their guardians and teachers to understand the method. Practice makes everything perfect. So, practice and master the technique of multiplication involving references.

Each example can be solved by a number of method and reference. The end result remains same but in between mathematical calculation varies little bit. This mental mathematical techniques, can be helpful in cracking all competitive examinations and also for cracking online examinations, involving numerical aptitude.

Dr Himanshu Shekhar

Multiplication with Reference: Part 3

Multiplication with Reference: Part 3

Dr Himanshu Shekhar

Kids! Multiplication with reference is explained in previous 2-posts (Multiplication with a Reference & Multiplication using Reference: Part 2) using 100 as reference. In fact, any power of 10 can be used as reference for quick mental multiplication, provided, both numbers are in the vicinity of the reference (however not mandatory, as explained with some examples below). 

In the current post, mental multiplication, explained in previous few posts, have been extended further to numbers, which are near to other powers of 10. As usual, the product is partitioned into two parts – left part and right part.

• The left part of the product is given by sum of both the numbers minus reference. It is written in previous 2-posts as subtracting cross-difference from any of the multiplier. 
• The right part of the product given by product of difference of multipliers from reference, taking proper sign (+ or -).
• The power of 10, used as reference indicates number of digits used for right part of the product. Number of digits for right part is equal to power of 10, used as reference.
• If right part is more than number of allocated digits, the provision of carry over to left part is there.
• If right part is less than number of allocated digits, leading zeros can be added.
• The numbers can be written as partial negative digits (shown by using underlined digits), as explained in previous post (Multiplication with Reference: Part 2).

12x17 = 12+17-10|2x7 = 19|14 = 19+1|4 = 204. (Reference 10; 1 digit for right partition)

12x17 = 12+7|2x7 = 19|14 = 19+1|4 = 20|4 + 204. (Technique used in previous 2-posts for left part: 1-digit for right partition)

12x17 = 17+2|2x7 = 19|14 = 19+1|4 = 20|4 + 204. (Technique used in previous 2-posts for left part: 1-digit for right partition)

14x16 = 14+16-10|4x6 = 20|24 = 20+2|4 = 22|4 = 224.

14x16 = 1x2|4x6 = 2|24 = 224 (Mental mathematics explained in post “Multiplication of Numbers with Compliment Digits)

8x6 = 8+6-10|-2x-4 = 4|8 = 48.

13x7 = 13+7-10|3x-3 = 10|-9 = 10|11 = 10-1|1 = 91. (Reference 10; 1 digit for right partition: -9 = -10+1 = 11)

1012x1004 = 1012+1004-1000|12x4 = 1016|048 = 1016048. (Reference 1000: 3-digits right partition)

998x987 = 998+987-1000|-2x-13 = 985|026 = 985026.

1016x992 = 1016+992-1000|16x-8 = 1008|-128 = 1007|1000-128 = 1007|872 = 1007872. (Reference 1000: 3-digits right partition: Negative carryover for right partition)

998x92 = 998+92-1000|-2x-908 = 90|1816 = 90+1|816 = 91|816 = 91816. (Reference 1000: 3-digits right partition: Carry over to left partition: One number too small from reference)

1013x111 = 1013+111-1000|13x-889 = 124|-11557 = 124-11|-557 = 113|-557 = 113-1| 1000-557 = 112|443 = 112443. (Reference 1000: 3 digits for right, Carry over jugglery to be understood).

1013x111 = 1013+111-1000|13x-889 = 124|-11557 = 124|12000-443 = 124-12|443 = 112|443 = 112443. (Reference 1000: 3 digits for right, Carry over jugglery to be understood).

991x87 = 991+87-1000| -9x-913 = 78|8217 = 78+8|217 = 86|217 = 86217.

9999x9983 = 9999+9983-10000|-1x-17 = 9982|0017 = 99820017. (Reference 10000: 4-digits right partition)

999999998x1000000018 = 999999998+1000000018-1000000000|-2x18 = 1000000016|-36 = 1000000016-1|1000000000-36 = 1000000015|999999964 = 1000000015999999964. (Reference 1000000000: 9-digits right partition: Negative carry over to right partition)

As, I believe in illustrating through examples, less English is written. However, the key points are indicated and training mind to solve multiplications is an art, to be mastered by individuals. Hope that this extension of multiplication with reference is liked by kids. I will post, multiplication with reference as multiples of powers of 10, as next post. Enjoy. Do solve and explore.

Dr Himanshu Shekhar

Multiplication using Reference: Part 2

Multiplication Using reference: Part 2

Kids! This section is added after getting motivation from the Daughter of Shri Dayanand Kumar, a senior scientist is DRDO. This part is an extension to previous post and many similar additional posts will come, as the topic has many diversions.

This post discusses mental multiplication using reference 100. In previous post, either both the multipliers were less than 100 or both were more than 100. In this section, those multiplications will be explained, where, one number is less than reference and other is more than reference.

For 108x98, differences from reference (100) are +8 and -2, respectively. The product has two parts. The left side digit will be 108-2 = 98+8 = 106. The right partition has two digits, given by product of these differences, +8x-2 = -16. This can be written as -16 = 84 -100 = -100+84 = 184. (underline denotes negative term). Now it has three terms, but only 2-digits are specified for right side. So, -1 is carried over to left side and left side digit becomes 106-1 = 105. So, 108x98 = 10584. Some more examples are solved.

108x98 = 108-2|+8x-2 = 106|-16 = 106|84-100 = 106|184 = 106-1|84 = 105|84 = 10584

108x98 = 98+8|+8x-2 = 106|-16 = 106|84-100 = 106|184 = 106-1|84 = 105|84 = 10584

112x97 = 112-3|+12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

112x97 = 97+12|+12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

107x88 = 107-12|+7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

107x88 = 88+7|+7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

112x88 = 112-12|+12x-12 = 100|-144 = 100|256 + 100-2|56 = 9856

112x88 = 88+12|+12x-12 = 100|-144 = 100|256 = 100-2|56 = 9856

119x83 = 119-17|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

119x83 = 83+19|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

This post is mainly to introduce the concept of negative part in a number in mental mathematics. The concept, if understood can be used with ease for solving the above multiplications in slightly different way also.

108x98 = 108x102 = 108-2|8x-2 = 106|-16 = 106|184 = 106-1|84 = 10584

112x97 = 112 x 103 = 112-3|12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

107x88 = 107x112 = 107-12|7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

112x88 = 112x112 = 112-12|+12x-12 = 100|-144 = 100|256 = 100-2|56 = 9856

119x83 = 119x117 = 119-17|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

67x 114 = 133x114 = 114-33|14x-33 = 81|-462 = 81|538 = 81-5|38 = 7638

More of similar examples can be solved and power of mental mathematics can be felt. Hope that introduction of negative part in a number is understood by kids. It needs practice as concept is entirely different from conventional approach. Enjoy it.

Dr Himanshu Shekhar

Multiplication with a Reference

Multiplication with a Reference

Kids! This post is to explain another mental mathematics technique, which can be mastered with practice.

As described in previous post of finding square of any 2-digit number, using a reference, this section is also utilizing a reference number to find product. Being decimal system of numbers, again powers of 10 (like 10, 100, 1000, etc) are taken as reference. For two digit numbers, 100 can be taken as reference. Since, reference added multiplication is explained, nearness to reference is one of the pre-requisites. Again, rather than writing in English, example is solved for explanation and illustration.

To find, 112x108, both the numbers are near 100, which is taken as reference. The product is partitioned into two parts. Right part has two allocated space and is given by product of difference of numbers from reference. First number is more by (112-100=) 12 and second is less by (108-100=) 8. So, last two digits (left two digits) of the product is 12x8=96. The right side of the product is given by cross-addition of difference. Right side is 112+8 = 108+12 = 120. So, product is 12096.

112x108 = 112+8|12x8 = 120|96 = 12096

Some more examples are shown, below.

116x104 = 116+4|16x4 = 120|64 = 12064

116x104 = 104+16|16x4 = 120|64 = 12064

108x108 = 108+8|8x8 = 116|64 = 11664

107x111 = 107+11|7x11 = 118|77 = 11877

107x111 = 111+7|7x11 = 118|77 = 11877

If both numbers are less than reference then also, the method is applicable. To find, 93x98, both numbers are near 100, which is taken as reference. The product is again partitioned into two parts. Right part has two allocated space and is given by product of difference of numbers from reference. First number is less by (93-100=) -7 and second is less by (98-100=) -2. So last two digits will be -2x-7= 14. The right side of the product is given by cross-addition of difference. Right side is 93-1 = 98-7 = 91. So, product is 9114.

93x98 = 93-2|-7x-2 = 91|14 = 9114.

93x98 = 98-7|-7x-2 = 91|14 = 9114.

99x89 = 99-11|-1x-11 = 88|11 = 8811

99x89 = 89-1|-1x-11 = 88|11 = 8811

95x94 = 95-6|-5x-6 = 89|30 = 8930

95x94 = 94-5|-5x-6 = 89|30 = 8930

88x97 = 88-3|-12x-3 = 85|36 = 8536

88x97 = 97-12|-12x-3 = 85|36 = 8536

77x96 = 77-4|-23x-4 = 73|92 = 7392

77x96 = 96-23|-23x-4 = 73|92 = 7392

The process is extended little further, where left side digit is more than 100 and two spaces are not sufficient to accommodate them. In that case, carry over provisions exits and hundreds place digits can be carried over to other side of partition. Some examples are illustrated below.

127x112 = 127+12|27x12 = 139|324 = 139+3|24 = 142|24 = 14224

133x116 = 133+16|33x16 = 149|528 = 149+5|28 = 154|28 = 15428

118x114 = 118+14|18x14 = 132|252 = 132+2|52 = 134|52 = 13452

119x111 = 119+11|19x11 = 130|209 = 130+2|09 = 132|09 = 13209

75x88 = 75-12|-25x-12 = 63|300 = 63+3|00 = 66|00 = 6600

84x88 = 84-12|-16x-12 = 72|192 = 72+1|92 = 73|92 = 7392

82x88 = 82-12|-18x-12 = 70|216 = 70+2|16 = 72|16 = 7216

The last multiplication can be solved by method explained in earlier post (Multiplication of Numbers with Compliment Digits) because unit place digits of both the numbers add to 10. For recapitulation of the method, explained in earlier post.

82x88 = 8x9|2x8 = 72|16 = 7216

The post explains a metal mathematics method for those multiplications, where numbers are near 100. It can be practiced and more extension of the method will be posted in subsequent posts, to extend the method to different references and different combinations of multiplicands.

These all are parts of metal mathematics, which can be used to crack competitive examinations. Please do post, your comments to improve the content.

Dr Himanshu Shekhar

Complimentary Multiplication : Extension 1

Complimentary Multiplication : Extension-1

It is observed that in the previous post, the calculation for two-digit numbers are understood by many kids. However, with higher number of digits, some more clarity is needed. This post is added to have more clarity for specially Type II Corollary of previous post.

If we have more than two digits, then decimal complimentation can be observed in several ways.

Unit place digits are complimentary, like 1239 and 1231, 437 and 433, 254 and 256, ..

Last two digits are complimentary, like 1239 and 1261, 437 and 463, 254 and 246, …

Last three digits are complimentary, like 1239 and 1761, 4998 and 4002, …

This trend can continue and pair of numbers having similar characteristics can be explored, obtained and considered. It must be noted that in the pair of numbers, leaving complimentary digits, other digits are the same.

The method resembles that explained in the previous post. Product will contain two parts. Left part will be same part of the two numbers (to be multiplied) and one more than the same part. Right part is product of complimentary digits and number of digits in right part is twice the number of digits in the complimentation. The multiplication by the method explained in previous post is carried out below.

1239x1231 = 123x124 | 9x1 = 15252 | 09 = 1525209

437x433 = 43x44 | 7x3 = 1892 | 21 = 189221

254x256 = 25x26 | 4x6 = 650 | 24 = 65024

1239x1261 = 12x13 | 39x61 = 156 | 2379 = 1562379

437x463 = 4x5 | 37x63 = 20 | 2331 = 202331

254x246 = 2x3 | 54x46 = 6 | 2564 = 62564

1239x1761 = 1x2 | 239x761 = 2 | 181879 = 2181879

4998x4002 = 4x5 | 998x002 = 20 | 001996 = 20001996

This post is initiated and created at the behest of Diptasha Das, a great all round achiever in academics and sports.

Any kid can make comments for clarity and share it with their friends for getting more feedback.

Dr Himanshu Shekhar

Multiplication of Numbers with Compliment Digits

Multiplication of Numbers with Compliment Digits

Kids! This post is for finding multiplication of numbers, fulfilling certain criteria.

In mathematics, Ancient India is credited with invention of Zero and that led to development of decimal system. This is because there are only 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to constitute entire mathematics. The topic in this post discusses simple method for multiplication of two-digit numbers, whose unit place digits are 10-compliment of each other and other digit is the same. This means that the last (Unit-place) digits of the multipliers add to 10. The steps are as follows:

• Partition the product into two parts.
• Left part is given by product of tens-place digit and one more than the 10s-place digits
• Right side is given by product of unit place digit and two places are allocated for this.
• If right side partition exceeds 100, carry over to left side is permitted.

Since it is mathematics and not literature, without much writing, some examples are solved directly for those multiplications, where unit place digits of multipliers are 10s compliment and other digits are the same.

Suppose 92x98 is to be obtained. Both numbers have 9 at tens place and sum of unit place digit is (2+8 =) 10. Left side of the product is 9x10 = 90 and right side of the product is 2x8 = 16. So, 92x98 = 9016.

92x98 = 9x10 | 2x8 = 90 | 16 = 9016

83x87 = 8x9 | 3x7 = 72 | 21 = 7221

64x66 = 6x7 | 4x6 = 42 | 24 = 4224

71x79 = 7x8 | 1x9 = 56 | 09 = 5609

52x58 = 5x6 | 2x8 = 30 | 16 = 3016

More examples, like 91x99 (= 9009), 93x97 (= 9021), 94x96 (= 9024), 81x89 (= 7209), 82x88 (= 7216), 84x86 (7224), 72x78 (= 5616), 73x77 (= 5621), 74x76 (= 5624), 61x69 (= 4209), 62x68 (= 4216), 63x67 (= 4221), 51x59 (= 3009), …. can be tried by kids and the power of quick mathematics can be explored.

This type of mathematical ease has many types of corollaries.

TYPE I: The process can be extended to three digit numbers also. 113x117 = 11x12 | 3x7 = 132 | 21 = 13221. 122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

TYPE II: If the last two digits of three digit number add to 100, the same process is repeated with last four digit earmarked for right hand side. 191x109 = 1x2 | 91x9 = 2 | 0819 = 20819, 198x102 = 1x2 | 98x02 = 2 | 0196 = 20196. 189x111 = 1x2 | 89x11 = 2 | 0979 = 20979. It can be extended to 4-digit numbers too. 1191x1109 = 11x12 | 91x9 = 132 | 0819 = 1320819, 1289x1211 = 12x13 | 89x11 = 156 | 0979 = 1560979.

Type III: The same technique can be used for finding square of a number with 5 at unit place. 25x25 = 2x3 | 5x5 = 6 | 25 = 625, 115x115 = 11x12 | 5x5 = 132 | 25 = 13225, 145x145 = 14x15 | 5x5 = 21025.

TYPE IV: The process can be extended to any number of digits, but the criteria of compliment of power of 10 is to be satisfied. This means, last few digits must add to 10, 100, 1000, 10000, 100000, and so on. 12399 x 12301 = 123x124 | 99x01 = 15252 | 0099 = 152520099, 129999x120001 = 12x13 | 9999x0001 = 132 | 00009999 = 13200009999.

Kids! Explore this and intimate concern, if any.

Dr Himanshu Shekhar