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