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