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