Multiplication and Division by Numerical "Trituration"

[dom_quixote]

Multiplication and Division by Numerical Trituration

Image: AI Google

I - Multiplication by Numerical Trituration

First Example

157 x 3 =

a = 100 x 3 = 300

b = 50 x 3 = 150

c = 7 x 3 = 21

157 x 3 = a + b + c

157 x 3 = 300 + 150 + 21 = 471

Second Example

45 x 23 =

a = 40 x 20 = 800

b = 40 x 3 = 120

c = 5 x 20 = 100

d = 5 x 3 = 15

45 x 23 = a + b + c + d

45 x 23 = 1035

Third Example

38.9 x 7

a = 30 x 7 = 210

b = 8 x 7 = 56

c = 0.9 x 7 = 6.3

38.9 x 7 = a + b + c

38.9 x 7 = 210 + 56 + 6.3

38.9 x 7 = 272.3

II - Division by Numerical Trituration

First Example

471 : 3 =

a = 400 : 3 = 100 remainder 171

b = 171 : 3 = 50 remainder 21

c = 21 : 3 = 7 remainder zero

471 : 3 = a + b + c

471 : 3 = 100 + 50 + 7

471 : 3 = 157

Second Example

471 : 30 =

a = 400 : 30 = 10 remainder 100

b = 100 : 30 = 3 remainder 10

c = 10 : 30 = 1/3 remainder zero

d = 70 : 30 = 2 remainder 10

e = 10 : 30 = 1/3 remainder zero

f = 1 : 30 = 1/30 remainder zero

471 : 30 = a + b + c + d + e + f

471 : 30 = [10 + 3 + 2] + [1/3 + 1/3 + 1/30]

471 : 30 = 15 + [10/30 + 10/30 + 1/30]

471 : 30 = 15 + [21/30]

471 : 30 = 15 + 0.7

471 : 30 = 15.7

Third Example

272.3 : 7

First Step: multiply by ten

2723 : 70

a = 2000 : 70 = 20 remainder 600

b = 600 : 70 = 8 remainder 40

c = 40 : 70 = 4/7 remainder zero

d = 700 : 70 = 10 remainder zero

e = 20 : 70 = 2/7 remainder zero

f = 3 : 70 = 3/70 remainder zero

2723 : 70 = a + b + c + d + e + f

2723 : 70 = [20 + 8 + 10] + [4/7 + 2/7 + 3/70]

2723 : 70 = 38 + [40/70 + 20/70 + 3/70]

2723 : 70 = 38 + 63/70

2723 : 70 = 38.9

 

[Gavran]

Multiplication and Division by Numerical Trituration

I - Multiplication by Numerical Trituration

First Example

157 x 3 =

a = 100 x 3 = 300

b = 50 x 3 = 150

c = 7 x 3 = 21

157 x 3 = a + b + c

157 x 3 = 300 + 150 + 21 = 471

II - Division by Numerical Trituration

First Example

471 : 3 =

a = 400 : 3 = 100 remainder 171

b = 171 : 3 = 50 remainder 21

c = 21 : 3 = 7 remainder zero

471 : 3 = a + b + c

471 : 3 = 100 + 50 + 7

471 : 3 = 157

Can you provide any reference where these methods are called "multiplication and division by numerical trituration"?

 

[dom_quixote]

Unfortunately, no. It's a joke, derived from an insight. Some people use this trick (including me) to solve mental calculations without using paper. Note that the process is fun, as it employs some mathematical properties and operations used in high school.

 

[FactChecker]

Ha! For those of us with bad memories, this has been the way to remember that 9*7 = 10*7-7 = 63 since 3rd grade. In fact, I truly believe that my bad memory gave me a head start in math. ;-)

 

[dom_quixote]

Dear FactChecker,

Thank you for the encouragement.
Colleague Gavran rightly found the application of the irreverent term "numerical crunching" strange.
Certainly, "numerical decomposition" is more appropriate to be discussed in a math class.
Like you, I also suffered a lot with math. The division operation, for example, done in the traditional way, causes numerical congestion in a small area of paper.
What I like most about the decomposition model is that it makes it possible to give a personal tone to the reasoning, as in your case of 9*7 = 10 * 7 -7 = 63.
Another point I can highlight is the "traceability of an error", in order to facilitate student learning.

 

[dom_quixote]

Example of Tedious Division

80 ÷ 12

a = 80 ÷ 12 = 6, remainder 8

b = 8 ÷ 12 = 8/12 or 2/3, remainder zero

80 ÷ 12 = a + b

80 ÷ 12 = 6 + 2/3 or 6,666