Two digit 9's shortcut

Mental Calculation, Mathematics, and Puzzles
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Life has got itself in the way of hobbies and it’s been a couple of years since I posted.

Working my way back up through Khan Academy with an objective of Calculus 5, Linear Alg, and Discrete math.

Grade 4 :slight_smile:

Two-digit multiplication…
Addition and subtraction 4 digits
simple division.

I’m a little out of practice so will be drilling my numbers for the next bit till they smooth out

on to the nines… I haven’t seen this shortcut around and there is almost certainly a general case
but for now, I like it.

79 * 89 = 7031

80 * 90 =	7200
-   170		 170
  +            1
	=============
	        7031
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I like this. Why the 1, tho? I can’t figure out how you figured out adding the 1. :stuck_out_tongue:

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I expect if you take a pencil and paper and doodle the answer will come.

29 * 39 = (29 + 1) * (39 + 1) - (29 + 1) - ( 39+1) + 1

             $$a^2*2ab+1$$

Looks an awful lot like a difference of squares calculation to me.
I suspect the answer is in here. :slight_smile:

Nope, it’s just the distributive property of math (damn I love GPT).

This is a neat mathematical trick that works because of the distributive property of multiplication over addition and subtraction. Here’s how it works:

  1. The expression 30 * 40 - 30 - 40 + 1 can be rewritten as (30 - 1)(40 + 1) - 30 - 40 + 1.
  2. This simplifies to (29)(41) - 30 - 40 + 1.
  3. Further simplifying, we get 29 * 41 - 69.
  4. Now, 29 * 41 is the same as 29 * 39 + 2 * 29, which simplifies to 29 * 39 + 58.
  5. Subtracting 69 from this gives us 29 * 39 - 11, which is 29 * 39.

So, 29 * 39 is indeed equal to 30 * 40 - 30 - 40 + 1. This trick works for any two numbers a and b where a = b - 1 and b = a + 1. The result is always a * b.

Let’s take the example of multiplying 17 and 23:

  1. The expression 18 * 24 - 18 - 24 + 1 can be rewritten as (18 - 1)(24 + 1) - 18 - 24 + 1.
  2. This simplifies to (17)(25) - 18 - 24 + 1.
  3. Further simplifying, we get 17 * 25 - 41.
  4. Now, 17 * 25 is the same as 17 * 23 + 2 * 17, which simplifies to 17 * 23 + 34.
  5. Subtracting 41 from this gives us 17 * 23 - 7, which is 17 * 23.

So, 17 * 23 is indeed equal to 18 * 24 - 18 - 24 + 1. This trick works for any two numbers a and b where a = b - 1 and b = a + 1. The result is always a * b.

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4

If we pretend that 79 × 89 is just 80 × 90, then we’ve over-counted by about 90 (by saying that 79 is like 80) and by another 80 (because we said that 89 is like 90).

Straightening the details with algebra: (80 – 1)(90 – 1) = 80 × 90 – 90 – 80 + 1, as claimed.

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image

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I expect you can apply this for +(-1) as easily making this a good trick for 2 digits ending in 1 but ones don’t usually confuse us the way 9’s do.

2 Likes
7

After all these examples, lol, thank y’all.

Here’s me having ChatGPT hold my hand through each property of multiplication and addition (just to ensure I’m properly disambiguated). :stuck_out_tongue:

1. (80 - 1)(90 -1) 
// Problem setup

2. = 80*90 + 80*(-1) + (-1)*90 + (-1)*(-1)
// Distributive Property

3. = 7200 + 80*(-1) + (-1)*90 + 1
// Multiplication (Identity for -1)

4. = 7200 - 80 - 90 + 1
// Simplification using Multiplication by -1

5. = 7200 - 170 + 1               
// Associative Property of Addition

6. = 7031                        
// Simplification

"""Breakdown 
with relevant properties 
of addition and multiplication 
noted for each step"""

1. (80 - 1)(90 - 1)                                     
// Problem setup: Initial expression.

2. = 80*90 + 80*(-1) + (-1)*90 + (-1)*(-1)              
"""Distributive Property: 
Multiplication can be distributed 
over addition or subtraction, 
defined as a(b + c) = ab + ac."""

3. = 7200 + 80*(-1) + (-1)*90 + 1                       
"""Multiplication (Identity for -1): 
Multiplying by -1 inverts the sign of the number, 
as per the definition a * (-1) = -a."""

4. = 7200 - 80 - 90 + 1                                 
"""Simplification using Multiplication by -1:
Applying the result of multiplication by -1 
to simplify expressions."""

5. = 7200 - 170 + 1                                     
"""Associative Property of Addition: 
The way numbers are grouped in an addition
does't affect the sum, 
represented as (a + b) + c = a + (b + c)."""

6. = 7031                                               
// Simplification