In progress. Here’s one for you, however.
To multiply anything by 9, multiply by 10 first and then subtract the number. After all, a number x (10-1) is the number times 10 minus the number x 1, or itself. This is the distributive law in action!
It should follow, then, that multiplying by 99 is easy. Multiply by 100 then subtract the number, for you are multiplying by (100-1). So, 83 x 99 would be 8217: (83 x100) – (83 x 1)= 8300-83=8217.
Multiplying by 11 can be done without great difficulty. To multiply by 11, multiply by (10+1), which is multiplying by 10, then adding the number (which is multiplying by 1). So, 123 x 11=(123 x 10) + (123 x 1)=1230 + 123 = 1353.
Multiplying by 50 is the same as dividing by 2 and multiplying by 100, since 100/2 is 50. That means that 98 x 50=(1/2) x 98, or 49 x 100=4900. In the same fashion as above, we can multiply by 51 or 49. The neat thing about multiplying by 51, is that for an even number less than 100, it is half that number with the number appended. Say what? 64 x 51 is half of 64 (that would be 32) with the number (64) appended. So 64 x51=3264.
Multiplying by 25 is dividing by 4 and multiplying by 100:
87 x25 is 21 3/4 x 100. But that is 2175, since 3/4 x 100 =75.
Try left to right multiplication: 23 x 7
Write 23 as 20 +3. So 7 x 23 is 7 x 20, or 140, plus 7 x3, or 21. Add 140 and 21 to get 161. In my head, 43 x6=”6 times 40 is 24o; 6 times 3 is/are 18, and 240 plus 18 is/are 258. ”
354 x 6= “6 times 300 is 1800; 6 times 50 is 300, and 1800 plus 300 is 21oo. 6 times 4 is 24. Answer is 2124.
498 x 8= 3200 plus 720 plus 64 or 3984. BUT, isn’t that also 8 x (500-2) or 8 x 500 (4000) minus 8 x 2, or 16? Same answer.
This one is really cool: Square any number x, ending in 5. The answer is (x)*(x+1), with 25 appended.
So, 85 squared is 8*9 with 25 appended, or 7225.
295 squared is 29 *30, with 25 appended, or 87025.
Multiply two numbers ending or beginning with the same digit, like 1:
31 x 31= 30*30 (or 900) +(3+6) x 10 (or 60) +1=961.
96 x97= 90 *90 (or 8100) + (6+7) 90 (or 1170) +(6*7)=9312.
This one came to me in the car on my way to a clinic in Safford:
The product of two numbers whose sum is an even number constant is maximal when the numbers are the same. For every unit away from that maximum number, the sum decreases by the square of that unit:
So, two numbers whose sum is 100 have a maximal product of 50*50 or 2500 (this can be proven with simple calculus).
51*49=2499; 52*48=2496; 61*30=2500-11 squared or 2379.
It works in reverse: 84 x 78 is 81 squared minus 9 or 6561-9=6552.
