What kinds of multiplication speed-ups and shortcuts do you have?
1. Introduction
Multiplication is one of my favorite things, mainly because I like big numbers and the fact that you can get much bigger amounts of "scrap" at the top of the question itself. Through the years, I've found numerous shortcuts and speed-ups, some suited for beginners and others, a few, suited only for experts or those who have that capability. Learn all about some of these powerful speed-ups and shortcuts to multiplication I have. Some can cut the time you need to do a question by 70% and others can cut the time by 10%. This would be very useful for many time-critical situations when math is a must. I've explained the processes as best as I can and editted the page for errors that I may have found.
Also, I can actually multiply just about any pair of two-digit numbers within ten seconds and in my head. This even includes some pairs of 3-digit numbers! I'm so fast at it as my math skills are very high and that I know all these shortcuts and speed-ups. I'm sharing them with you. Most of which are easy to deal with and they're also very effective with the old-fashioned paper-and-pencil method [especially the copy trick]. The stats for each are based only on using the paper-and-pencil method. See the corresponding footnotes on how it may be with mental computation.
2. The zero trick:
type: speed-up
reliability: 
- no limitations
understandability: 
- very high
time saving: 
- small gain
usefulness: - absolutely a must
difficulty: - nearly absent
required skill: - beginner
overall: 44 of 48 points,
This interesting trick is not recognized by many, however, it applies such basic concepts. Also, you could learn to avoid "zero confusion" as well. Let's take this math question as an example:
180
265
---
First off, you could work it out normally:
4
4
180
265
-----
1
900
10800
36000
-----
47700
You could also eliminate the zero and later add in the number of zeros dropped:
54
265
18 1
----
2120
2650
----
47700
Remember, you can work it in any order due to the commutative property of multiplication. Note how you don't have "zero confusion". If you had a bigger one like:
1
1
4800000
220000
-------------
11
96000000000
960000000000
-------------
1056000000000
Sometimes it's rather pointless to write all those zeros. This example shows you the best method:
1
1
48 5
22 4
----
11
96
960
----
1056000000000
Then from there add the number of zeros to get your 1,056,000,000,000 answer. To find the number of zeros, simply count them then jot the number to the side. If you see zeros after the number and there's no other nonzero digits, count the leading zeros [they are also known as place holders]. It also works with decimals, rather, count the digits and subtract instead of add. If you get a negative 4 and nothing else, and that your final answer was 13631, you'd place the dot as follows: 1.3631. Since there were 9 zeros [5 from the first number and 4 from the second], you'd just jot down the 5and the 4 to the side and circle it and add the 9 zeros at the end of doing it the simpler way. This trick is often useful, however, you can actually create more zeros than counted!
3. The five trick:
type: speed-up and shortcut
reliability: 
- medium-high
understandability: - medium-high
time saving: - very small gain*
usefulness: - high
difficulty: 
- medium
required skill: - advanced
overall: 29 of 48 points,
Let's refer to the question first mentioned:
4
4
180
265
-----
1
900
10800
36000
-----
47700
Here, rather than having to add 1 zero at the end, you can make it 2! This trick only works for little numbers as I've given. You could multiply the 265 by 2 to get 530 then divide 180 by two to get 90. Now you have two zeros to add later rather than one and 53×9 should be a breeze:
2
53 1
9 1
---
47700
And you got the same exact answer too! 53×9 is so much easier than 265×18! This is often much quicker, by even 50% faster, only when you're very good at multiplying and dividing by two! Yet, even with this, you can still speed up multplication! Unfortunately, this trick only works when the other number has a "nonfive" digit that is even. If it is odd, it doesn't work.
4. The copy trick:
type: super shortcut and speed-up
reliability: - has no limitations
understandability: - maximum
time saving: - very big gain
usefulness: - absolutely a must
difficulty: - very easy
required skill: - beginner
overall: 46 of 48 points,
The copy trick is extremely effective if the digits are the same in the bottom number. It's the most powerful mathematical shortcut I have, hence the awesome ratings! Let's consider:
214
313
---
Well, you'd probably work it out in a much longer way:
21
21
376
313
------
1
1128
3760
112800
------
117688
If I were to do this question on paper, I could do it twice as fast as it would to work it out otherwise. Here's the hint: all you need to do multiply once and copy and add. That's right copy! Since you already did 376×3, you only need to copy your previous answer. Yet, if there's a one, simply copy the top number! Doing just that, all you really need to do is multiply one number, and add. If you try 111×111, it should be finished in the matter of even ten seconds! If you do 532×777, you only need to do the first seven then copy throughout the last of the multiplication then add. However, what if you had:
744
627
---
If you remember right, you can flip them around! Turn that into:
627
744
---
and multiply just the 4 and the 7! That saves a lot of time. However, even with these three on your agenda, you could still speed up multiplication!
5. The transfer trick:
type: speed-up
reliability: 
- very low***
understandability: - medium
time saving: 
- indeterminate**
usefulness: - very high
difficulty: - very hard
required skill: 
- expert
overall: 20 of 40 points,
The transfer trick is not recommended for beginners and those who lack such mental math skills, but it can help with creating something that would allow you to use the other tricks listed above. Let's consider this question:
345
154
---
Doesn't seem like you can do much, but have you thought of 154 divided by 2 and transferring it? Doing so, you can turn that into:
69
77 1
--
You can transfer a 2 from 154 to 345 and create a zero and yet, you're getting doublets and one fewer digit to deal with! Which do you think is easier:
22
12
345
154
-----
111
1380
17250
34500
-----
53130
OR
6
69
77 1
----
11
483
4830
----
53130
In the top one, you're more vulnerable to flaws and errors from multiplying than you are in the bottom one. Yet, you even got the same answer! However, only use this trick if you're very good at multiplying and dividing by 2's and 3's and can spot uses for it. Well, by now, you would probably be thinking that there's another way to speed up multiplication. Well, there is still another one.
6. The baby-digit trick:
type: speed-up
reliability: 
- too many limitations
understandability: - medium
time saving: - very small gain
usefulness: - very high
difficulty: - indeterminate†
required skill: - indeterminate†
overall: 16 of 32 points,
This one involves simple, baby digits [a baby digit is a 0, a 1, or a 2]. Try to find ways to covert the multiplication quickly to something really easy to do. Consider this question:
122
444
---
In this case, you can covert the 4 4's into 4 1's. Since multiplying by 1 and 2 with 4 is simple enough you can convert it into:
488
111
---
... and it would really speed it up. Yeah, you can apply the transfer trick to doing so, but you're creating simple, easy-to-use 1's. Just copy the first number and add:
488
111
-----
121
488
4880
48800
-----
54168
That is so much quicker! Yet it's much easier as you just need to copy. Note: This speed-up, in general, is not useful for beginners and those who are advanced in arithmatic. Use this speed-up in cases in which something is readily available.
7. The zero-in-the-center trick:
type: speed-up and shortcut
reliability: - no limitations
understandability: - maximum
time saving: - big gain
usefulness: - very high
difficulty: - easy
required skill: - beginner
overall: 43 of 48 points,
If you have zeros, they also come in handy for yet, a sixth speed-up trick:
608
695
---
Here, you can apply the five trick as well. However, for numbers with a zero in them, you should put those at the bottom. Also, apply the five trick:
12
13
139 1
304
-----
1
556
41700
-----
422560
Rather, zeros are quick. Just skip them, write in a zero, and go to the next digit. Try to find ways you can introduce a zero if it's readily available. Try to introduce 1's and 0's if possible and if 2's are easy enough to you, add those.
8. The same-number trick
type: mental math shortcut
reliability: - medium-high
understandability: - low
time saving: - little change‡
usefulness: - medium-high
difficulty: - medium
required skill: - intermediate
overall: 25 of 48 points,
This trick is very useful when dealing with 2-digit numbers being multiplied and that they are identical, best effective with doing it in your head. Consider this question:
73
73
--
You could work it out normally, and since none of the above tricks can be used at all, you would seem out of luck:
2
73
73
----
219
5110
----
5329
There is a trick to actually speeding this up. For those doing mental math, this trick would be mighty useful. Consider the pattern with square numbers, or numbers that are a result of multiplying the same two numbers. In order counting from 1 to 20 in square numbers, you have 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, then 400. Note how the ones have the 1-4-9-6-5-6-9-4-1-... pattern involved. Also, note the tens place. After you see the 9, the increments increase by 1. Consider something you might know, 12×12. To apply this trick, round your numbers to the nearest ten. 12 rounds to 10. Square 10 [or multiply it by itself] to get 100. Note that after every 10 steps in the square number count, the tens count by 2 more than the previous. Since 1 is in the tens place, 10×2 is 20. For every extra step, you add 20. Since 12 is 2 higher than 10, you add 20×2 or 40 giving you 140. Then, the final step is to add the square of the difference in the ones from your rounded answer. In the question given with 73, you'd round it to 70 and square that to get 4900. Since 70×2 is 140 and that you're doing 3 extra steps, 14×3 is 42 and knowing the zero trick, you'd get 420 as you're replacing lost zeros. 4900+420 is 5320, very close to the answer. The final step is to take the square of the ones and add it. 3×3 is 9 and 5320+9 is 5329, the answer you got! For questions where you round up instead, instead of adding the difference, you subtract instead. Consider 56×56. You could round down and consider 50 then adding 600 then 36, or you can round up and consider 60. Either way works. Let's consider the rounding up method. 60×60 is 3600. Since you're 4 steps before you subtract 120×4, or 480 to get 3120. Since you're 4 off with the ones, you add 16 which gives you the result 3136 and you'd get the same answer if you rounded down instead. Note that if you multiply 2 negative numbers, you'll always get a positive, which is why you always add when dealing with the ones.
Note: This doesn't work well with 3 digit numbers, although it still does work. Anything above 125 should be worked out normally unless another trick can be applied.
9. The split-same-number trick
type: mental math shortcut
reliability: - medium-high
understandability: - low
time saving: - little change‡
usefulness: - medium-high
difficulty: - medium
required skill: - intermediate
overall: 24 of 48 points,
The eighth trick to add to your collection is called "the split same number trick". The concept is exactly like that of the previous trick, except with one extra step. This trick will only work if the difference between the numbers being multiplied is always an even number. Consider this question:
31
39
--
The number that is inbetween these two is 35. Apply the "same number trick" with 35, but take the difference of one of the numbers from 35 and square it. Subtract it from the original. You know that 20×20 is 400, but 19×21 is 400-1, or 399, 18×22 is 400-4, or 396. Note the pattern? It's also the same with something you should've known since 3rd grade: 5×5=25, 6×4=25-1=24, 7×3=25-4=21, 8×2=25-9=16, and furthest out of the multiplication table would be 9×1=25-16=9.
Footnotes:
* This has a small gain, but it's also based almost entirely on your skill. An expert would get about 8 stars, but a newbie will have 1 or 2 stars.
** This speed-up requires great skill. Those who use it wisely can get 8 or 9 stars, but, even someone who is intermediate on math would get only 2 stars! This is indeterminate for this reason.
*** This is of almost no use from beginners to those who are advanced in general arithmatic, however, masters would find this of high use, about 7 stars. The limitation is your skill and it's very strict, thus reliability gets a very low rating.
† These are indeterminate as it's dependant on the transfer trick, something very hard to use. This same thing applies to the baby-number trick. Also, because 2 items have been grayed out, the overall value is practically indeterminate.
‡ For mental processing, this can be 8 stars as it's far easier to use this process for mental math instead of just doing it on paper. This also applies for the split same number trick as well.
