Mathematical Tricks and Shortcuts
Created | Updated Jan 20, 2003
Mention math to many people and some will inevitably shudder in terror.
All those numbers! (An infinite supply!)
All those operators! (How they seem to multiply!)
However, there is a certain beauty in the way all those numbers work together, and how certain shortcuts can lead one to the right answer for a problem in a very short (or at least shorter) period of time. Now, some of the tricks I will mention here you may never use at all, while some of them can be quite useful. In any case, one can at least appreciate the wonderful way in which certain apparently difficult problems break down into amazingly simple operations; operations that many people can do in their heads.
And now, on with the show!
Some multiplication shortcuts:
Multiplying by 11
11*735=?
1. Write down the ones digit (___5).
2. Now, add the two right-most digits (3+5), and write down that answer (8) along with the ones digit (__85).
3. Continue adding two digits together, moving to the right (3+7). Since 3+7 equals 10, we will have to carry. Write down the zero with the previous two numbers and carry the 1. Now we have (_085) and a one to carry.
4. Continue the two-digit adding, substituting a zero now that we are at the front of the number (7+0). Add the one we carried (giving us 8) and place at the front of the number (8085). Thus we have 11*735=8085.
(This principles of this shortcut can be used on any size number multiplied by 11, just continue the two digit addition from right to left.)
Multiplying by 25
25*36
1. Divide the number being multiplied by 4. If it comes out evenly, (which in this case it does: 36/4=9), simply multiply that answer by 100. 24*36=900
2. If it doesn't come out evenly, (say the problem was 25*37: 37/4=9.25), then write down the number of times 4 goes in fully (9), and find the remainder (1 in this case). A remainder of one tells you to put a 25 after the quotient (9), giving us 24*37=925
(Should the remainder have been a two, you would put a 50 in, and a 3 would mean a 75 should be put in.)
Multiplying by 50:
50*23
This is very much a derivative of the previous shortcut (or the previous shortcut is very much a derivative of this one, whichever way you prefer).
1. Divide the number being multiplied by 2 (23/2=11.5)
2. Multiply the quotient (11.5) by 100 (11.5*100=1150).
This is your answer. (50*23=1150)
Multiplying Two Numbers Ending in Five:
85*35
1. Add the tens digits together (8+3=11). If the sum is odd, write down a 75. If even, write down a 25.
2. Divide the tens digit sum by 2 (11/2=5.5). Remember the whole number (5).
3. Multiply the tens digits (8*3=24) and add the whole number from step 2 (24+5=29).
4. Place step 3's answer in front of the 25 or 75 and you have the answer. (85*35=2975)
Squaring Numbers Close To (But Less Than) 1000
993*993=993^2
1. Subtract the number from 1000 (1000-993=7)
2. Subtract step one's answer (7) from the number being squared (993-7=986)
3. Write step 2 down. Square the difference in step one (7^2=49) and make it a three-digit number by adding the appropriate number of zeroes in front of it (049)
4. Take the number figured in step 3 and place it behind what you wrote down already. Thus 993^2=986049
To be continued...
All those numbers! (An infinite supply!)
All those operators! (How they seem to multiply!)
However, there is a certain beauty in the way all those numbers work together, and how certain shortcuts can lead one to the right answer for a problem in a very short (or at least shorter) period of time. Now, some of the tricks I will mention here you may never use at all, while some of them can be quite useful. In any case, one can at least appreciate the wonderful way in which certain apparently difficult problems break down into amazingly simple operations; operations that many people can do in their heads.
And now, on with the show!
Some multiplication shortcuts:
Multiplying by 11
11*735=?
1. Write down the ones digit (___5).
2. Now, add the two right-most digits (3+5), and write down that answer (8) along with the ones digit (__85).
3. Continue adding two digits together, moving to the right (3+7). Since 3+7 equals 10, we will have to carry. Write down the zero with the previous two numbers and carry the 1. Now we have (_085) and a one to carry.
4. Continue the two-digit adding, substituting a zero now that we are at the front of the number (7+0). Add the one we carried (giving us 8) and place at the front of the number (8085). Thus we have 11*735=8085.
(This principles of this shortcut can be used on any size number multiplied by 11, just continue the two digit addition from right to left.)
Multiplying by 25
25*36
1. Divide the number being multiplied by 4. If it comes out evenly, (which in this case it does: 36/4=9), simply multiply that answer by 100. 24*36=900
2. If it doesn't come out evenly, (say the problem was 25*37: 37/4=9.25), then write down the number of times 4 goes in fully (9), and find the remainder (1 in this case). A remainder of one tells you to put a 25 after the quotient (9), giving us 24*37=925
(Should the remainder have been a two, you would put a 50 in, and a 3 would mean a 75 should be put in.)
Multiplying by 50:
50*23
This is very much a derivative of the previous shortcut (or the previous shortcut is very much a derivative of this one, whichever way you prefer).
1. Divide the number being multiplied by 2 (23/2=11.5)
2. Multiply the quotient (11.5) by 100 (11.5*100=1150).
This is your answer. (50*23=1150)
Multiplying Two Numbers Ending in Five:
85*35
1. Add the tens digits together (8+3=11). If the sum is odd, write down a 75. If even, write down a 25.
2. Divide the tens digit sum by 2 (11/2=5.5). Remember the whole number (5).
3. Multiply the tens digits (8*3=24) and add the whole number from step 2 (24+5=29).
4. Place step 3's answer in front of the 25 or 75 and you have the answer. (85*35=2975)
Squaring Numbers Close To (But Less Than) 1000
993*993=993^2
1. Subtract the number from 1000 (1000-993=7)
2. Subtract step one's answer (7) from the number being squared (993-7=986)
3. Write step 2 down. Square the difference in step one (7^2=49) and make it a three-digit number by adding the appropriate number of zeroes in front of it (049)
4. Take the number figured in step 3 and place it behind what you wrote down already. Thus 993^2=986049
To be continued...
