Home » 19 July 2009
Mathematic Tricks

Math can be terrifying for many people. This list will hopefully improve your general knowledge of mathematical tricks and your speed when you need to do math in your head.
1. The 11 Times Trick
We all know the trick when multiplying by ten – add 0 to the end of the number, but did you know there is an equally easy trick for multiplying a two digit number by 11? This is it:
Take the original number and imagine a space between the two digits (in this example we will use 52:
5_2
Now add the two numbers together and put them in the middle:
5_(5+2)_2
That is it – you have the answer: 572.
If the numbers in the middle add up to a 2 digit number, just insert the second number and add 1 to the first:
9_(9+9)_9
(9+1)_8_9
10_8_9
1089 – It works every time.
2. Quick Square
If you need to square a 2 digit number ending in 5, you can do so very easily with this trick. Mulitply the first digit by itself + 1, and put 25 on the end. That is all!
252 = (2x(2+1)) & 25
2 x 3 = 6
625
3. Multiply by 5
Most people memorize the 5 times tables very easily, but when you get in to larger numbers it gets more complex – or does it? This trick is super easy.
Take any number, then divide it by 2 (in other words, halve the number). If the result is whole, add a 0 at the end. If it is not, ignore the remainder and add a 5 at the end. It works everytime:
2682 x 5 = (2682 / 2) & 5 or 0
2682 / 2 = 1341 (whole number so add 0)
13410
Let’s try another:
5887 x 5
2943.5 (fractional number (ignore remainder, add 5)
29435
22189271
4. Multiply by 9
This one is simple – to multiple any number between 1 and 9 by 9 hold both hands in front of your face – drop the finger that corresponds to the number you are multiplying (for example 9×3 – drop your third finger) – count the fingers before the dropped finger (in the case of 9×3 it is 2) then count the numbers after (in this case 7) – the answer is 27.
5. Multiply by 4
This is a very simple trick which may appear obvious to some, but to others it is not. The trick is to simply multiply by two, then multiply by two again:
58 x 4 = (58 x 2) + (58 x 2) = (116) + (116) = 232
6. Calculate a Tip
If you need to leave a 15% tip, here is the easy way to do it. Work out 10% (divide the number by 10) – then add that number to half its value and you have your answer:
15% of $25 = (10% of 25) + ((10% of 25) / 2)
$2.50 + $1.25 = $3.75
7. Tough Multiplication
If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer:
32 x 125, is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000
1000-Abacus
8. Dividing by 5
Dividing a large number by five is actually very simple. All you do is multiply by 2 and move the decimal point:
195 / 5
Step1: 195 * 2 = 390
Step2: Move the decimal: 39.0 or just 39
2978 / 5
step 1: 2978 * 2 = 5956
Step2: 595.6
9. Subtracting from 1,000
To subtract a large number from 1,000 you can use this basic rule: subtract all but the last number from 9, then subtract the last number from 10:
1000
-648
step1: subtract 6 from 9 = 3
step2: subtract 4 from 9 = 5
step3: subtract 8 from 10 = 2
answer: 352
10. Assorted Multiplication Rules
Multiply by 5: Multiply by 10 and divide by 2.
Multiply by 6: Sometimes multiplying by 3 and then 2 is easy.
Multiply by 9: Multiply by 10 and subtract the original number.
Multiply by 12: Multiply by 10 and add twice the original number.
Multiply by 13: Multiply by 3 and add 10 times original number.
Multiply by 14: Multiply by 7 and then multiply by 2
Multiply by 15: Multiply by 10 and add 5 times the original number, as above.
Multiply by 16: You can double four times, if you want to. Or you can multiply by 8 and then by 2.
Multiply by 17: Multiply by 7 and add 10 times original number.
Multiply by 18: Multiply by 20 and subtract twice the original number (which is obvious from the first step).
Multiply by 19: Multiply by 20 and subtract the original number.
Multiply by 24: Multiply by 8 and then multiply by 3.
Multiply by 27: Multiply by 30 and subtract 3 times the original number (which is obvious from the first step).
Multiply by 45: Multiply by 50 and subtract 5 times the original number (which is obvious from the first step).
Multiply by 90: Multiply by 9 (as above) and put a zero on the right.
Multiply by 98: Multiply by 100 and subtract twice the original number.
Multiply by 99: Multiply by 100 and subtract the original number.

Tricks with Arithmetic

What is 26 × 34? What about 37 × 13? There are some neat little tricks you can remember which will help you do these and other calculations in your head in seconds. Here we give examples and explain why they work. The Difference of Squares Multiplying 26 by 34 might not seem to have anything to do with square numbers, but look at it like this: 26 × 34 = (30 – 4)(30 + 4) = 30² + (30 × 4) – (4 × 30) – 4² The middle two terms cancel, giving 26 × 34 = 30² – 4², and we know that 30² = 900 and that 4²= 16, so our answer is therefore 884. That was much easier than trying to do a huge multiplication! This method works in general; for two numbers a and x, (a – x)(a + x) = a² – x². So, to give another example, 17 × 23 = 20² – 3² = 391. Know Numbers My sister once said “Numbers are my friends.” She’s never been able to live it down, of course, but she may have had a point. If you get to know some interesting habits of a few numbers, mental arithmetic certainly becomes a lot simpler. For instance, it’s very useful to know that 17 × 3 = 51, and not only because it reminds you that 51 isn’t prime. For example: 24 × 17 = (8 × 3) × 17 = 8 × (3 × 17) = 8 × 51 = 408. To return to the second problem at the beginning of the article, the number 37 is interesting because 37 × 3 = 111. To write this more usefully, 37 × 3 = (100 × 1) + (10 × 1) + (1 × 1). So, 37 × 3a = a ((100 × 1) + (10 × 1) + (1 × 1)) = 100a + 10a + a . This means that 37 × 12 = 400 + 40 + 4 = 444, and so 37 × 13 = 444 + 37, which is 481. Now try 37 × 42. General points * Try factorising the numbers involved; 512 might be a lot easier to deal with if you remember it’s a power of 2. * Check your short cuts. 37 × 12 can’t be much different from 40 × 10 = 400, so if you get something wildly different, there’s been a slip at some point. * Keep practising! Here are some mental arithmetic magic tricks I have found that you can use to surprise your family. 1. if a number is divisible by 3 then so are the numbers based on mixing up the digits of the original number. For example, consider 123 which can be evenly divided by by 3. Then 132, 213, 231, 312 and 321 (which are obtained by mixing up the digits 1, 2 and 3 that make up 123) are all divisible by 3. This is called a permutation of the digits of a number. Check it for yourself! 2. to make up a number that is divisible by 4, make up a number and tag on the end any 2 digit number divisible by 4. For example, I make up the number 111111111, and now I tag 16 (which is divisible by 4) on the end to get 11111111116. This number is divisible by 4. Check it for yourself! An interesting trick follows on from this one. The following numbers can all be evenly divided by 4: 116, 1116, 11116, 111116, and so on . . . Not what you would expect! 3. if a number is divisible by 6, then any shuffle of its digits will give you a new number divisible by 6 as long as the last digit is even. For example, 1272 is divisible by 6. Permutations of its digits while keeping the last digit even gives me 2172, 2712, 1722, 7122, 7212 which can all be evenly divided by 6. Check it for yourself! 4. to make up a number that is divisible by 8, the process is very much like point 2. above. Make up a number and tag on the end any 3 digit number divisible by 8. For example, I make up the number 777777, and now I tag 016 (which is divisible by 8) on the end to get 777777016. This number is divisible by 8. Check it it for yourself! Another interesting trick follows on from this one. The following numbers can all be evenly divided by 8: 1016, 11016, 111016, and so on . . . Again, not what you would expect! 5. if a number is divisible by 9 then so are the numbers based on mixing up the digits of the original number. For example, consider 189 which is divisible by 9. Then so are 198, 819, 891, 918 and 981. Check it it for yourself! 6. if a number is divisible by 11, then permutations of its odd digits and/or its even digits will give you a new number also divisible by 11. For example, consider 154 which is divisible by 11. Then so is 451 (obtained by swapping its first and third digits). Another example, consider 1122 which is divisible by 11. Then so is 1221 (obtained by swapping its second and fourth digits). Check it it for yourself! 7. if a number is divisible by 12, then any permutations of its digits (except for the last 2) will give you new numbers also divisible by 12. For example, 14652 is divisible by 12. Then so are 16452, 41652, 46152, 61452 and 64152. Check it it for yourself! You would have to agree that such tricks do look like arithmetic magic which you can do in your head. In case you are wondering ‘why is it so?’ The trick lies in the test that determines whether a number is divisible by another.

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