Multiplication Tricks

Being able to perform arithmetic quickly and mentally can greatly boost your self-esteem, especially if you don’t consider yourself to be very good at Math. And, getting comfortable with arithmetic might just motivate you to dive deeper into other things mathematical. This article presents nine ideas that will hopefully get you to look at arithmetic as a game, one in which you can see patterns among numbers and pick then apply the right trick to quickly doing the calculation. The tricks in this article all involve multiplication. Don’t be discouraged if the tricks seem difficult at first. Learn one trick at a time. Read the description, explanation, and examples several times for each technique you’re learning. Then make up some of your own examples and practice the technique. As you learn and practice the tricks make sure you check your results by doing multiplication the way you’re used to, until the tricks start to become second nature. Checking your results is critically important: the last thing you want to do is learn the tricks incorrectly. 1. Multiplying by 9, or 99, or 999 Multiplying by 9 is really multiplying by 10-1. So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81. Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414. One more example: 68×9 = 680-68 = 612. To multiply by 99, you multiply by 100-1. So, 46×99 = 46x(100-1) = 4600-46 = 4554. Multiplying by 999 is similar to multiplying by 9 and by 99. 38×999 = 38x(1000-1) = 38000-38 = 37962. 2. Multiplying by 11 To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges. Let me illustrate: To multiply 436 by 11 go from right to left. First write down the 6 then add 6 to its neighbor on the left, 3, to get 9. Write down 9 to the left of 6. Then add 4 to 3 to get 7. Write down 7. Then, write down the leftmost digit, 4. So, 436×11 = is 4796. Let’s do another example: 3254×11. The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794. One more example, this one involving carrying: 4657×11. Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7). Going from right to left we write down 7. Then we notice that 5+7=12. So we write down 2 and carry the 1. 6+5 = 11, plus the 1 we carried = 12. So, we write down the 2 and carry the 1. 4+6 = 10, plus the 1 we carried = 11. So, we write down the 1 and carry the 1. To the leftmost digit, 4, we add the 1 we carried. So, 4657×11 = 51227 . 3. Multiplying by 5, 25, or 125 Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number. 12×5 = (12×10)/2 = 120/2 = 60. Another example: 64×5 = 640/2 = 320. And, 4286×5 = 42860/2 = 21430. To multiply by 25 you multiply by 100 (just add two 0’s to the end of the number) then divide by 4, since 100 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4. 64×25 = 6400/4 = 3200/2 = 1600. 58×25 = 5800/4 = 2900/2 = 1450. To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times. 32×125 = 32000/8 = 16000/4 = 8000/2 = 4000. 48×125 = 48000/8 = 24000/4 = 12000/2 = 6000. 4. Multiplying together two numbers that differ by a small even number This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6. Let’s say you want to calculate 12×14. When two numbers differ by two their product is always the square of the number in between them minus 1. 12×14 = (13×13)-1 = 168. 16×18 = (17×17)-1 = 288. 99×101 = (100×100)-1 = 10000-1 = 9999 If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4. 11×15 = (13×13)-4 = 169-4 = 165. 13×17 = (15×15)-4 = 225-4 = 221. If the two numbers differ by 6 then their product is the square of their average minus 9. 12×18 = (15×15)-9 = 216. 17×23 = (20×20)-9 = 391. 5. Squaring 2-digit numbers that end in 5 If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself. 35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225. To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer. 85×85: Calculate 8×9 = 72 and write down 7225. 6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10 Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer. An illustration is in order: To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20. Multiply together the last digits: 2×8 = 16. Write down 16. The product of 42 and 48 is thus 2016. Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4. Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224. A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224 7. Squaring other 2-digit numbers Let’s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40. Double this product: 40×2=80, then add a 0 to it, getting 800. Add 800 to 2564 to get 3364. This is pretty complicated so let’s do more examples. 32×32. The first part of the answer comes from squaring 3 and 2. 3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits. Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12. Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024. 56×56. The partial product comes from 5×5 and 6×6. Write down 2536. 5×6x2 = 60. Add a zero to get 600. 56×56 = 2536+600 = 3136. One more example: 67×67. Write down 3649 as the partial product. 6×7x2 = 42×2 = 84. Add a zero to get 840. 67×67=3649+840 = 4489. 8. Multiplying by doubling and halving There are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do. Let’s say you want to multiply 14 by 16. You can do this: 14×16 = 28×8 = 56×4 = 112×2 = 224. Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180. 48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.) 9. Multiplying by a power of 2 To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2x2×2. 15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240. 23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184. 54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432. Practice these tricks and you’ll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you’ll get to choose which one is easiest for a particular problem. Multiplication can be a great sport! Enjoy.

Math Beauty

Beauty of Mathematics !!!!!!! 1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10= 1111111111 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888 Brilliant, isn't it? And look at this symmetry: 1 x 1 = 1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321 Now, take a look at this... 101% From a strictly mathematical viewpoint: What Equals 100%? What does it mean to give MORE than 100%? Ever wonder about those people who say they are giving more than 100%? We have all been in situations where someone wants you to GIVE OVER 100%. How about ACHIEVING 101%? What equals 100% in life? Here's a little mathematical formula that might help answer these questions: If: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Is represented as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26. If: H-A-R-D-W-O- R- K 8+1+18+4+23+ 15+18+11 = 98% And: K-N-O-W-L-E- D-G-E 11+14+15+23+ 12+5+4+7+ 5 = 96% But: A-T-T-I-T-U- D-E 1+20+20+9+20+ 21+4+5 = 100% THEN, look how far the love of God will take you: L-O-V-E-O-F- G-O-D 12+15+22+5+15+ 6+7+15+4 = 101% Therefore, one can conclude with mathematical certainty that: While Hard Work and Knowledge will get you close, and Attitude will get you there, It's the Love of God that will put you over the top!

Math Magic

Trick 1: Number below 10

Step1: Think of a number below 10. Step2: Double the number you have thought. Step3: Add 6 with the getting result. Step4: Half the answer, that is divide it by 2. Step5: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought. Answer: 3 Trick 2: Any Number

Step1: Think of any number. Step2: Subtract the number you have thought with 1. Step3: Multiply the result with 3. Step4: Add 12 with the result. Step5: Divide the answer by 3. Step6: Add 5 with the answer. Step7: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought. Answer: 8 Trick 3: Any Number

Step1: Think of any number. Step2: Multiply the number you have thought with 3. Step3: Add 45 with the result. Step4: Double the result. Step5: Divide the answer by 6. Step6: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought. Answer: 15 Trick 4: Same 3 Digit Number

Step1: Think of any 3 digit number, but each of the digits must be the same as. Ex: 333, 666. Step2: Add up the digits. Step3: Divide the 3 digit number with the digits added up. Answer: 37 Trick 5: 2 Single Digit Numbers

Step1: Think of 2 single digit numbers. Step2: Take any one of the number among them and double it. Step3: Add 5 with the result. Step4: Multiply the result with 5. Step5: Add the second number to the answer. Step6: Subtract the answer with 4. Step7: Subtract the answer again with 21. Answer: 2 Single Digit Numbers. Trick 6: 1, 2, 4, 5, 7, 8

Step1: Choose a number from 1 to 6. Step2: Multiply the number with 9. Step3: Multiply the result with 111. Step4: Multiply the result by 1001. Step5: Divide the answer by 7. Answer: All the above numbers will be present. Trick 7: 1089

Step1: Think of a 3 digit number. Step2: Arrange the number in descending order. Step3: Reverse the number and subtract it with the result. Step4: Remember it and reverse the answer mentally. Step5: Add it with the result, you have got. Answer: 1089 Trick 8: x7x11x13

Step1: Think of a 3 digit number. Step2: Multiply it with x7x11x13. Ex: Number: 456, Answer: 456456 Trick 9: x3x7x13x37

Step1: Think of a 2 digit number. Step2: Multiply it with x3x7x13x37. Ex: Number: 45, Answer: 454545

Trick 10: 9091

Step1: Think of a 5 digit number. Step2: Multiply it with 11. Step3: Multiply it with 9091. Ex: Number: 12345, Answer: 1234512345

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.