What are some useful mental math tricks?

Question

Ghada Eweda · Answer

Great Post.

You can read whole bunch of articles on the following website MathsEquation on mental math tricks.

Link here.Learn Cool Tricks to Solve Maths EasilyThereafter, while you perform any mathematical calculation,there exist certain patterns.If you can find the pattern and memorize each pattern.Of course not in a day. Soon you will find yourself in a position of performing mental math in front of audience.

Regards

Noufal Ali · Answer

very intresting subject.. thank you for invitation.Mental math involves three basic techniques:

Recognizing when numbers have ideal properties, and using pre-worked out shortcuts to solve the problem.
Rearranging the problem so intermediate calculations are easier.
Breaking hard problems into simpler, more manageable parts, then combining the results back later to get the answer.

You'll need a repertoire of tricks so you can recognize when to apply each technique.First, I want to buy your attention with some low hanging fruit. Here are two tricks you can use to do math faster than a calculator:--Multiplying 2-digit numbers by 11 [1]:To multiply any two digit number by 11, say 43 x 11, just sum the digits (4 + 3 = 7) and put it between the 4 and 3 to get the answer: 473. Easy, right?If the sum is greater than 10, just add 1 to the first digit. For example 49 x 11, 4 + 9 = 13, so stick the 3 between the 4 and 9 (439), and increment the 4 to get 539.You can use the same trick for 3-digit numbers [2]. For example 417 x 11:4+1 = 51 + 7 = 8Stick them between the 4 and 7 to get 4587.Squaring 2-digit numbers that end in 5 [3]:Two rules to remember:1) The answer begins by multiplying the first digit by the next higher digit. 2) The answer ends in 25.So to compute 35 x 35, multiple 3 by the next higher digit (4) to get 12, and end in 25, so 1225. Same with 65 x 65, which is 6 x 7 = 42, and end in 25 to get 4225.--These are both examples of using pre-worked out shortcuts to solve problems. They're the same kind of shortcuts we use for problems like 23 x 100. We don't literally try to add 23100 times; we just know to add two zeros to the end of 23 to get the answer. Shortcuts exist for all kinds of number properties. Part of being good at mental math is being able to identify when you can use a shortcut.Let's move on to basic arithmetic, which requires some unlearning.--Addition & SubtractionDespite what your 1st grade teacher taught you, you always work from left to right. Instead of trying to visualize a large problem in your head, working from right to left, it's much easier to break that problem into smaller addition problems, combining the results back from left to right.For example, to calculate 328 + 267, you would break the problem into smaller parts: 328 + 200 + 60 + 7.Mentally, it's much easier to arrive at the answer: 328 + 200 = 528528 + 60 = 588588 + 7 = 595The same principle applies with subtraction. To calculate 563 - 328, we rearrange the problem as 563 - 300 - 20 - 8:563 - 300 = 263263 - 20 = 243243 - 8 = 235If your subtraction problem requires borrowing, round the number up to a multiple of ten. Subtract the rounded number, then add back the difference.For example, to calculate 84 - 28, we would round 28 up to 30: 84 - 30 = 54. We can then add back 2 to get 56.MultiplicationAgain, we always work from left to right, breaking problems into simpler more manageable parts. For example, to calculate 56 x 42, we would rearrange the problem as 56 x 40 + 56 x 2:56 * 40 = 560 * 4 = 500 * 4 + 60 * 4 = 2000 + 240 = 224056 * 2 = 50 * 2 + 6 * 2 = 100 + 12 = 1122240 + 112 = 2352There are often many ways to break up a problem. For example, to calculate 46 x 42, you could break it up as 40 x 42 + 6 x 42, or 46 x 40 + 2 x 46. The first method simplifies to 1840 + 92, while the other simplifies to 1680 + 252. The latter is a harder problem to solve. So how do you decide which number to break up?Generally, you want to choose the number that will produce the easier addition problem. In most cases, this means breaking up the number with the smaller last digit, because it usually produces a smaller second number for you to add [4].For example, lets try breaking up 81 x 59:Breaking the larger number: 81 * 50 + 81 * 9 = 4050 + 729Breaking the smaller number: 59 * 80 + 1 * 59 = 4720 + 59In this example, breaking the smaller number gives you the simpler addition problem.The Subtraction Method (Multiplication)If the number you want to multiply ends in 8 or 9, it's often easier to round up, then subtract. For example, to calculate 59 x 13, it's easier to round 59 up to 60, then subtract 13 from the final answer:59 x 13= (60 - 1) x 13 = 60 x 13 - 13= 780 - 13= 767Similarly, to compute 58 x 13, you would just round up to 60, then subtract 26 from the final answer:58 x 13= (60 - 2) x 13= 60 x 13 - 13 x 2= 780 - 26= 754DivisionInstead of covering division in this answer, I'll instead refer you to Chapter 4 of Arthur Benjamin's excellent book "Secrets of Mental Math":http://www.amazon.com/Secrets-Me...--Let's move on to the fun stuff!Squaring any 2-digit number:To square any two digit number, say 13 x 13, rearrange the multiplication problem so that you are multiplying two numbers that also add up to 26, but is easier to calculate in your head. So in this case, 10 x 16 = 160. Then we take the distance from 13 (in this case 3, since 10 and 16 are both 3 away from 13), square it (3 x 3 = 9), and add it to 160 to get the final answer, 169.So to calculate 47^2, we rearrange the problem to be 44 x 50 + 3^2. We can do 44 x 50 in our head, 2200. Then add 3^2 to get 2209.Why does this trick work? Because of the following algebraic observation:A^2 = (A + d) x (A - d) + d^2A^2 = A^2 - dA + dA - d^2 + d^2A^2 = A^2Squaring any 3-digit number:You can apply the same trick above for squaring 3 digit numbers. Instead of rounding to the nearest multiple of 10, you round to the nearest multiple of 100. So for example, to square 193, you would do 200 x 186 + 7^2. You can do 186 x 200 in your head. Just do 186 x 2 = 372, and add two zeros to get 37200. Then add 7 x 7 to get 37249.Cubing any 2-digit number:To cube numbers in our head we can use a similar trick. Here's our equation:A^3 = (A - d) * A * (A + d) + d^2 * ASo to cube 13 in our head, we rearrange the problem to be:10 * 13 * 16 + 3^2 * 13We chose 3 as the value for d, since it lets us multiply by 10.--Now all you need is practice! Hopefully I've shown that anyone can do mental math. You don't need to be a savant, or have an IQ of 185 to square numbers in your head. It's all about breaking hard problems into simple problems, and solving them one step at a time. Just like life.

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