Mental Math

Calculating things in your head can be a difficult task. If you can't remember what you've worked out or simply don't know how to solve a problem then it can be very challenging and frustrating. I'm going to try and give a few tips on how to do it more easily. My own mental calculation skills are below my general math ability due to problems with short term memory, but with a few shortcuts I can often calculate things scarily fast.

Addition

A useful trick when adding lots of small numbers is to clump together the ones that add up to multiples of 10. For example, if you have to add 2 + 3 + 5 + 7 + 9 + 11 + 8, that can be rearranged as (3 + 7) + (9 + 11) + (2 + 8) + 5 = 10 + 20 + 10 + 5 = 45.

Subtraction

A useful trick when subtracting numbers is to begin with the smaller value and mentally skip your way up the difference, with jumping points at recognizable boundaries, such as powers of 10. For example, to subtract 67 from 213 I would start with 67, then add 3 + 30 + 100 + 13. Try this once and you see how easy it is. Sounding out your thoughts it would be "three, thirty-three, one hundred thirty-three plus the remaining 13 is one hundred forty-six".

Multiplication

When multiplying it is very important to pick the correct sums to do. If you multiply 251 by 323 straight off it can be very difficult, but it is actually a very easy sum if approached in the right way. 251x3 + 251x20 + 251x300 is a scary prospect, so you have to work out the simplest method.

Rounding

One of the first things to do is to look if the numbers are near anything easy to work out. In this example there is, very conveniently, the number 251, which is next to 250. So all you have to do is 323x250 + 323 - much easier, but 323x250 still doesn't look too simple. There is, however, an easy way of multiplying by 250 which can also apply to other numbers. You multiply by 1000 then divide by 4. So 323x1000 = 323,000, divide by two and you get 161,500, divide by 2 again and you get 80,750. Now this may not seem easy, but once you've got used to it dividing by four (or other low numbers) in that way becomes natural and takes only a fraction of a second. 80,750+323 = 81,073 , so you've got the answer with a minimum of effort compared to what you would otherwise have done. You can't always do it this easily, but it is always useful to look for the more obvious shortcuts in this style.

An even more effective way in some circumstances is to know a simple rule for a set of circumstances. There are a large number of rules which can be found, some of which are explained below.

Factoring

If you recognize that one or both numbers are easily divisible, this is one way to make the problem much easier. For example, 72 x 39 may seem daunting, but if taken as 8 x 9 x 3 x 13, it becomes much easier.

First, rearrange the numbers in the hardest to multiply order. In this case, I'd go with 13 x 8 x 9 x 3. Then multiply them one at a time.

13 x 8 = 10 x 8 + 3 x 8 = 80 + 24 = 104
104 x 9 = 936
936 x 3 = 2808

Same First Digit, Second Digits Add to 10

Let's say you are multiplying two numbers, just two two-digit numbers for now (though the rules could be adapted for others) which start with the same digit and the sum of their unit digits is 10. For example, 87x83. You multiply the first digit by one more than itself (8x9 = 72). Then multiply the second digits together (7x3 = 21). Then stick the first answer at the start of the second to get the answer (7221). If you want to know how this works it is proved (and plenty of other techniques given)here.

Squaring a Number That Ends with 5

This is a special case of the previous method. Discard the 5, and multiply the remaining number by itself plus one. Then tack on a 25 (which as in the previous section, is 5x5). For example, 45x45. Discarding the 5 leaves us with 4. Multiplying 4 by itself plus one gives us 20 (4x5 = 20). Tacking on a 25 yields 2025, so 45x45=2025.

Just Over 100

This trick works for two numbers that are just over 100, as long as the last two digits of both numbers multiplied together is less than 100. For example, for 103 x 124, 3 x 24 = 72 < 100, so this trick will work. For 117 x 112, 17 x 12 = 204 > 100, so it will not.

If the first test works, then the answer is:

1[sum of last two digits][product of last two digits]

Examples:

108 x 109 = 1[8+9][8x9] = 1[17][72] = 11,772
105 x 115 = 1[5+15][5x15] = 1[20][75] = 12,075
132 x 103 = 1[32+3][32x3] = 1[35][96] = 13,596

This trick works for numbers just over 200, 300, 400, etc. with one simple change:

[product of first digits][(sum of last two digits) x first digit][product of last two digits]

Examples:

215 x 204 = [2x2][(15+4)x2][15x4] = [4][19x2][60] = [4][38][60] = 43,860

For numbers just over 1000, 2000, etc., use the following:

[product of first digits]0[(sum of last two digits) x first digit]0[product of last two digits]

Examples:

2008 x 2009 = [2x2]0[(8+9)x2]0[8x9] = [4]0[17x2]0[72] = [4]0[34]0[72] = 4,034072
- - 2008 x 2009 = 4,034,072

For each order of magnitude (x10), add two zeroes to the middle.

Division

Again there are many possible techniques, but you can make do with the following or research your own. All numbers are the products of primes (you can make them by multiplying together prime numbers). If you are dividing you can divide by all the prime products of the number you are dividing by to get the answer. This means that 100/24 = (((100/2)/2)/2)/3. Although this means you have a lot more stages to do they are all much simpler. 100/2 = 50 , 50/2 = 25 , 25/2 = 12.5 , 12.5/3 = 4⁵/₃₀ = 4¹/₆ = 4.166666666recurring

Also, another helpful trick is, when you have to muliply and then divide by a number, always divide first, until you've reached numbers that are relatively prime, and then multiply. This keeps numbers from being too large. For example, if you must do (18 * 115)/15, it is much easier to divide 115 by 5 and 18 by 3, and then multiply them together to get 23 * 6 = 138.

Multiply by the Reciprocal

Division is equivalent to multiplying by the reciprocal. For instance, division by 5 is the same as multiplication by 0.2 (1/5=0.2). To multiply by 0.2, simply double the number and then divide by 10.

Division by 7

The number 1/7 is a special number, equal to $0.\overline{142857}$ . Note that there are six digits that repeat, 142857. A beautiful thing happens when we consider integer multiples of this number:

$\frac{1}{7} = 0.\overline{142857}$

$\frac{2}{7} = 0.\overline{285714}$

$\frac{3}{7} = 0.\overline{428571}$

$\frac{4}{7} = 0.\overline{571428}$

$\frac{5}{7} = 0.\overline{714285}$

$\frac{6}{7} = 0.\overline{857142}$

Note that these six fractions of seven contain the same six digits repeating in the same order ad infinitum, but starting with a different number. But how is this useful when dividing by seven? Consider the problem 207/7. First, we can convert this to 200/7 + 7/7. We know that 7/7 equals one, so the answer will be 200/7 + 1. But what is 200/7? It is simply 2/7 times 100, and from the above, we know that $\frac{2}{7} = 0.\overline{285728}$ , so by moving the decimal point, we know that $\frac{200}{7} =28.\overline{571428}$ . All that remains is to add the one from 7/7, giving us $\frac{207}{7} =29.\overline{571428}$ .

Division by 9

The fraction 1/9 and its integer multiples are fairly straight forward - they are simply equal to a decimal point followed by the one-digit the numerator repeating to infinity:

$\frac{1}{9} = 0.\overline{11}$

$\frac{2}{9} = 0.\overline{22}$

$\frac{3}{9} = 0.\overline{33}$

$\frac{4}{9} = 0.\overline{44}$

$\frac{5}{9} = 0.\overline{55}$

$\frac{6}{9} = 0.\overline{66}$

$\frac{7}{9} = 0.\overline{77}$

$\frac{8}{9} = 0.\overline{88}$

To solve a problem such as 367/9, we reduce it to

$\frac{367}{9} = \frac{300}{9} + \frac{60}{9} + \frac{7}{9}$

$\frac{367}{9} = 33.\overline{33} + 6.\overline{66} + 0.\overline{77}$

First add $33.\overline{33} + 6.\overline{66} = 40.0$ . Then add $40.0 + 0.\overline{77} = 40.\overline{77}$ .

Estimation

The best way to make estimation quickly in mental math is to round to one or two significant digits (that is, round it to the nearest place of the highest order(s) of magnitude), and then proceed with typical operations. Thus, 1241 * 15645 is approximately equal to 1200 * 16000 = 19200000, which is reasonably close to the correct answer of 19415445. In certain cases, one can even round to simply the nearest power of ten (which is useful when making estimations with much error and large numbers).

Other mental math

Perhaps one of the more useful tricks to mental math is memorization. Although it may seem an annoyance to need to memorize certain math facts, such as perfect squares and cubes (especially powers of two), prime factorizations of certain numbers, or the decimal equivalents of common fractions (such as 1/7 = .1428...). Many are simple, such as 1/3 = .3333... and 2^5 = 32, but speed up your calculations enormously when you don't have to do the division or multiplication in your head. For example, trying to figure out 1024/32 is much easier knowing that that is the same as 2^10/2^5, or which, subtracting exponents, gives 2^5, or 32. Many of these are memorized simply by frequent use; so, the best way to get good is much practice.

I haven't got time to write any more at the moment (hopefully some other people will be able to contribute though) so I wont add any more for now, but the ideas I have shown can often be applied to more areas and help in most mental math.

I haven't mentioned addition or subtraction, which seem to be strange things to overlook, but there are much fewer shortcuts for these activities. If anyone edits this I suggest that is the first thing to talk about.

Text