## Favorite mental math tricks/shortcuts

For the discussion of math. Duh.

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Wixx
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### Favorite mental math tricks/shortcuts

Anyone know any cool or useful shortcuts or tips?

Some of my favorites:

If you know the squares of numbers, this one can be useful:
Ex: 12x14 = (13x13) - 1 = 168
24x26 = (25x25) - 1 = 624

To figure out approx one's salary:
Wage/hr * 2000 = Salary in one year
Ex: $8/hr =$16,000/yr (aka don't drop out)

x% of y = y% of x

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.
65×65, 6x7 = 42. So the answer is 4225

I might post some more later. Anyone else have any good ones?

Sungura
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### Re: Favorite mental math tricks/shortcuts

I often round up or down when doing calculations, then account for it at the end.

I also like the multiplication by doubling. Lovely little system for doing crazy problems quickly by hand.
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Frimble
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### Re: Favorite mental math tricks/shortcuts

amysrabbitranch wrote:I also like the multiplication by doubling. Lovely little system for doing crazy problems quickly by hand.

I believe this is called the Russian peasant method. There is a poster explaining it in the maths department at my college.
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### Re: Favorite mental math tricks/shortcuts

Sales tax is 9% here, so that's a calculation I have to make a lot. 100%+10%-1%. Not terribly sophisticated, but it makes me happy.

Factoring numbers there's also something I use:
Say you want to figure out whether 6239 (random number not divisible by 2, 3, or 5) is divisible by seven. Mental long division can be really hard to do, keeping all the places and whatnot, so instead think of a multiple of seven that ends in either one or nine, then add or subtract accordingly:
6239+21=6260.
Now divide by ten (that won't remove its divisibility by seven if it has it), then divide by 2, 3, or 5 if that will let you lose another digit. (Smaller numbers are always better.) Repeat the adding/subtracting multiples and stupid-easy division until you get to a number you can easily recognize as either a multiple of seven or not.
6260/10=626
626+14=640
640/10=64 -->clearly not a multiple of 7. Therefore, neither is 6239.

Um, whee! I'm like a lightning factorer by now.
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btilly
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### Re: Favorite mental math tricks/shortcuts

In Los Angeles sales tax is 8.5%. So the price of whatever you bought plus 25% gives you tax and a decent tip on your meal. Useful for figuring out what each person owes when you eat out together.

I also occasionally show off by using continued fractions to turn a decimal back into a fraction. For instance someone gives you a decimal like 0.764705882352941. You can find the fraction as follows if you have a calculator:

$0.764705882352941 = \frac{1}{\displaystyle 1.30769230769231} = \frac{1}{\displaystyle 1 + \frac{1}{\displaystyle 3.24999999999998}} = \frac{1}{\displaystyle 1 + \frac{1}{\displaystyle 3 + \frac{1}{4}}} = \frac{1}{\displaystyle 1 + \frac{4}{\displaystyle 13}} = \frac{13}{\displaystyle 17}$
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Klotz
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### Re: Favorite mental math tricks/shortcuts

I'm a fan of 2^10=10^3

Sungura
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### Re: Favorite mental math tricks/shortcuts

Frimble wrote:
amysrabbitranch wrote:I also like the multiplication by doubling. Lovely little system for doing crazy problems quickly by hand.

I believe this is called the Russian peasant method. There is a poster explaining it in the maths department at my college.

I learned it as being an Egyptian method from my history of math class. Our webpage is down (all the notes and such were on Moodle) but this link explains it as well: http://www.saintjoe.edu/~karend/m441/m441-1.html
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### Re: Favorite mental math tricks/shortcuts

Squaring numbers using (x+y)2=x2+2xy+y2.

For example, 392=1600-80+1=1521, and 482=2500-200+4=2304. It makes mentally computing 2-digit squares quite quick.
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Blatm
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### Re: Favorite mental math tricks/shortcuts

You can use the same idea to take square roots to a few digits effectively.

example:
2994 = (54 + b)^2 = (50+4)^2 + 108b + b^2 = 2916 + 108b + b^2
78 = 108b + b^2

Now, because b < 1, b and b^2's most significant digits will be on different orders of magnitude, so you're guarenteed 1 correct digit of the root like this by ignoring the b^2, but you can guess another reliably.

78 ~= 108b
39/54 ~= b ~= 0.72 (A guess based on the fact that 39/54 is just slightly less than 4/5.5 = 8/11, and 7*11 = 77)

And indeed the square root of 2994 is ~54.7175, compared to our guess of 54.72.

Another way to find square roots which I find less effective because it gets messy quickly, but might work, is with the Newton approximation method. (For those who don't know, it says that if q is a reasonable guess for the root of a function, then q - f(q)/f'(q) is a significantly better guess, so, for example, you would take the f(x) = 2994 - x^2 to find the above root.)

antonfire
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### Re: Favorite mental math tricks/shortcuts

Here's "another" quick divisibility test which works any number relatively prime to the base we're working in (I think I've mentioned it elsewhere on this forum). It's really the same thing as Quixotess's trick, just phrased differently.

If k is the inverse of 10 modulo n, repeatedly chop off the last digit, multiply it by k, and add it to what's left. For example, the inverse of 10 modulo 7 is -2. So, to see if 6239 is divisible by 7: 6239 -> 623-2*9=605 -> 60-2*5=50, which is not divisible by 7, so neither is 6239.

I recently noticed that you can do 7 and 13 at the same time by using -9, since 10*(-9) is 1 modulo both 7 and 13. -9 might seem harder than -2, but it's actually about as easy, since it's -10+1, so you can subtract the last digit from the 3rd to last, and add it to the 2nd to last.

For examlpe, 6239 -> 552 -> 37, which is not divisible by 13 or 7, so neither is 6239.

11661 -> 1157 -> 52, which is divisible by 13 but not 7, so the same is true for 11611.
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### Re: Favorite mental math tricks/shortcuts

When I read that I got all excited, thinking that maybe I had randomly picked a prime number, but 6239 / 17 = 367, so, poo.
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### Re: Favorite mental math tricks/shortcuts

Klotz wrote:I'm a fan of 2^10=10^3

Also 10^6=2^20, 10^9=2^30, and so on. Pretty convenient.

Klotz
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### Re: Favorite mental math tricks/shortcuts

Yeah I combine that with my knowledge of the laws of exponents.

eijkaibjck
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### Re: Favorite mental math tricks/shortcuts

To make squares, I use one of two methods.

The first one is the one that Skeptical Scientist (and probably Newton) mentioned.

I decompose, say, 42, in a=40 and b=2 and by virtue of a^2+b^2+2ab=(a+b)^2 I get, um, 1604

The other one is adding and substracting the same quantity, multiply the simmetrical numbers and add the offset quantity squared.

For instance, 58^2=60*56+2^2=3364, I hope.

This works because (a-b)*(a+b)=a^2-b^2.

Adding b^2 we get a^2, which was our target.
Ok, I take it back.

Rgeminas
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### Re: Favorite mental math tricks/shortcuts

Actually, you did the wrong maths on the first one. Actually, it is 1600 + 160 + 4...

lgonick
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### Re: Favorite mental math tricks/shortcuts

1. In doing everyday calculations, I try to treat everything as a common fraction. 15 is 3/2, 25 is 1/4, etc. Forget the decimal point and put it back at the end. Makes a lot of multiplication and division easier.

2. When computing discounted prices, use the distributive law. If something is 15% off, just take 85% (roughly 5/6, and multiplying by 5 is like dividing by 2!) of the original price. No subtracting clumsy quantities...

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### Re: Favorite mental math tricks/shortcuts

It baffles me that people use calculators to factor tips. 10% and half that again.

$57.72 would be$5.77 + ~$3...$9.

Similar for kilometers to miles, half plus 10%.

172 km ~= 86 + 17 ~= 103 miles.
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### Re: Favorite mental math tricks/shortcuts

Not hugely brilliant, but pretty useful for me, maybe because I tend to wander around with 4 other dudes and that makes it 5 to split bills etc.:

x / 5 = 2 * (x/10)

In case your mind and mine work differently, for mental calculations I seem to be much faster (and more reliable) at multiplying than at dividing, specially for largish numbers. Thus, I'm much more at ease multiplying by 2 than dividing by 5.

It also works the other way (x / 2 = 5 * (x/10)). This is less useful, but for me it is still sometimes better than division.
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### Re: Favorite mental math tricks/shortcuts

andvaranaut wrote:Not hugely brilliant, but pretty useful for me, maybe because I tend to wander around with 4 other dudes and that makes it 5 to split bills etc.:

x / 5 = 2 * (x/10)

I do this but with multiplication. I have trouble multiplying by 5 (but not dividing by 5), so I use x * 5 = (x * 10) / 2.
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SimplySemantics
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### Re: Favorite mental math tricks/shortcuts

Personally my favorite (though there has already been alot of work with squares on this thread) is the multiplication of two even or two odd numbers. Let's say you want to find 83 times 57.

To start, you find the average, which we can denote "a". This is a simple one, (83+57)/2 is just 70. Then we can take the difference from the number and either of the other ones, and it's just 13, which we can denote "b". Simple enough.

Now, we can deduce that the result will be a^2-b^2. The reason is that (a-b)(a+b) = a^2-b^2, with simple factoring. Since we have the original numbers in the the factored out version (70-13)(70+13) we can just take 70^2 - 13^2, a far easier calculation. Thus we have 4900 - 169, which is simply 4731. You can have an even and an odd number I suppose, but then you get decimal averages, and those aren't nice.

It is useful now and then With a little bit of practice it takes very little time to do in your head, under 5 seconds if the original average of the two numbers is a multiple of 10.

EDIT: spelling.

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### Re: Favorite mental math tricks/shortcuts

Torn Apart By Dingos wrote:
Klotz wrote:I'm a fan of 2^10=10^3

Also 10^6=2^20, 10^9=2^30, and so on. Pretty convenient.

Am I missing something?

(2^10 = 1024) != (10^3 = 1000)

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### Re: Favorite mental math tricks/shortcuts

It's approximate, not exact, but it's a common way to go between powers of 2 and powers of 10 for order of magnitude/back of the envelope calculations.
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### Re: Favorite mental math tricks/shortcuts

andvaranaut wrote:Not hugely brilliant, but pretty useful for me, maybe because I tend to wander around with 4 other dudes and that makes it 5 to split bills etc.:

x / 5 = 2 * (x/10)

In case your mind and mine work differently, for mental calculations I seem to be much faster (and more reliable) at multiplying than at dividing, specially for largish numbers. Thus, I'm much more at ease multiplying by 2 than dividing by 5.

It also works the other way (x / 2 = 5 * (x/10)). This is less useful, but for me it is still sometimes better than division.

Our minds must work differently: the first part is normal enough, but the second...
I'm pretty sure most people do this in reverse, ie. prefer to halve 10*n rather than multiply by 5 directly.
I find it quite surprising someone would do the opposite.
You must really hate division

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### Re: Favorite mental math tricks/shortcuts

troyp wrote:Our minds must work differently: the first part is normal enough, but the second...
I'm pretty sure most people do this in reverse, ie. prefer to halve 10*n rather than multiply by 5 directly.
I find it quite surprising someone would do the opposite.
You must really hate division

Actually, I kind of do , but it depends on the particular number. It is handy to have both possibilities in mind

Anyway, there are a lot of variations based in that, whenever there is a 5 or a 2 involved, you can slip in the other number to get a 10 somewhere. Each of us will have a favorite.
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troyp
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### Re: Favorite mental math tricks/shortcuts

Actually, I kind of do , but it depends on the particular number. It is handy to have both possibilities in mind

Actually, I guess it might be quicker sometimes if you were used to it and it was a multiple of 5 you knew by heart.
I can't remember ever doing it myself.

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### Re: Favorite mental math tricks/shortcuts

I'm in Italy and now my girlfriend is in Canada. To figure out what time is in Toronto I have to calculate ItalianTime-6 mod 24...
I realized that it was easier on an analog clock, where i could read that time on the "wrong" side of the minute-hand thanks to the fact that 6 is an half of 12.

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### Re: Favorite mental math tricks/shortcuts

I just noticed (probably for the 3rd or so time) that n2 and (n+1)2 are always separated by n+(n+1). This fells out of the cool "lucky seven" method of factoring squares. I named it the lucky seven method because I noticed it while playing with dice probabilities.

If you take a square and start subtracting 2n, incrementing up from n=1, at some point you'll have exactly n left, which is the square root.
Examples: 4-2(1)=2
9-2(1)-2(2)=3
196 -2-4-6-8...-2(12)-2(13)=14
_ _ To speed this up you can start at easier squares (+n), like "subract 650 and start at n=26" or "-930 and n=31"
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samian
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### Re: Favorite mental math tricks/shortcuts

vat is 17.5% here. I always thought that was a neat number to pick.

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### Re: Favorite mental math tricks/shortcuts

17.5%: 10% + half + half again, yes, I've found it quite easy (although extortionate!). Sadly, though, it's usually added on for us. I like to think we'd all be smarter if we had to calculate VAT every day.

The way I do 7s, for example:

34681. Multiple of 7. To find out, recursively take the last digit, double, subtract from remainder:
34681 -> 3468-2 = 3466
3466 -> 346 - 12 = 334
334 -> 33 - 8 = 25. Fail. Not a multiple of 7.

Easier with 11, for digits abcd. Multiple of 11 iff a+c = b+d. For fewer than 4 replace a,b, etc with 0.

I also use the difference of squares for multiplying numbers.

Hefty One
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### Re: Favorite mental math tricks/shortcuts

There's the 1/7 trick, I guess.

1/7=.142857
2/7=.285714
3/7=.428571
4/7=.571428
5/7=.714285
6/7=.857142

same numbers, different orders
Just need to remember 14 x 1 -- 14 x 3 and then 14 x 4 + 1 -- 14 x 6 + 1

Useful since the rest of the numbers, 1/10 - 9/10 are easy to remember/use basic multiplication to get the decimal equivalent.

NathanielJ
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### Re: Favorite mental math tricks/shortcuts

Hefty One wrote:There's the 1/7 trick, I guess.

1/7=.142857
2/7=.285714
3/7=.428571
4/7=.571428
5/7=.714285
6/7=.857142

same numbers, different orders

How have I never noticed that before...? Is this part of a more general rule about repeating digits in different bases?
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### Re: Favorite mental math tricks/shortcuts

For tips, I do: Divide by 3, divide result by 2, all the while rounding down to the nearest "nice" number (multiples of 10 cents). It works out to about 16% tips, which I feel is fair.

E.g: $10.77. Round down to$10.50 because it divides nicely by 3 to get $3.50, divided by 2 gives$1.75. Actual fraction of total bill: 16.2% Hey, generosity!

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### Re: Favorite mental math tricks/shortcuts

My tricks are not nearly as cool, but when multiplying by 9, (for single digits), remove one from the number to get the first digit, then the second digit you get by working out what the difference is between the first digit and 9. eg 7*9 first digit is 6. 9-6 is 3 so the second digit is 3. therefore 9*7=63
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### Re: Favorite mental math tricks/shortcuts

Gojoe wrote:My tricks are not nearly as cool, but when multiplying by 9, (for single digits), remove one from the number to get the first digit, then the second digit you get by working out what the difference is between the first digit and 9. eg 7*9 first digit is 6. 9-6 is 3 so the second digit is 3. therefore 9*7=63

This is a good trick, and you can do it without thinking on your hands. However, it's a temporary trick until such time as you memorize your times tables past 9.

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### Re: Favorite mental math tricks/shortcuts

I know my 9's timetables, but only due to this lol. I am a 19 year old working at a computer software company, and I use this trick mentally whenever I have to multiply anything by 9 lol.
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### Re: Favorite mental math tricks/shortcuts

Despite having memorized my times table, I still use this trick. 9x7? Well, it'll be 6..3. 63.

This goes on in my head every single time.
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### Re: Favorite mental math tricks/shortcuts

Our physics teacher taught us a neat trick, it came up when dealing with simple harmonic motion.

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### Re: Favorite mental math tricks/shortcuts

For multiplying two-digit numbers by 11, stick the sum of the digits in between them (and carry if necessary). So for example 53x11 is 583 because 5+3=8.

I also use this for approximately dividing by 11, which I remember having to do once to get percentages from a 110 point sum of two tests.

So, for example, if a person got 74 points, I'd subtract the first digit from the second digit (with borrowing from the ones digit) to get 7. 73.7, in other words, which is 1.1*67, "obviously", giving 67% as the final score. If the first digit is smaller, 79, for example, I wouldn't borrow, and instead just get 2, so 79.2, which is 1.1*72, giving 72% as the final score.

Okay, so this perhaps isn't the easiest thing ever, but it was still faster to do than noting the (nonetheless very useful) fact that the repeating portion in the decimal expansion of n/11 (for 0<n<11) is just 9n.
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Blatm
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### Re: Favorite mental math tricks/shortcuts

Estimating logs mentally:

First, memorize the following:
log2 = 0.301
log3 = 0.477
log4 = 0.602
log5 = 0.699
log6 = 0.778
log7 = 0.845
log8 = 0.903
log9 = 0.954

It also helps to know

log1.5 = 0.176
log2.5 = 0.398
log e = 0.434 (for conversions)

Given some number (let's use 6,969,284,682), that's roughly 6.969 * 10^9. Its log will be 9 + log 6.97, which is 0.03 less than log 7, so 9.845 - 0.03(0.845 - 0.778) = 9.845 - 3*0.068/100 = 9.845 - 0.204/100 = 9.843. The actual answer is 9.843188.

I suggest learning log 1.5 and log 2.5 because the last step was assuming that the log curve is made up of line segments. Normally, it's good enough, but from about 1 to 3, the curve bulges out considerably, so the approximations are further off. For example, if we tried to calculate log 1.5, we would do 0.5(0.301 - 0) = 0.150, which is off by more than 1 part in 6.

I also suggest remembering that log e = 0.434 for conversion purposes. One of the ways I used this recently was to estimate the size of factorials. If we wanted to guess the order of magnitude of 100!, we would take the log. But instead of taking the (base 10) log, first take it in base e. ln(100!) = ln100 + ln99 + ln98 + ... + ln1, which can be estimated by integrating lnx, which gives 100(ln100 - 1) - [1(ln1 - 1)]. I'm going to ignore the +1 until the end. 100(ln100 - 1) = 100(log100/loge - 1) = 100(2/0.434 - 1) = 100(100/21.7 - 1). 21.7 * 4 = 80 + 2*4 - 0.3*4 = 86.8, which is 13.2 off of 100. 13.2 is 11.2/22.7 + 2/22.7. The first term is about 0.5, the second about 0.1, so 2/0.434 is about 4.6. From this, I can guess that 100! is about e^361, or 10^(361*log e) = 10^(180.5 - 18.05 - 3.61 - 1.805 - 0.361) (taking 0.434 as 0.5 - 0.05 - 0.01 - 0.005 - 0.001)= 10^156.6. 100! is actually 10^157.97.

I saw a way to estimate logs and exponentials with knowledge of the ratios of musical intervals somewhere, but I can't seem to find it at the moment.

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### Re: Favorite mental math tricks/shortcuts

Blatm wrote:Estimating logs mentally:

First, memorize the following:
log2 = 0.301
log3 = 0.477
log4 = 0.602
log5 = 0.699
log6 = 0.778
log7 = 0.845
log8 = 0.903
log9 = 0.954

I read something like this in Feynman's book. I was pretty impressed so I found it worthwhile to memorize this table. I recommend it to anyone.