Arithmetic / Math shortcuts

Today I learned that my co-worker figures a tip by multiplying by .2 on her cell phone (calculator function).

She was a bit dumbfounded when I pointed out that it was easier for me to multiply by 2 and then divide by 10, or vice versa, than to take my phone out of my pocket. She IS of an age where she can multiply by two in her head with no trouble, it just never dawned on her that this was usable to figure 20%. I then showed her how she could use division by two and an addition to figure a 15% tip for just OK service.

This got into a bit of discussion of similar shortcuts. One of my favorites is multiplying by 1.8 (for C-F temperature conversion) by first doubling, then subtracting 10% of that product…this I can do in my head, but not “real” multiplication by 18.

I frequently need to convert between inches and millimeters. Knowing that 1mm is very close to .040 inches is very handy, because multiplication and division by 4 is pretty easy, sometimes I do resort to two divisions/multiplications by 2 though.
Basically, 4 is a much handier number for top-of-my head conversion than 25.4 While the later is exact, the former gets me close enough that I know I need a 10mm wrench when the 3/8" won’t quite fit over the bolt head.

Recently, on these boards, a metrication proponent cited the convenience and logic of a system where 1 liter of water has a mass of 1Kg…just another of the many people that don’t seem to realize a pint of water weighs exactly a pound* I don’t need to use this much, but it does make it trivial to remember that water density is 8 lb/gal…which I do use a lot.

Finally, it seems that most folks (the ones who can do it at all) figure “price with tax” by calculating price * tax_rate + price. Which is how I’d still do it in my head, or with pencil and paper. Using a calculator, though, this requires punching in the price twice. I find it a little easier to figure as: price * (1+tax_rate), as the addition of 1 can be done in my head. When I was working retail in college, this one completely mistified one of my managers “Kevbo does it wrong, but still gets the right answer!”

So what little tricks or approximations do other dopers use to make simple calculations even simpler?

** yet will still refer to 16 oz. beer bottles as “pounders”, presuming this refers to “pounding them down”
*

These probably aren’t terribly unique or special skills, but it’s part of how I astound my colleagues with my mental calculator.

When adding or subtracting numbers, I add enough of one number to the other to make it a round, easy to manipulate number, then dump the rest of the first number onto it. Example: 68 + 47, move 2 over for 70+45, move 30 over for 100+15, then dump the rest on it for 115.

Similarly, for multiplying, I break one number into its components. 35 x 11 is 30 x 11 = 330 and 5 x 11 = 55, thus 330 + 55 = 385.

Granted, I’ve done speed tests with someone else using a calculator, and I’m usually as fast as or slower than the calculator, so it’s not a preferable task when one is available. But that I can do so when one isn’t available makes a world of difference.

Multiplying by 11…for a 2 digit number, xy, the resultant is x (x+y) y,
ie, 23 x 11 is 2 (2+3) 3 = 253. If the x+y is great 10, then carry it over, just like normal addition.

Me too.

Tell me if I got this: a 9mm bullet x 4 / 100 = roughly .36"? A 30 caliber bullet (actually 30/100) / 4 is roughly 7.5 mm? Better test: .223 x100 / 4 = roughly 5.5? (Officially 5.65, right?)

Stole this from one of Feynman’s books: to square numbers near 50, square 50 (2500) then subtract 100 times the difference, then add in the square of the difference. So 48[sup]2[/sup] is 50[sup]2[/sup] - 100*2 =2300. Plus 2[sup]2[/sup]=2304.

47[sup]2[/sup] = 50[sup]2[/sup] - 100*3 + 3[sup]2[/sup] = 2209.

53[sup]2[/sup] = 50[sup]2[/sup] + 100*3 + 3[sup]2[/sup] = 2809.

And so forth.

When I was just a wee Balance, I learned to multiply by 9 on my fingers. Hold your hands out in front of you and, counting from the left, fold down the finger that corresponds to the number you’re multiplying by. (E.g: To multiply by 4, fold down your left index finger.) Count the fingers to the left of the folded one–that’s your 10s digit. Then count the fingers to the right, and that’s your 1s digit. (I still do this automatically, even though I know my 9x table perfectly well.)

To expand on Kevbo’s tip tactic, I generally ignore decimals and trailing zeroes when doing multiplication. Once I have the product, I just move the decimal one place to the left for each decimal place in the original numbers, and one to the right for each trailing zero.

When converting from binary to hex, split the binary number into sets of 4 digits, then convert each into a hex digit separately. Thus, 11010011->1101 0011->D3, or A4->1010 0100->10100100. When converting between binary and decimal, I usually find it faster to go through hex on the way.

When I need to “store” a number while doing other calculations, I sometimes encode it in binary on my fingers with a folded finger for a 0 and an extended finger for a 1. You can store values up to 1023 this way. 132, for example, would be both middle fingers extended…although you might get some funny looks for storing that one in the hand register. :wink:

If you want a faster, practical coversion between C and F, just double and add 30. Over the human comfort range, that gets you within a couple of degrees of the correct answer, and it’s a lot quicker.

Those are quite close. There is no agreement on wether to use the minor or the major diameter of the rifling when naming a calibre… so a .357 and a 38 calibre use exactly the same diameter bullets for example. In some cases the metric name will use the major diameter, and the imperial designation the minor. This is more difference than the above approximations.

There are about pi seconds in a nano-century.

Very handy when explaining to someone how the exponential time solution they came up with is going to take millennia (at least) to run.

Sales tax (VAT) in the UK is 17.5%. Although it’s always included in the price, sometimes I need to work it out. Someone pointed out to me it’s easily done; take 10% and 5% and 2.5%. So for a price of £35.99, the VAT is £3.60+£1.80+£0.90. Approximately:)

A physics teacher told me that a great loss, since even young kids use calculators now, is the ability to roughly check an answer. If you have an exam question with a set of unmanageable numbers, eg 78.86 x 0.3 x 1250 / 266, my estimated answer would be about 130 (80 / 3 x 5). If my calculator delivered an answer of 1112 or 11.12 I’d know I’d made an order of magnitude error, but this is much harder to spot if you can’t guesstimate the answer.

I learned long ago to do rote addition and multiplication from left to right. You get a useable answer in much less time.

Also, (1+x)^n = 1 + nx, for small values of x.

Also, remember the sequence 1, 1.25, 1.6, 2, 2.5, 3.2, 4, 5, 6.4, 8. These are 10 raised to the powers 0, 0.1, 0.2, … 0.9. So if you need log10(700) you can figure log10(7) must be about 0.85 and log10(700) must be about 2.85. And my calculator says 2.845098…

Finally, multiply that log by 2.3 and you have the natural log, if you need that.

I floored one guy in New Mexico who claimed to have been some sort of engineer during his alleged stint in the Air Force by pointing out to him that to multiply by 10, 100, etc, you need only move the decimal point the same number of zeros.

Rules of thumb that are not *too * accurate but close enough for government work:

For Celsius to Fahrenheit, take the temp, add 15, then double it. For kilometers to miles, divide in half, then add 10% of the original number.

See #7 above. This is the same formula, but I think it’s easier to double first, then add 30 than to add 15 and double.

What about shifting the decimal place?

For example, if I needed to know what 156 times 5 was, I simply take half of 156 ( 78) and then add a zero to get 780. Why take half? Because 5 is like 0.5, or half.
I do the same for .25s, .33s, etc. Anyone else?

In Wisconsin, the tax is 5.6%. So I normally take the tax, multiply by three and figure out my exact tip from there.
So if the tax is $3.68, I’ll figure 4*3=12. So I’ll put my tip somewhere between $10 and $15 depending on the service.

Yeah, I figured this out in 2nd grade. There were drills where we lined up at the chalk boards and the teacher would read out 2 numbers to add or subtract. You would write down the answer and put down the chalk. Whoever put down their chalk first with the right answer won.

I noticed that by doing the math left to write I could start writing down the answer as the second number was being read out. Right after the last digit was read, I’d finish the answer and put down the chalk while the other kids were just getting to figuring out the 2nd digit.

Of course this made be a “problem kid”. Nevermind that I got the right answer quickly using a method of my own invention. It wasn’t “the right way” and that was that. That told me a lot about the education system at a very young age.

For those of us not using metric yet:

To convert between ounces and pounds, or prices per those units, multiply/divide by 2 four times.

I don’t do it quite this way, but what I’ve often done is similar: I move to the number at double the distance from 50 in the same direction, divide by 2 and multiply by 100, and then add the square of the original difference.

Thus: 48[sup]2[/sup] is 46/2*100 + 2[sup]2[/sup] = 2304.

47[sup]2[/sup] = 44/2*100 + 3[sup]2[/sup] = 2209.

53[sup]2[/sup] = 56/2*100 + 3[sup]2[/sup] = 2809.

I apply this idea more generally to squaring numbers X that can be thought of as halfway between two numbers whose product is “nice” to calculate (for example, as above, if one of the factors is 50): to get X^2, you pick an A such that (X-A) * (X+A) happens to be “nice”, and then you add A^2 to that result. So, for example, to calculate 97^2, you go to 100 * 94 + 3^2 = 9409. To calculate 18^2, you go to 20 * 16 + 2^2 = 324. That sorta thing.

Here’s a handy way to add 1 + 1:

First, define the natural numbers as the set N, such that N is the smallest set satisfying the following axioms:

  1. The number 1 is in N.

  2. For any X that is in N, the successor of X (X’) is in N.

  3. There is no X such that X’ = 1.

  4. If X does not equal 1, then there exists Y in N such that Y’ = X.

  5. If S is a subset of N and 1 is in S and X in S implies X’ in S, then S = N.

Now, define addition as follows:

If A and B are both in N and B = 1, then A + B = A’. If B does not equal 1, then posit C, such that B = C’.

Then from (1), (2), and (4), A + B = A + C’.

Now, define the number 2 as the successor of 1:

Let 2 = 1’.

Thus defined, 2 is in N by (1) and (2)

Okay. Now, hypothesize that 1 + 1 = 2, and prove:

From the definition of addition, let B = 1. Therefore, A = 1.

1 + 1 = A’.

A’ = 2.

Therefore, 1 + 1 = 2.

This is especially handy when you don’t have a calculator.

For those who find this to be a bit much, don’t worry. Axiom 5 is redundant given the previous mention of “smallest set satisfying…”. Upon accounting for that, I’m sure you’ll agree this shortcut is quite pleasingly compact…

(Actually, if you drop the unused axioms 3 and 5/“smallest set”, you get a nice general proof that applies in all sorts of structures (reals, complex numbers, surreals, cardinals, integers mod 20, Diophantine polynomials, whatever). Best of all, you get a proof which could be used by even those who don’t subscribe to the rank heresy of excluding 0 from the natural numbers. :)).