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January 17

mental arithmetic - how are you supposed to find a reciprocal quickly

EDIT: Summary - "The question is, given that 16% is $500K, what is the whole" (thanks Meni Rosenfeld for this phrasing)

Original phrasing:

I'm watching shark tank,when they say like "$500K for 16%" I have to divide 100 / 16 and then multiply by 500K to get the valuation, but I don't know what 100/16 is. I have to put it into google. I know some numbers like obviously 25% goes 4 times (so money times 4), 10% goes ten times, 5% goes twenty times. For other numbers, like 16%, how am I supposed to know? Do people kind of memorize more numbers, or how do they know what to multiply by? (how do you do it in your head if you hear 7% or 16% or 12.5% etc.) 212.96.61.236 (talk) 20:49, 17 January 2015 (UTC)[reply]

Yes, I memorize them: 25%=0.25=1/4 so 12.5%=0.125=1/8.
1/7 is close to 1/8 and once upon a time I calculated 1/7≈0.14 (because 7×14=98, close enough to 100), that is 14 percent.

However I still don't remember what 1/16 is in decimal, anyway it's too long to use in full-precision calculation. If I were to calculate 500/16 I would probably replace 500 with 400, then 400/16=400/4²=100/4=25, then add 1/4 of that (to get back from 400 to 500).The fourth part of 25 is about 6, so the result is approx. 25+6=31. --CiaPan (talk) 22:40, 17 January 2015 (UTC)[reply]

16²/₃ percent is one sixth, so for 16% or 17% you can just multiply by 6 and make a slight adjustment. Dbfirs 23:20, 17 January 2015 (UTC)[reply]

THIS IS REALLY USEFUL!!!! I never "memorized" (noticed) what 1/6th was (I would just have put it into a calculator just now). Would pretty much everyone else here say that they did know offhand that 1/6 is 16²/₃%? (That you "know" it just like you know a quarter is 25% or a third is 33.3%?) Am I the only one who hadn't 'memorized' it yet, i.e. is it like not knowing the order of the months? :) I'm curious whether it's something everyone else knows offhand. --212.96.61.236 (talk) 22:46, 18 January 2015 (UTC)[reply]

This is considered basic, yes. I expect most of the commenters here knew that. One thing though, I'd generally think about it as 16.66...% rather than 16²/₃%. -- Meni Rosenfeld (talk) 22:57, 18 January 2015 (UTC)[reply]

You weren't brought up on pounds shillings and pence, Meni. Why would you think of two thirds as as

\lim _{n\to \infty }\sum _{i=0}^{n}{\frac {6}{10^{i}}}

?. Dbfirs 00:02, 19 January 2015 (UTC)[reply]

Not sure I understood the question, but - for the same reason that we're talking about percentages in the first place. I usually work in decimal. -- Meni Rosenfeld (talk) 13:18, 19 January 2015 (UTC)[reply]

Those with mad Unicode skilz spell it 16⅔% —Tamfang (talk) 06:35, 19 January 2015 (UTC)[reply]

It may help to factor the denominator into its prime factors. $500k*100/16 = $500*100/4/4 =$500k*25/4 = $125*25k = $675*5k = $3375k. This reduces the number of quotients you have to memorize.--Jasper Deng (talk) 23:30, 17 January 2015 (UTC)[reply]

Jasper, which quotients have you got memorized for example? 212.96.61.236 (talk) 02:02, 18 January 2015 (UTC)[reply]

1/2 = .5, 1/3 = .333..., 1/4 = .25, 1/5 = .2, 1/7 = .14... and all other fractions that can be expressed as the product of an integer and only one or more of those can be deduced.--Jasper Deng (talk) 03:34, 18 January 2015 (UTC)[reply]

Jasper, was that supposed to be exhuastive? Surely you have .1 memorized, .25 memorized, if nothing else. But do you have .125 memorized, .75 memorized, .08 memorized, or .06 memorized, for example? Walk me through how you do .08 if you would do it in your head right now (same as your examples which you have memorized.). 212.96.61.236 (talk) 17:47, 18 January 2015 (UTC)[reply]

You have an error, 125*5 = 625, not 675. The final result should be $3125.
In this case, it's simpler to do division rather than multiplication. 500/16 = (500/4)/4 = (125/4) = 31.25. Multiply by 100 since you had %, and you get $3125.

-- Meni Rosenfeld (talk) 11:18, 18 January 2015 (UTC)[reply]

Am I missing something here? Why does anyone need to divide 100 by 16? But it's always interesting how people do these. For this specific example, $500K at 16%, I think "Oh, that's $1000K at 8%, so that's 80K." On the other hand, if it were $1234K at 16%, I'd divide the number by 5 (for 20%) and then subtract 1/5 of that (to get to 16%). I spent many bored hours in school figuring out shortcuts like that. --jpgordon^{::==( o )} 16:38, 18 January 2015 (UTC)[reply]

The question was not what is 16% of $500K. The question is, given that 16% is $500K, what is the whole. The context is investment in a company - given that $500K were invested for 16% of the company, what is the post-money valuation. -- Meni Rosenfeld (talk) 20:32, 18 January 2015 (UTC)[reply]

Yes, this is exactly correct. It's clearer the way you just said it, as several people didn't know what I was talking about (I don't blame them, I didn't contextualize well) I've put your phrasing at the top. Thanks. --212.96.61.236 (talk) 22:41, 18 January 2015 (UTC)[reply]

You're just taking a reciprocal, I call it "dividing 100 by 16%" but it's just the reciprocal. Like, say I offered you $50K for 10% of your house, and you were trying to figure out if it's a good deal. You know that if you're paying $50K for 10% then 100% would be $500K. (because the reciprocal of 10% is 10, so you multiply $50K by 10). So you know that "values" your house at $500K and if the whole thing is worth more than that it's a bad deal, if it's worth less than that it's a good deal. If I had said $50K for 20% you'd be like, okay, so that values it at 50 * 5 = $250 (because 20% goes into 100% five times), if I had said 5% you'd have said, wow, okay, $50K * 20 = $1 million (because 5% is 1/20th). Because the reciprocal of "5%" is 20. Another way to say that is 100/5 = 20. So we're just talking about how many times a percentage goes into 100. 10% goes ten times, 5% goes 20 times, 33.3% goes 3 times, 3% goes 33.333 times, and so forth. And I was wondering how people did this in their head for other percentages.. . . 212.96.61.236 (talk) 17:47, 18 January 2015 (UTC)[reply]

Reciprocals are like times tables, powers of 2, things you pick up through use. It's easier for me to write down the ones less than 20 that I don't know or can't quickly write down: 13, 17 and 19. All others are known or can be written down from them. After that converting to and from percentages is just dividing and multiplying by 10 or 100, which is easy. Multiplying by something like 5 or 500 is also straightforward: times by 10 and half it, or double twice and add the original. Often when dividing in particular you can multiply by two to simplify before doing any real calculations. E.g. 500/14 = 250/7 = 35 5/7 = 35.714. Or you calculate 25/7 = 3 4/7 = 3.5714 then multiply by 10 (the sevenths, 1/7, 2/7 etc. are easily remembered as they are the same six digits 142857 rotated).--JohnBlackburne^words_deeds 18:11, 18 January 2015 (UTC)[reply]

Take a flexible piece of paper of length 50 units. Count the number of times this wraps around a circle of radius 1 unit (about 8). Then multiply by 10. So 80K. This also works for calculating tips at restaurants. Sławomir Biały (talk) 20:49, 18 January 2015 (UTC)[reply]