[HN Gopher] What use is mental math in 2022?
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What use is mental math in 2022?
Author : ColinWright
Score : 91 points
Date : 2022-02-28 15:51 UTC (7 hours ago)
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| ejb999 wrote:
| To prevent showing the world how truly ignorant a person you are,
| like this classic: https://www.youtube.com/watch?v=krHkjdnniDE
|
| from an editor of the New York Times, and 'the most trusted named
| in journalism' Brian Williams, not to mention all the other
| people along the way in producing this piece who couldn't do
| basic math, that allowed this 'story' to make it onto the air
| without anyone sensing that it was off by a factor of about a
| million.
| christiangenco wrote:
| This mistake is so interesting to me. Here's a great
| explanation of what's probably happening:
| https://youtu.be/6egeUxIEQnM
|
| tl;dr: mentally we take off "millions" as a unit, do the
| calculation, then add "millions" back on. This works for
| addition, subtraction, and inequality but fails for
| multiplication and division.
| ejb999 wrote:
| the fact that this basic math error made it thru so many
| people and still made it onto the air is very telling - and
| scary.
| conductr wrote:
| The math mistake actually doesn't surprise me. The lack of
| critical thinking to say, wait, is that right? Is a
| complete whiff.
| jihadjihad wrote:
| I'm thankful every day that throughout my education it was
| drilled into me _never_ to circle my answer at the end of a
| math problem until I first re-read the problem and asked if my
| answer made any sense.
|
| I feel for the math teachers out there who receive homework
| where a sink faucet spews 70.4 gallons of water per second.
| mtlguitarist wrote:
| One funny incident I had on this topic was when I was TAing
| calculus one student proudly proclaimed that there's no
| gravity on Mars while doing velocity problems.
|
| This didn't stop them from using the gravitational constant
| provided in the assignment or from getting the correct answer
| even, but it was funny regardless.
| MontyCarloHall wrote:
| That one is especially egregious because you don't need any
| mental math to realize the absurdity of the statement ("the
| $500 million Bloomberg spent on his failed presidential
| campaign could have given every American over a million dollars
| each"). Realizing that statement is absurd just requires a
| basic sense of how the world works.
|
| As an analogy, imagine someone said that "a cow that weighs 500
| million milligrams could give every American more than a
| kilogram of meat each." You don't need to know that 500 million
| milligrams is 500 kilograms and that this clearly would not be
| enough to give 300 million Americans a kilo of meat apiece --
| you just have to have an elementary enough understanding of the
| world that it's obvious a single cow cannot feed an entire
| country.
| rainonmoon wrote:
| The analogy falls a little flat because most people can
| conceptualise a cow easily enough, but most people have zero
| intuitive concept of how much 500 million dollars is.
| jodrellblank wrote:
| It falls a little flat because of that, but it should still
| be a quantity which feels suspicious:
|
| - One person could have enough to give every American $1M
| without first taking $1M from every American. Did you and
| everyone you know buy $1,000,000 of Washington Post or
| Wallstreet Journal subscriptions, for example?
|
| - One person could spend $1M-per-American without anything
| very interesting or noteworthy happening to the people
| taking that money in payment.
|
| - One person could spend enough for $1M/American on a
| Presidential run and lose; did Hillary and Trump and the
| other candidates also spend that much (WOW!) or did they
| spend nothing by comparison and still beat Bloomberg? (also
| wow!).
|
| - One Presidential run could cost so much; $1M/American is
| a lot more than Americans earn every year and therefore pay
| in tax every year, and handwaving debt away it's therefore
| many times more than government annual spending on
| _everything_ ; infrastructure, military, social services,
| schooling, etc. Suspiciously high.
|
| - A million per American is in the region of 340 Trillion,
| several times the world's GDP.
|
| "There are 10^11 stars in the galaxy. That used to be a
| huge number. But it's only a hundred billion. It's less
| than the national deficit! We used to call them
| astronomical numbers. Now we should call them economical
| numbers." - Richard Feynman
| booleandilemma wrote:
| It always amuses/scares me how much of a separation there is
| between people who _think_ and people who just parrot others.
| The people in that video just copied what they saw people
| commenting on twitter. It's ok though, they have nice looks and
| voices and it's impolite to call them idiots.
|
| It's how we end up with eggcorns. People just say things
| without understanding what it is they're saying.
|
| When your words don't have to adhere to reality, you can end up
| saying some ridiculous things, and the other parrots in the
| room will nod and agree with you.
|
| https://en.m.wikipedia.org/wiki/Eggcorn
| exolymph wrote:
| Early-stage covid was so revealing of this -- who actually
| thinks, versus downloading updates from NPR and CNN.
| BolexNOLA wrote:
| Interesting 2 choices there.
| ALittleLight wrote:
| Once I was in a meeting where someone was presenting from a
| spreadsheet. As the meeting was insanely boring I was
| entertaining myself double checking the math in the spreadsheet
| and spotted a formula error thanks to mental math. I don't recall
| that it changed anything, but the amount of smugness I felt when
| I pointed out the error and the presenter confirmed it was so
| great that I have a smug smile on my face even now, five or six
| years later, just thinking about it.
| curiousllama wrote:
| Happens all the time! Just be careful with this stuff - often,
| the small mistakes in unimportant stuff are just a junior
| futzing up something small. If it derails the meeting, you may
| be ruining some faceless 23 YO's performance review for an
| otherwise-innocuous bug.
| treeman79 wrote:
| Was in physics class and the professor put a problem on board
| to keep the entire class busy for the hour. She had grading or
| something. I immediately shouted out the answer at first
| glance. Did that several times.
| conductr wrote:
| It happens. I work in corporate finance and am presenting
| boring spreadsheet stuff daily. I/My team makes mistakes too
| and I don't always catch them before the presentation.
| Sometimes there's literally no time it's like dev on prod
| without tests.
|
| Anyways, what you did is very common. Many people are actively
| looking for mistakes in the math instead of focusing on the
| content of the presentation. And will derail my entire meeting
| when they find one. Just word of warning , don't be that guy.
| Usually it's some unimportant metric (or someone would have
| noticed) and directional correctness facilitates the same
| conversation. For example, oh you're right overtime is running
| at 10% instead of 8% is usually not a material fact when your
| department is 30% over budget and some productivity metric is
| where the businesses focus needs to be (eg. widgets/hour).
|
| Instead, bring it to my attention after the fact. Or shoot me a
| email while in the meeting. Only verbalize it if it's crucial
| to the conversation or impacts the decision being made. It's a
| bit like expecting bug free software, it's just impossible.
|
| Not saying you handled it wrong perse, most people do this.
| Just suggesting how I have been on the other side of this and
| prefer for others to act. I find this more professional as I
| wouldn't turn up to your product launch party and just tell
| everyone in the room about the GitHub issues log. Or, as
| another example, if I were your manager, I wouldn't call you
| out publicly but assume you prefer a private feedback session
| when you screw something up.
| kragen wrote:
| If it isn't important whether the numbers are real or fake,
| why show them? To intimidate people with a baseless
| appearance of mathematical rigor? If so, I'd say the people
| who point out that they're fake are doing a service to the
| decisionmakers in the meeting, though it's unsurprising that
| the presenters don't appreciate it.
| conductr wrote:
| I have to answer in the context I'm familiar with, finance.
|
| People start aimlessly hypothesizing if you don't show them
| the nonissues.
|
| Also, management is conditioned to look at certain key
| things in combination because they are in fact
| interconnected.
|
| So for example, is the core issue is profit is lower than
| budgeted. I will show them sales are on plan. I will show
| them non labor is on plan. Then I will say now labor is the
| problem. Overtime is immaterial , but our hours/widget KPI
| seem way high. We call the plant manager and find out an
| automation process is down. Because it's so costly, we fly
| in a support tech that night to fix it. Seems simple enough
| but in an enterprise this could involve a dozen people and
| move slowly unless I craft the story of how much this issue
| is costing us. And since I'm not in the plant, I didn't
| even know the problem that I was pointing out to begin
| with. I probably don't even know who the plant manager or
| his boss is. So it's not like I can just pick up the phone
| and figure out the whole story (but generally would try
| to).
| kragen wrote:
| Thank you for explaining! In that case it seems like it
| would be worthwhile to sort out the mistakes in the
| calculation in order to figure out which of the seemingly
| implausible figures are in fact correct and which are
| just calculation errors?
| treeman79 wrote:
| For a couple of years I would always include an obvious UI
| fix in my presentations with senior management. I always
| got my plans approved the first time. Management was always
| so happy to contribute something that they didn't question
| anything I actually wanted. They never caught on.
|
| This was for CEO of a fortune 100
| conductr wrote:
| Yeah on the flip side of my comment I've known some
| finance guys that bury in mistakes just to see if anyone
| catches them. Obviously not on super official filings.
| But internal management stuff. One boss of mine early on
| in my career gave out prizes. It was their way of making
| interactive
| nayuki wrote:
| Related: https://blog.codinghorror.com/new-programming-
| jargon/#5 , https://softwareengineering.stackexchange.com
| /questions/1681...
| curiousllama wrote:
| You've never shipped with known bugs?
|
| If someone asked you "If it isn't important whether the
| software works or not, why ship it?", what would your
| reaction be?
|
| Note: > Directional correctness facilitates the same
| conversation
| Jensson wrote:
| A math error in a spreadsheet can change the entire
| conclusion to be drawn from the data, it is very
| different from known bugs users can work around.
| kragen wrote:
| I would say that most software doesn't have to be correct
| to be useful. There are exceptions, like anything to do
| with cryptography.
|
| You might say that the same can be true of calculations;
| after all, calculations about the contingent universe (as
| opposed to, say, number theory) are always based on data
| incorporating some uncertainty. But there are two key
| differences:
|
| 1 I don't put up a slide showing all the intermediate
| values of the variables in my program in order to make an
| argument for something. Instead, I try to show my chain
| of reasoning in enough detail to be convincing and to
| expose any relevant potentially incorrect assumptions or
| reasoning steps I've made so that others can find my
| errors, without including trivial or irrelevant details.
| Including trivial and irrelevant details makes it harder,
| not easier, to find relevant incorrect assumptions or
| reasoning steps.
|
| That's why I question the motivation of people who
| include those in their slides: it sounds like they're
| attempting to make their argument so complex that there
| are no obvious errors, rather than so simple that there
| are obviously no errors.
|
| 2 A 10% uncertainty in an input datum can be traced
| through the calculation from beginning to end and will
| often result in a 10% or 21% uncertainty in the result,
| or even less. (And where that's _not_ the case the
| sensitivity should be called out.) A calculation that 's
| simply incorrect--for example, treating millions as
| thousands, or forgetting to divide by the relevant
| denominator--commonly produces results that are off by
| multiple orders of magnitude.
| ALittleLight wrote:
| One reason I think these meetings are sometimes so boring
| is because there is a sense that we are looking at
| numbers that don't matter. That sense is reinforced if
| you discover that the values of the numbers we are seeing
| literally don't matter.
|
| I also think there is a difference between software and
| spreadsheets. At least, in the role I was in, software
| let us do things and spreadsheets let us reason about the
| business.
|
| If there was a bug in software, say, 5% of requests to
| save a configuration failed, and that failure triggered
| an automatic retry, and the result of this bug was
| elevated configuration save latency at the 95+
| percentile, we might decide to ship that bug (depending
| on other factors). On discovering the bug we would try to
| understand its impact and then we would reason about the
| impact on the software's goals. For example, we might say
| that people rarely change configurations, and an
| additional 100ms of latency for 5% of customers is
| something we can live with until such time as we solve
| the underlying cause.
|
| What we would not do, in any software org that I've been
| apart of, is say "Hey, we notice some of our operations
| are failing sometimes, anyway - let's ship it!" To me,
| that seems analogous to what you are saying. You know
| there are errors in the spreadsheets but you ignore them
| on the assumption they are inconsequential, reasoning
| that if they were consequential you or your team would
| have noticed them.
|
| There are ways for bugs to exist in software that don't
| block what the customer is trying to do. Maybe the bug
| makes it harder to accomplish a workflow, or makes it
| take longer, or looks silly, or something like that - and
| those are bugs it may be okay to ship with (depending on
| context).
|
| If values are wrong in a spreadsheet, then the only way I
| could think that doesn't matter is if the values are
| unimportant. That is, either the values are important and
| it matters if they are wrong, or it doesn't matter if the
| values are wrong because they are unimportant. If the
| values are important, let's correct them. If they are
| unimportant, let's not discuss them.
| kragen wrote:
| Ego. It's a hell of a drug.
| singularity2001 wrote:
| Conclusion (also from own experience): Mental math in 2022 can
| be a career boost and significantly increase your appreciation
| among peers.
| karaterobot wrote:
| Since he said "use" and not "practical use" or "lucrative use" or
| even "productive use", I'll answer for myself: I really enjoy
| passing time making fermi estimates and back-of-the-envelope
| calculations (sans envelope).
|
| For example, any time I go to a new restaurant, I try to work out
| the annual profits based on the time of day, cost of a dish,
| number of people, staff, location, etc. I don't want to open a
| restaurant myself, but it's fun to try to think of all the inputs
| I can, and do some arithmetic in my head to get an answer. It's
| an enjoyable way to make time pass while waiting for my meal to
| arrive. For me, it beats reading Twitter.
| mimimi31 wrote:
| I often pass the time while e.g. running or sitting on a train
| by approximating square roots of and squaring random numbers or
| calculating the day of the week for random days. I don't think
| I've ever used these skills for a practical purpose.
| milansm wrote:
| It would be great if you could get a correct (or close) answer
| every time if you made a guess, and where you made an error. I
| assume you would get better after a while and would be even
| more fun.
| corry wrote:
| Curious if others have had this experience too - many successful
| founder CEOs I've known through my career (working closely with
| them) are excellent at ball-parking / order-of-magnitude / back-
| of-napkin thinking off the top of their mind.
|
| There's something about it that IMO is an excellent 'tell' for a
| certain kind of intelligence especially for startups. I imagine
| part of it is that it's a good sign of 'technical skills' like
| fluency with math, and fluency with modelling scenarios. The
| other part is it's a 'demonstration of fitness' to others. Like,
| wow, our CEO is SMART and thinks FAST - great!
| squidbot wrote:
| Mental math is useful (vital) when playing poker or other odds
| based skill games! You'd be thrown out of any casino if you broke
| out your phone every hand :D
|
| And, this sort of goes with the article, it's really useful for
| negotiations/haggling where you may not want to use a calculator.
| [deleted]
| mathattack wrote:
| I find that it is hard to discuss strategy in real time with
| people who can't do mental math. They just can't keep up if they
| need to go to Excel every time there's a number.
| klyrs wrote:
| > When I'm on Zoom with a client, I can't say "Excuse me a
| second. Something you said gave me an idea, and I'd like to pull
| out my calculator app."
|
| You can't? Have you thought about investing in a separate device
| for doing calculations? A good scientific calculator is tens of
| dollars, and would probably blow your calculator app out of the
| water in terms of usability (mmm, those tactile buttons). They're
| even solar powered so you never need to worry about charging.
| Just because phones _can_ replace handheld electronics, doesn 't
| mean they _must_.
|
| Don't get me wrong, mental math is great, and in my work, I do a
| lot of off-the-cuff estimation with orders of magnitude and
| limiting behavior. But a calculator is way more precise, and can
| handle way more complexity than my poor little head. I tend to
| use a repl in my favorite language. But it sounds like the author
| is using their phone for zoom and can't switch windows for some
| reason.
| jmyeet wrote:
| We tend to think of the second one as "back of the evenlope"
| estimation, which is both pervasive and important in software
| engineering and related fields (eg capacity planning, resource
| allocation, etc). I wouldn't necessarily attribute this to mental
| math however. You can still be good at this without mental math
| (IMHO).
|
| I might rephrase it to something like: mental math is a
| reflection your ability to use numbers as one would use tools.
| So, for example, when you learn probability you use lots of
| examples of games of chance or you boil it does to coin tosses,
| drawing stones from bags or rolling dice. Part of doing this is
| being able to (among other things) correctly negate
| probabilities, which tends to be far more obvious if you have a
| foundation in set theory.
|
| Example: expected values. If you roll a d6 you expect to roll a 1
| approximately once in every six rools. You get an array of
| possibilites from this based on how you choose to look at it (eg
| the probability of rolling a 1 after N rolls or the probability
| distribution of M 1s from N rolls). A common question comes up is
| "what is the probability I don't roll a 1 in N rolls?" and those
| naive in probability will often get this wrong. The answer is of
| course 1-(5/6)^N. And while you do learn that in probability,
| even if you don't specifically learn it I find that people who
| are comfortable with numbers as tools (as evidenced by mental
| math) will tend to figure it out anyway, or at least a good
| approximation of it.
|
| Edit: corrected "not rolling" to "rolling". My bad.
| leephillips wrote:
| To see that your answer for "what is the probability I don't
| roll a 1 in N rolls?" is incorrect, consider what happens as N
| gets large. The probability - 1, which should be intuitively
| wrong.
| kragen wrote:
| Probably they meant (1- 5/6 )n.
| leephillips wrote:
| Also wrong.
| kragen wrote:
| Yes, but less implausibly so. At least it gives the right
| answers for _n_ = 0 and _n_ - [?].
| ekanes wrote:
| I used it all the time in business. You're talking to someone,
| and can know what you want in real-time, without saying, "ok I'll
| run the numbers and get back to you." Sometimes that doesn't
| matter, but lots of business is done casually/socially, and
| finishing the conversation/negoatiation while it's still over
| drinks at the bar vs on a call later can very much matter.
| curiousllama wrote:
| Data issues!
|
| I never bother to pull out my calculator to see how much
| 639/42536 is, but good mental math/intuition lets me
| thoughtlessly go "oh, that's only 1.5%" rather than "600+ seems
| like a lot"
| yummypaint wrote:
| Mental math can be like an extention of your natural intuition
| and vise versa. Ideas that wouldn't otherwise arise can
| spontaneously come to you because you have the feeling in the
| back of your mind that A<<B, A=B, etc.
|
| Sometimes it seems a problem has dozens of possible solutions.
| Mental math is great for whittling away the less optimal ones
| quickly.
|
| It has been said that the best way to have a good idea is to have
| alot of ideas. Mental math is the natural companion to testing
| and refining those ideas in an intuitive but quantitative way.
| jancsika wrote:
| > Of course nobody knows exactly how many piano tuners there are
| in New York, but you could guess about how many piano owners
| there are, how often a piano needs to be tuned, and how many
| tuners it would take to service this demand.
|
| You don't want to know how often a piano _needs_ to be tuned. You
| want to know 1) how often the average piano owner pays to get
| their piano tuned, and 2) how often the outliers (like Julliard)
| pay to get theirs tuned. For #1, _my own ears_ doubt there 's
| much if any relationship to how often they need to be tuned.
|
| I feel like this is a common problem with this approach--
| something about starting the journey leads one to choose an
| imagined ideal contingency rather than one that is practical.
| Almost like the deeper you go, the more likely the thing you're
| guesstimating will be a fantasy.
|
| Edit: clarification
| jodrellblank wrote:
| > " _Of course nobody knows exactly how many piano tuners there
| are in New York, but you could guess about how many piano
| owners there are_ "
|
| You could guess how many piano tuners there are. Why is a guess
| at piano owners expected to be any more accurate?
|
| (I know that's in the blog, and original question, not just
| your comment).
| jancsika wrote:
| Because a meaningful guesstimate for tuners needs a workable
| sample size, and it would generally be too close to zero to
| tell if any exist.
|
| At the time Fermi lived a metric shitton of abodes housed a
| piano-- that's almost certainly why he used that example. You
| could just peruse a block of abodes to get a decent estimate
| (or simply think of one in your head). But in general there
| are fewer tuners than there are pianos, so even at the time a
| rough estimate for a city block could return zero.
|
| Even today, if I walk a city block I'll find six buildings
| that house a piano. But I'd probably need a 20-mile radius
| before one of two piano tuners come into range.
|
| Another question occurs to me-- when I peruse the pianos in
| the block, what is the chance I'll actually see one of the
| tuners doing the tuning? Smart tuners leave their card inside
| the piano with a list of dates of tuning-- if I have a chance
| I'll sneak into one of the houses of worship and see what the
| frequency is.
|
| I'll bet it's more likely I'll spot some deer meandering on
| the way.
| spaetzleesser wrote:
| "Almost like the deeper you go, the more likely the thing
| you're guesstimating will be a fantasy."
|
| Like with every other method you need to know when to stop.
| jancsika wrote:
| Seems like people stop. Just rank speculation that the last
| level they go to before they did was probably not a great
| choice for a branch.
| melony wrote:
| Interviewing for trading jobs. It's the leetcoding of prop shops
| jake_morrison wrote:
| In consulting, I have had to train myself to not give specific
| answers in a sales call. Any number for cost or delivery date
| that you give a customer will become stuck in their head, and you
| will have to fight to change it to match reality.
| 1e-9 wrote:
| It makes you look like a genius rather than like this:
| https://www.youtube.com/watch?v=DHFB40WOMOo
| asdff wrote:
| Because you never always have a device on you. Sometimes its
| dead. Sometimes its charging at a dock across the house. Its
| ultimately a lot clunkier to pull up a calculator app and tip tap
| at a faux 4 function calculator than it is to just think.
| nofunsir wrote:
| New Math - Tom Lehrer (1965)
|
| https://www.youtube.com/watch?v=UIKGV2cTgqA
| jasfi wrote:
| The main benefit is that I know mental math makes my brain sharp.
| I can't believe the difference it makes, and that it is not more
| widely known and practiced. However, it takes discipline to make
| it a habit, which isn't a very popular thing.
|
| Any practical utility is a distant second.
| vsareto wrote:
| I don't doubt that estimations are useful, but you don't
| necessarily need to provide an estimation. Why not just say,
| "I'll need to look into the impact of that change and I can get
| back to you". What are you gaining by introducing an opportunity
| for your head math to be wrong? You can still keep the
| conversation velocity by committing to providing an estimate
| later, especially a more accurate one. You might be making a
| careless error because you're doing the math during a meeting.
| gowld wrote:
| > You can still keep the conversation velocity by committing to
| providing an estimate later,
|
| conversation velocity, yes, but not project velocity, which is
| much more important.
| Jensson wrote:
| That is good in theory but most people will have forgotten most
| of the conversation by the time you get back to them with your
| calculated values. People don't care that much about what
| others do, in order to grab their attention you need to be well
| prepared and answer things quickly or you lose them.
| MathCodeLove wrote:
| It's not quite that binary. There are times where your approach
| is the best one, likely more times than not. But there are also
| situations in which it'd be advantageous to provide _some_
| concrete figure /estimate in the moment, even if its not
| completely accurate.
| spaetzleesser wrote:
| I find it infuriating when people can't come up with a ballpark
| number. You can and should check the numbers later but don't
| stop a discussion.
| vsareto wrote:
| I'd rather delay for an estimate than give something I'm not
| confident in. I'm most confident in estimations I've had time
| to think over, instead of trying to ballpark something. If I
| can ballpark something, it's usually because I happen to have
| done it before or it's an otherwise easy problem.
|
| I don't know why one is considered a superpower over the
| other though (superpower is often a code word for "skills I
| wish everyone had because I have them"). When your loose
| estimations propagate to others, there's always a chance of
| it being taken out of context too.
|
| If it's only you and a single person as your client in the
| conversation, usually things won't be taken out of context.
| If it's a larger org, it might become more than just an
| estimate.
|
| This might even be framed as optimistic estimates vs.
| pessimistic estimates. One is faster but more error-prone,
| the other slower but more accurate. Different tools for
| different situations.
| mkr-hn wrote:
| A lot of people have the unfortunate experience of being
| pressed for estimates and guesses only to be held to them as
| though they were 100% certain, and possibly punished for
| being wrong.
| kragen wrote:
| I think a "ballpark estimate" here means "between one hour
| and one month", not "between two and four days". That's
| more like second base than the ballpark.
|
| Ballpark estimates for software tasks are useful in
| situations like https://xkcd.com/1425/: before ImageNet,
| determining whether a photo contained a bird really would
| have been a multi-year research project with uncertain
| success, while determining if EXIF places a photo within a
| national park was a relatively simple GIS query. But this
| is not obvious to muggle stakeholders.
|
| Of course, for things like bug fixes, scientific
| discoveries, and market research, often even estimation of
| that level of precision is unavailable. It's true that that
| can be infuriating, but that's just the way it is. Will
| this new model sell to a thousand customers or a billion?
| You can't tell in advance. Will this bug take five minutes
| or five months of effort to fix? You can't tell until you
| find the cause. When will we discover the next new class of
| antibiotics? It may already have happened or it may take a
| century.
| RcouF1uZ4gsC wrote:
| > Why not just say, "I'll need to look into the impact of that
| change and I can get back to you".
|
| When you say that, do you then end the conversation about the
| change. Or do you keep on taking about the specifics of the
| change. If you keep on talking, and the change turns out to be
| infeasible, then you just wasted a bunch of time. However, if
| you stop talking and the change turns out to be feasible, you
| just wasted time that you and the other person had set aside.
| [deleted]
| lacrosse_tannin wrote:
| Combine this with memorizing a small amount of facts about the
| world, like "the population of the united states". If you know
| the population of new york, you're one step closer to the answer
| to that piano tuner example question.
| astroalex wrote:
| I have always been terrible at mental math. Numbers are goop to
| me. I can create systems which process the goop, but when I try
| to play with the goop directly, it's like the numbers are
| mentally "slippery" -- I can't get traction.
|
| I did okay in math in college (almost completing a math minor in
| addition to my cs major), but I continue to struggle with doing
| arithmetic in my head. I also have a hard time memorizing and
| processing dates and times. I can't even quickly recall the order
| of the months!
|
| However, once I move up a level of abstraction, I'm totally fine.
| For example, I successfully implemented some relatively complex
| date & time logic (with timezone messiness) on a recent work
| project. I almost think that my lack of intuition around dates
| and times helped me there, because it forced me to logic my way
| through each and every part of the system.
|
| Just don't make me touch the goop directly!
| lmkg wrote:
| Are you aware of "Dyscalculia?" It's similar to dyslexia, but
| with numbers. Less-studied, so less-well-understood. On the
| surface, it sounds similar to what you describe: no issue with
| math per se, but numbers in particular try to run away from
| you.
| ddingus wrote:
| Do people not need estimate scale order of magnitude check and
| any number of other common tasks?
|
| I hold a number of mental models that I used to make all sorts of
| decisions. These don't need to be precise oh, but I often need
| them to be fast and or handy.
|
| Sometimes one needs the math just to think about things.
|
| Always needing a machine transfer the data in and out input and
| all of that is a real pain in the ass sometimes. Who wants that
| dependency?
| spaetzleesser wrote:
| Being able to do quick basic math is also very useful outside of
| business. A lot of people have no idea how much financing of
| something like a car will really cost them. Or the decision buy
| vs rent. Obviously there are a lot of other factors and you will
| want to check your numbers before making a final decision but I
| find it very useful to make a quick calculation as a baseline.
| lordnacho wrote:
| Having anything in cache allows you to check for errors and
| requirements.
|
| So for example if you're going to be away for 7 days and you need
| to spend $100 a day in notes, you might be able to avoid hitting
| a withdrawal limit, eg if you bank for some reason limits your
| weekly to $500.
|
| Basically the estimate serves as an input into your requirements
| for the next step, and tells you whether there's something to
| worry about.
|
| Same as if you're painting your house and you need to know
| whether you need to go buy a new bucket of paint or the existing
| one will do.
| Jtsummers wrote:
| Free resource that has come up a few times on HN in the past:
| _Street-Fighting Mathematics_ by Sanjoy Mahajan.
|
| https://mitpress.mit.edu/books/street-fighting-mathematics
|
| Click on "open access" to get to the PDF link.
| sitkack wrote:
| If the number you are calculating will be used as a lower
| bound, then rounding all the items up to the nearest 10s place
| and then using a little factoring can get one to the go/nogo
| solution.
|
| "can't be more than X"
|
| or
|
| "can't be less than Y"
|
| are powerful devices that allow one to keep thinking, rather
| than making tasks to be done later. But like the article
| mentions in the comments, if precision is warranted, then one
| needs to slow down and double check their work.
|
| As mentioned in the preface to Street Fighting Mathematics,
| "How to Solve It" by George Polya is another very approachable
| book. This tl;dr is pretty good [1] (aside, this book should be
| in the public domain by now)
|
| [1] https://math.ucr.edu/~res/math138A-2012/polya.pdf
|
| There are other books that train folks on how to do mental
| mathematics like
|
| * The Trachtenberg Speed System of Basic Mathematics
| https://www.abebooks.com/products/isbn/9780285629165/3011837...
|
| * Secrets of Mental Math
| https://www.abebooks.com/products/isbn/9780307338402
|
| I found this neat publication from the American Mathematical
| Society on Back of the Envelope calculation.
|
| https://www.ams.org/publications/TEXT-33-chap-1.pdf
| MandieD wrote:
| Mental math can give you a quick idea of whether or not something
| is worth investigating more closely.
|
| I play a guessing game at the grocery store after I load my cart
| onto the belt and am waiting for the checker to finish. I'm
| usually within 5% when there's a large variety of things and
| produce is involved, and have been alerted that I needed to check
| the receipt more carefully than usual.
|
| This habit helps me be more confident at restaurants in a place
| where I'm a second language speaker - and when discussing pricing
| with vendors in that second language.
| frontman1988 wrote:
| Practising mental maths at a young age probably also helps in
| improving working memory. Brains of young people are so
| malleable, it's very important to do the hard work even if it's
| drudgery cause it helps improve the thinking capabilities.
| ogogmad wrote:
| Do people just memorise their times tables up to 100? Is that
| what it takes?
| ColinWright wrote:
| It's a sense of what numbers are. As a decimal, 1/6 is
| 0.1666.., so that will mean that 6 times 16 is around 100.
|
| How many seconds in a day? It's 60 times 60 times 24, but 24 is
| around 25, which is 100/4. So it's 60 times 15 times 100, and 6
| times 15 is about 6 times 16, which is 100, so it's 10 times
| 100 times 100, which is 100,000.
|
| But that's too big by about 4% (because we used 25 instead of
| 24) and another 6% or 7% (from using 16 instead of 15) so it's
| about 11% smaller.
|
| So seconds in a day is about 88,000.
|
| All small numbers, few facts, but being comfortable in using
| them quickly with a side order of sanity checking.
| [deleted]
| tombert wrote:
| I can't speak for anyone else, but for my mental math (which
| admittedly is only so-so, not spectacular), I make pretty
| liberal use of factoring. Want to multiply a number by 15?
| Multiply it by ten, then multiply it by five, and add them
| together.
|
| Usually that's good enough to get me a rough number, if I need
| a precise number I whip out my phone.
| ColinWright wrote:
| Or if you want to multiply by 15, multiply by ten, then add
| on half again.
|
| So 34 times 15 is 340 plus half of 340, which is 170, so the
| answer is around 500 (actually 510).
|
| But if you want 34 (specifically) times 15 there are other
| ways to go.
|
| Or (34 times 15) is (17 times 30). Six times 16 is around
| 100, so three times 15 will be half that, or around 50. So 34
| times 15 is around 500.
|
| Or 34 is about 100/3, so multiply that by 15 is 100/3 times
| 15, which is 100 times 15/3, which is about 500.
|
| Being good at mental arithmetic and estimation isn't about
| being able to do sums quickly and accurately, it's about
| being happy with rough calculations, retaining a sense of how
| accurate you are so you can fix it later (if necessary).
| jonsen wrote:
| > So 34 times 15 is ...
|
| We who knows our powers of two do 34 ~= 2^5
| 15 ~= 2^4 2^5 * 2^4 = 2^9 = 512
| shadowofneptune wrote:
| Memorizing the tables past 12 is something stressed in older
| education, but isn't necessary. It really comes down to knowing
| strategies to simplify the calculation. If you can round the
| operands to a more easily used number, say 23 becomes 22 or 20,
| it's much simpler to do the mental math.
| chousuke wrote:
| Do people actually do that? I never bothered and made up my own
| method instead.
|
| 10x is trivial. 9x is 10x - x, which is also trivial. 5x is
| also trivial because it's 10x / 2. For the rest I just
| memorized whatever stuck the easiest and calculate the rest.
| For example, 7*4 is not automatic for me but it becomes 7*2*2
| which is easy.
| raghavtoshniwal wrote:
| If you're quick at simple calculations early on in your life, it
| probably has a compounding effect on the other aspects of your
| life. Not sure how you'd prove it but alot of people who have a
| Math/Science acumen, do so because they're above average early on
| and it gets reinforced. Mental math could give one that edge.
| Workaccount2 wrote:
| I am excellent at arithmetic (mental math) and can do "beat the
| calculator" party tricks.
|
| I fall flat on my face with actual math (algebra onwards).
|
| Thankfully, despite the teachings, the engineering I do is
| heavier on mental math skills. Anything more advanced and we
| use tools to do it.
| sockpuppet69 wrote:
| People who have more experience in a subject are relatively
| better at it.... Revolutionary.
| cinntaile wrote:
| His point was that it creates a snowball effect if you have
| positive experiences early in life. No need to be snarky.
| unfocussed_mike wrote:
| That is not quite what raghavtoshniwal meant now is it?
|
| The key word is "compounding", surely.
|
| For example, I think young children need a few good ways to
| be taught about managing and understanding very large
| numbers, about combinational explosion, precision and
| estimation; essential in today's world.
|
| I think kids from about seven or eight should be taught
| things about tree structures (decision trees, classification
| trees, parts diagram enclosures) but also about fractals --
| about measuring coastlines, etc.; perhaps even show them
| Cantor's Comb.
|
| They could be shown the liquid-nitrogen potato experiment to
| help them understand power series in future.
|
| I feel 2020 taught us that young people need to be taught
| about the rice-on-chessboard problem, about the Small World
| experiment, etc.
|
| There are good few kid-friendly ways to teach concepts like
| this which _compound_ in building understanding that current
| adults do not have.
|
| Kids in the UK are already taught some quite innovative ways
| to estimate, multiply and divide; maths has changed a bit
| since I was a kid. But there's a long way to go.
| ravi-delia wrote:
| Maybe UK kids are smarter, but I feel like fractals are a
| lot for 7 to 8. Maybe 10 or 11? At least let them know how
| decimals work first, otherwise fractal dimension wouldn't
| even make sense.
| unfocussed_mike wrote:
| (UK kids are not smarter!)
|
| Just to add when I mean fractals at 7 and 8, I don't mean
| the hard maths or the programming, for sure.
|
| But some ways to imagine the context of the hard maths.
|
| Like how to _recognise_ fractals (or just self-
| similarity) in nature, or explore drawing some by hand,
| that is a really crucial thing that a seven or eight year
| old should be introduced to. Why are leaves like trees?
| Why is broccoli simpler than it looks? River deltas,
| snowflakes, you know...
|
| Kids are shown lots of these things -- and they often
| notice self-similarity without prompting -- without being
| told that there is a unifying theory. The unifying theory
| is amazing in and of itself, and can be demonstrated with
| a Logo turtle.
|
| (Which brings me onto another thing... where the heck are
| all the logo turtles)
|
| There's a long and broad tradition in the UK of the
| "Christmas Lecture" now (started with the Royal Society
| which is televised; it's what a TED talk is, but better).
| All-ages family learning, made fun and accessible.
| petercooper wrote:
| It's super useful as a way to have confidence in
| computer/calculator generated values. I have quite often seen
| people perform calculations and claim their answer is correct but
| it's out by orders of magnitude because they didn't bother to
| even think about the basics of what their answer should be.
| sokoloff wrote:
| Another super-power is dimensional analysis. People make sloppy
| mistakes all the time by ignoring this. (
| https://en.wikipedia.org/wiki/Dimensional_analysis )
|
| Between that and simple order of magnitude approximations, it's
| distressing how often seemingly carefully prepared materials
| contain errors that can be spotted in a 5 minute reading.
| lanstin wrote:
| A similar thing that drives me nuts is when people report
| quantities that only have interest as a flux over some time,
| like eg tax cuts or investment costs or whatever and they just
| report the dollars. Is the tax cut 1 trillion USD / year or
| decade or century? It isn't just 1 trillion USD that is a
| meaningless statement. If we switch to AWS we save how much per
| month? Not just how much.
| gowld wrote:
| More generally, the slightly gauchely named "Street-Fighting
| Mathematics" http://streetfightingmath.com/
|
| http://web.mit.edu/6.055/
|
| * Divide and conquer * Abstraction * Symmetry and conservation
| * Proportional reasoning * Dimensional analysis * Easy cases
| (plugging simple values into complex formulas) * Lumping
| (discrete approximations of continuous functions) *
| Probabilistic reasoning * Springs (approximate complex systems
| as simple oscillators)
| jihadjihad wrote:
| > it's distressing how often seemingly carefully prepared
| materials contain errors that can be spotted in a 5 minute
| reading.
|
| It really is remarkable how many mistakes like this happen. One
| I've seen _a lot_ is not knowing how percentages work,
| particular when it comes to changes between them. Going from
| 10% of something up to 15% is _not_ a 5% increase--it 's a _50_
| % increase.
| LordDragonfang wrote:
| The issue is that, especially with very small percentages,
| the absolute percentage change is often more important to the
| discussion than the relative change, especially when we're
| talking about risk.
|
| In particular I remember there were a number of breathless
| articles about covid vaccine side effects where they were
| talking about a 50% increase in incidence - from 2 in 100,000
| to 3 in 100,000*. That's not something the average person
| needs to factor into their risk model, and headlining it as a
| 50% increase makes it seem more significant than it is.
|
| *not the actual numbers, but around the right order of
| magnitude, fit to the example.
| rland wrote:
| That still trips me up from time to time. I'm not sure why
| anyone ever uses percentage increase to communicate anything.
|
| Just say the quantity increased "from 0.0015 to 0.002" or
| something.
|
| It's particularly prevalent in news announcement of
| scientific studies. "New drug decreases x by 350%!"
| eindiran wrote:
| Percentages are useful because they communicate scale
| without deep context. Who knows what we are talking about
| at all with regard to 0.0015? What units are we using? Most
| people don't have the context necessary to evaluate what it
| means that we started at 0.0015 foos and are now up to
| 0.002. But the percentage abstracts most of that away: on
| some metric we care about, this intervention increased the
| output by 33%. That immediately suggests the scale of the
| effect, without requiring anyone to walk out into the
| weeds.
|
| Obviously this can and is constantly abused, but it is
| undeniably useful.
| thfuran wrote:
| If everyone in the room implicitly understands all that
| elided context, sure. But I think there are very few
| contexts in which stripping away all context is useful
| when the audience doesn't actually understand it. Then
| you're not really just streamlining things, you're
| removing all meaning. Is 50% more Foo a lot? Who the hell
| knows. But it _sounds_ like a big deal.
| jonsen wrote:
| Just solve this one, and you will love percentages forever:
|
| Is x% of y the same as y% of x?
| jihadjihad wrote:
| This is a great trick to use if you're trying to find out
| a percentage mentally and get stuck. For example, if you
| can't immediately come up with 14% of 50, it turns out
| that you can just do 50% of 14 and arrive at the same
| answer thanks to the commutative property.
| pjot wrote:
| In my head I convert percentages into simple
| multiplication.
|
| So 14% of 50 becomes 1.4 * 5 == 7
|
| Or to simplify even more...
|
| 1 * 5 = 5
|
| 4 * 5 = 20 -> remove the "0"
|
| 5 + 2 = 7
| leephillips wrote:
| Is that easier than half of 14?
| lanstin wrote:
| Or trying to figure out the basis for the percentage. 90%
| faster! And how about a percentage of percentages. The APR
| will go up by 10%. Condition A saved 10% but Condition B
| saved 50% more. (Is that 15% then?)
| sokoloff wrote:
| My pet peeve is something that used to 200 units and is
| now 300 units is accepted to be "150% larger!" despite
| that being non-sensical mathematically. (IMO, it's "50%
| larger" or "150% as large" but not "150% larger".)
| xwdv wrote:
| Percentages were created for clickbait.
| whoomp12342 wrote:
| when all of our calculators are SAAS on the cloud, and then the
| stock market crashes and all cloud vendors go offline, you will
| be thankful you memorized your multiplication tables! or atleast
| thats what I will tell the kids
| [deleted]
| pyfan878 wrote:
| It's useful for the estimation part of system design.
| jihadjihad wrote:
| All good points, and the only thing I'll add is that, for me, the
| utility of mental math (and putting it into practice often) is
| that it strengthens your mathematical intuition--you start to
| know, and even _feel_ how numbers work. Having strong
| mathematical intuition that is reinforced by even basic
| /rudimentary mental math abilities is indeed a real superpower.
| gowld wrote:
| > you start to know, and even feel how numbers work.
|
| > mathematical intuition ... is indeed a real superpower.
|
| What is this power used for?
| teawrecks wrote:
| Once you are familiar with Fourier series expansion, you
| start seeing everything as a signal that can be broken up and
| manipulated in interesting ways. From silly stuff like a
| talking piano (https://youtu.be/-6e2c0v4sBM), to more
| complicated and useful inventions like jpeg, real time
| dynamic light rendering (spherical harmonics), and countless
| others.
| thrashh wrote:
| Saving you a shit ton of work down the road because you on a
| quick glance caught a weirdness.
| thomascgalvin wrote:
| Math is all around us, and it's nice to have a gut-level
| check on whether or not something makes sense.
|
| Say you're at a restaurant, and you and your other both
| ordered $25-ish dollar entrees, and $10-ish drinks, and split
| a $15-ish appetizer, and the bill comes closer to $150 than
| $100, something was probably mixed up.
|
| That kind of quick approximation gives you a nudge to check
| real numbers.
| watwut wrote:
| Basically, you will be able to "feel" when numbers are wrong
| - without doing calculations or making an effort.
|
| Then you of course have to check before you say something.
| But without that intuition you would not checked.
| mkr-hn wrote:
| The replies suggest a need to separate mental math from
| mental arithmetic. I can't do mental arithmetic despite a
| lifetime of trying to develop the ability, but still have no
| trouble with the mathematic intuition based on expected costs
| described in other comments. It's easy enough to round things
| to 0 or 5 and sum it up for comparison even if doing the
| actual numbers in my head just won't happen.
| spaetzleesser wrote:
| For deciding if it makes sense to buy a house or finance a
| car. Or if a high deductible health plan makes sense. It also
| helps with cooking. Math is all around you.
| jihadjihad wrote:
| Good question. For me, in my line of work, it allows me to
| very quickly and on the spot (similar to John's blog post)
| confirm or deny if something passes the sniff test. If
| someone says "but that will cost X!", it is a superpower to
| easily refute or corroborate that. Along those lines, if
| someone throws out that doing Y will _easily_ provide Z
| amount of value, simply being able to work through the
| assumptions the person made when coming up with Z, and
| offering up your own Z ' if necessary, is very advantageous
| in a live, synchronous setting like a meeting with a client.
| peteradio wrote:
| Error spotting during otherwise very quick meetings.
| ajkjk wrote:
| Everything, oh my god. (imo, it changes the whole way you
| think about the world).
| lanstin wrote:
| I think that is just called understanding the world.
| BizarroLand wrote:
| Even in that context, having a basic mental math model
| improves the quality of your results when you're trying
| to understand the world.
|
| For instance, a job offer. One job offers you $28/hr, the
| other offers you $55,000 a year with 2 weeks paid
| vacation time
|
| Simple math shows that $28/hr, with 2 weeks unpaid
| vacation time, is $56,000/year. ($28*2000), and can be
| ~$2.3k more if you don't take vacation.
|
| Understanding of the world and yourself may make you
| realize that earning $1k-~$3.3k less a year would be
| worth it to have a set income that doesn't change even
| when you take vacations, but it's nice to be able to
| quickly compare the stark math so that you don't
| immediately think "big number better".
|
| That's not the best allegory to explain the utility of
| math, but the basic concept has rung true in many
| situations anecdotally in my life.
| lanstin wrote:
| I totally agree. I find people that can't ball park
| various numerical things are perpetually astonished by
| things that are perfectly obvious if you see the numbers.
| Jensson wrote:
| We primarily understand the world in terms of numbers and
| equations. Human intuition for people only sees
| individuals, it doesn't see the scale of a population or
| a country. For example the war in Ukraine, people talk a
| lot about Putin and Zelenskyy and the emotional parts,
| but the best way to see what really happens is to look at
| a lot of logistics spreadsheets and how the numbers
| evolves. Then you understand better what Russia can do,
| how reasonable their threats are, get a feeling for
| timeframes etc.
|
| On the other side of the spectrum, many people don't even
| understand percent. So if an article says "1% of corona
| patients died", they don't understand what that actually
| means. They understand some people died, but not really
| what the probabilities are or if they should be worried.
| They don't really understand much at all about the world
| as even the simplest of analytical articles are beyond
| their understanding. They can pick up the emotions in the
| article and understand that 1% dying is bad, but if you
| told them that 1% dying isn't a big deal they would trust
| you as well, as the number doesn't tell them anything its
| just how you say it.
| ColinWright wrote:
| > ... many people don't even understand percent.
|
| A worryingly (to me) large number of people can't tell
| you what 3% of 200 is.
|
| Should we care? Should people have a clue about this?
| Rates of interest are quoted in percentages. Deaths from
| diseases are quoted in percentages. Growth rates of
| economies are quoted in percentages.
|
| If you have no idea about them, how can you make properly
| informed decisions?
|
| And yet people survive, so maybe it really doesn't
| matter.
| Jensson wrote:
| > And yet people survive, so maybe it really doesn't
| matter.
|
| They survive thanks to people who knows math. We can't
| run modern society without a lot of people who knows
| math, we would regress centuries, every natural science
| requires people who are good at math. There is a reason
| why the people who are decent at math are valued in every
| modern society, even soviet understood that part.
| BizarroLand wrote:
| Also, if you can't do 3% of 200, you can do 200% of 3 and
| get the right answer. (6). As far as I know this seems to
| hold true for all (percentage of integer) combinations.
| ColinWright wrote:
| X% of Y is always Y% of X, and that's because (X times Y)
| equals (Y times X).
| Jtsummers wrote:
| Percentages are just fractions: x% of y =
| x/100 * y = y/100 * x = y% of
| x
|
| It's not a "seems to hold true", it is true.
| vsareto wrote:
| Most people would just grab a calculator on their phone
| to do it. We invented calculating machines because doing
| arithmetic ourselves is annoying and error-prone. It's
| the smart move because we know the calculator is better
| than our heads for arithmetic.
|
| I think we should care if kids can't do it because they
| learned it recently, but who cares if an adult does it in
| their head vs. uses a calculator correctly?
| ColinWright wrote:
| The submitted article says it:
|
| > _" being able to do quick approximations in mid-
| conversation is a superpower."_
|
| So here's something to think about:
|
| * People who can do quick, rough estimations in their
| heads say that they find it useful, and that the
| experience is that they have a sort of superpower.
|
| * People who can't do it say that there's nothing special
| about it, and using a calculator is just as good.
|
| Perhaps people in the second group don't have the
| information needed to make an accurate assessment.
| vsareto wrote:
| I don't know, if your superpower only saves a minute or
| so during a conversation compared to pulling out a
| calculator and getting an estimate that way, I wouldn't
| really call that a superpower.
|
| The math doesn't add up for me, and I'm hesitant to
| believe anyone who says it feels like a superpower
| because that sounds like an exaggeration. What does that
| comparatively make people who use calculators to do
| estimates? Simple primates using an iPhone?
|
| Plus the article's reasoning is really bogus to me too:
| "When I'm on Zoom with a client, I can't say "Excuse me a
| second. Something you said gave me an idea, and I'd like
| to pull out my calculator app.""
|
| This feels like a forced problem to justify his
| conclusions. It goes away if the other party is willing
| to wait for you.
| lanstin wrote:
| "Siri what is three percent of 200"
|
| I don't know if the calculations are essential (tho I
| guess they are) but having the feeling for the numbers
| and the ways things become inevitable because of numeric
| truths and relationships is essential for understanding
| reality.
| ThatGeoGuy wrote:
| If you can get used to applying Bayes Theorem in your head as
| a second nature, it helps greatly as a quick dummy check with
| reality. Obviously this doesn't mean calculating hundreds of
| probabilities in your head at once, but it can be very useful
| as a litmus test for all manner of things.
|
| Knowing roughly what the value will come out to be if your
| priors are ~10, ~100, or ~1000x apart is very helpful when
| trying to discern if you're just being led down a wrong path.
|
| Generally speaking doing Monte-Carlo simulations in your head
| would be nothing more than a stress test on your brain, but
| the simple multiplications and divisions used in reasoning
| under uncertainty can take you quite far, and with very
| little effort. You can even double check your work in the
| future if you really felt you did a bad job, and you'll be
| able to point out exactly where the error was (to yourselves
| or others) because it's more formalized than "I had a gut
| feeling."
| reactspa wrote:
| In Thinking Fast and Slow, Kahneman describes a mental math
| exercise [1] which can turn on System 2. I love using it to warm
| up my brain sometimes.
|
| If anyone can share links to mental math resources that can be
| used instrumentally to improve brain performance, please do.
|
| [1]
| https://psychology.stackexchange.com/questions/16531/kahnema...
| gnicholas wrote:
| Last week I was overcharged for a large takeout order at a
| restaurant. They had billed me twice for one of the items, and
| when the total came up I asked for an itemized receipt. If I
| weren't at least somewhat confident in my ability to do mental
| math, I might have just let it go and figured their prices had
| gone up (as so many have recently.
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