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Math Government United States Politics

At Issue In a Massachusetts Town, the Value of Two-Thirds 449

An anonymous reader writes "In Truro, Massachusetts (a town on Cape Cod), a zoning decision came up for vote, where the results were 136 for, 70 against. The vote required 2/3 approval to pass. The Town Clerk and Town Accountant believe that since .66 * 206 is less than 136, the vote passes. However, an 'anonymous caller' noted that a more accurate value of 2/3 would require 137 (or perhaps even 138 votes) for the measure to be considered passed. The MA Secretary of State and State Attorney General are hard at work to resolve this issue." Updated 20100422 23:55 by timothy: Oops! This story is a year old (rounding up), which I didn't spot quickly enough. Hope they've got it all worked out in the meantime.
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At Issue In a Massachusetts Town, the Value of Two-Thirds

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  • It should at least be rounded, if not just simply rounded up (i.e., ceiling). It's talking about people; you can't have 3.5 people, so if you want "more than 3" people then you need to go up to 4 people.

    • by lgw ( 121541 ) on Thursday April 22, 2010 @06:16PM (#31947676) Journal

      Rounding is not relevent here. They need 2/3 * 206 votes to pass. 137 is less than that value. 138 is more than that value.

      137 votes fails to be more than 2/3 of 206. Why would rounding even be a topic for discussion?

      • Hmmm, after looking at it again it looks like they *are* rounding up/ceiling.

        The issue is that they were rounding up an integer that was too small..

        (in other words, you're right, hehe)

        • Re: (Score:3, Insightful)

          by Bigjeff5 ( 1143585 )

          No, they definitely didn't round up, they truncated a number that was never accurate to begin with. 206*2/3 takes a half second longer to punch in than 206*0.66, if that. Why were they ever using 0.66 to begin with?

          The law says 2/3, use 2/3. It's not hard.

          • by flajann ( 658201 ) <.fred.mitchell. .at.> on Thursday April 22, 2010 @09:10PM (#31949752) Homepage Journal
            They would need 138 votes. 137 is less than 2/3rds. 0.66 != 2/3. Plain and simple.

            But I think it's amusing to say the least -- splitting hairs on a vote.

            Really, the who notion of voting is severely flawed from a mathematical point of view. One extra vote makes all the difference between whether or not a bill is implemented. What is the intrinsic importance of making it 2/3rds? Why not 3/4ths? 1/2? 5/8ths? What is the significance of 2/3? Seems arbitrary.

            But then, that is the difference between law and mathematics, I suppose. 20 years and 364 days old, you're too young to drink, it's illegal, and there are sanctions. 20 years and 365 days -- 21 years old, and it's perfectly legal. But what is the significant difference in a person at 20 years 364 days vs. 20 years 365 days? Is there some sort of "maturity switch" that is magically flipped? Do the gods of time descend upon you and bestow you with something special?

            We humans make so much ado over meaningless arbitrary demarcations. Life situations are fuzzy and spread out, not the digital of "on/off". It all seems rather a bit silly! Splitting arbitrary hairs without real meaning.

            • by retchdog ( 1319261 ) on Thursday April 22, 2010 @09:39PM (#31950022) Journal

              Under my proposed law, during the years between 16 and 21 noninclusive, the probability of being chargeable, upon discovery, with underage drinking shall be determined by interpolation through a truncated logistic function. n'hey.

            • by TheLink ( 130905 ) on Thursday April 22, 2010 @10:10PM (#31950322) Journal
              While drawing these arbitrary lines is silly, it is often far sillier to not draw them in the universe we live in.

              Making silly arbitrary decisions is a necessary part of life. Life situations aren't that fuzzy except at the quantum level. Even little things like which hand to use, whether to breath in or out. And even if the Many Worlds Interpretation is correct, it's not that fuzzy in each path of the universe.

              Say a car is about to hit you, you could jump either left or right to save yourself. The neurons in your brain are going to have to make a decision. Say you jump right, you think all the neurons participating in the decision wanted to go right? I doubt it, some would have wanted to go left. But you cannot satisfy all of them. You can't go both left and right, unless you wait for the car to split you in two.

              Back to your question, there is no magical maturity switch. Some people never even become mature. So what? With our current technology we are not able to practically put you 60% in jail and 40% out of jail at the same time, just because you are actually "60% mature".

              And it's costly to put in all the shades of gray for the different percentages of "maturity". Some countries do cater for a few categories: juvenile prisons, probation, etc.

              So there are very many arbitrary lines in laws: when it's legal to abort a fetus/baby, when does a child become an adult.

              There's definitely much silliness that should perhaps be fixed. For example, in many countries you might be legally considered old enough to sign up as a soldier, but not do other "adult things". This to me is silly. If you are going to be old enough to kill others and risk your own life, you should be considered old enough to do the other adult stuff. Otherwise, you shouldn't be considered old enough to be a soldier (unless the country is in such a bad/desperate state that you might as be allowed to be a soldier).
              • Re: (Score:3, Insightful)

                by metacell ( 523607 )

                There are even better reasons to draw sharp, arbitrary lines - to make the outcome of the law predictable. If the line between between old-enough-to-drink and not-old-enough-to-drink was fuzzy, or the court was required to decide if you were mature enough to drink, it would be almost impossible for the individual to determine when it was safe to drink.

      • Really? Long division has been lost to the ages?

      • by Gudeldar ( 705128 ) on Thursday April 22, 2010 @06:26PM (#31947884)
        I think somebody needs to teach them that .66 != 2/3
      • by Aeiri ( 713218 ) on Thursday April 22, 2010 @06:52PM (#31948216)
        I just did it myself in Python and it is a rounding issue:

        206/3*2 = 136
        206./3*2 = 137.333333333334

        If you round the division down then do the multiplication, you get 136.
        • Re: (Score:3, Interesting)

          by b4dc0d3r ( 1268512 )

          I don't know much about python, but if you're using Integer data types, you're taking the wrong approach. I assume Python implicitly converts 206 to an integer, and that division of integers results in integers. 206.0 would be currency if not an actual float, then, making the results floats.

          But the best way to calculate fractional multiplication is get all of your multiplications done first, then do a single division. The last thing you want is a rounding error, so you do it last and it's the last thing

          • Re: (Score:3, Funny)

            by Stray7Xi ( 698337 )

            I don't know much about python, but if you're using Integer data types, you're taking the wrong approach... 206 * 2 / 3

            This story would be much more interesting if they did use integer division as 2 / 3 * 206. In fact since the law didn't declare the type for number_of_votes, I suggest we fall back to fortran conventions.

      • by Anonymous Coward on Thursday April 22, 2010 @07:23PM (#31948602)

        I don't believe in rounding. I truncate. It helps to always win 2/3 majority votes:

        2/3 is 0.66666... Truncated to integer = 0

        Total voters: 206
        To win the vote, we require at least: 206 * 0 = 0

        Therefore any number of affirmative votes constitutes a 2/3 majority.

        Numbers don't lie!

      • by Anonymous Coward on Thursday April 22, 2010 @11:24PM (#31950942)

        Y'all screwed up.

        2/3 majority in parliamentary procedure is taken as meaning there are at least twice as many votes for than against. That avoids the whole fractional vote issue, which is a nonsensical concept.

        In this case there were 70 against, which means there would have to be at least 140 for. Thus the motion fails.

        C'mon, people. This thing has to have happened more than a few times in the course of history.

    • by spazdor ( 902907 ) on Thursday April 22, 2010 @06:16PM (#31947678)

      Yeah, this strikes me as a pretty trivial problem to solve.

      If the process requires the approval of 2/3 of the voters or more, then the lowest whole number that satisfies this requirement is the lowest number of votes which can pass the motion.

      Fucking duh, Massachusetts.

    • by jedidiah ( 1196 )

      You would think that an "accountant" would know that 2/3 is actually .66666... and gets rounded to .67 at two digits.

      • by TheCarp ( 96830 )

        Actually, thats not even correct. Its a repeating number, but, its wrong to round in the middle of an operation. You always round the final value, not the intermediary value. You take 2/3 and then round, not round and then multiply.


        • How about, you multiply by 2 and then, at the end, divide by 3.

          That leaves the repeating decimal part at the very end of the calculation, where you can "round the final value" like you suggest.

          Whereas your suggestion "take 2/3" means you start out with the repeating decimal whose approximation was the source of the problem.

          In other words, the root of the problem is the idea of an absolute-precision decimal rendition of "2/3". You haven't escaped it. By doing this division first, you're stuck arbitrarily cho

      • pft, clearly you've been taught wrong. 2/3 is really .666666. Duh.

        • Re: (Score:3, Informative)

          by Bigjeff5 ( 1143585 )

          pft, clearly you've been taught wrong. 2/3 is really .666666. Duh.

          You're both wrong, 2/3 is two divided by three. To make 2/3 of any number, you multiply it by two and divide it by three. It's not hard, and no decimal will ever be as accurate.

          The 100% accurate answer is 137 and 1/3.

      • You would think any dumbass would know how to multiply fractions on a calculator - it doesn't take any fancy functions, just a very very basic understanding of what a fraction is.

        When it says 2/3, you use 2/3, not a decimal of anything. (206*2)/3, that's it. Done. The answer is 137.3~, and it's as accurate as is humanly possible. The only number of voters that satisfies the 2/3 majority requirement is 138.

    • Re: (Score:2, Insightful)

      by Anonymous Coward

      The scariest part is their accountant is the one who things .66 is 2/3. I'll bet their books are ALL MESSED UP

    • by duh P3rf3ss3r ( 967183 ) on Thursday April 22, 2010 @09:56PM (#31950166)
      This is so incredibly simple that I can't believe I'm reading scores of responses about significant digits and rounding, etc. For a motion to be passed by a 2/3 majority, at least twice as many people have to vote in favour as those who vote against. Since 136 is less than 70*2, the vote fails. No calculator required, no consideration of significant digits. It's the kind of thing a reflective schoolchild should be able to reason out, frankly.

      I think this is a symptom of a generation raised with calculators...
    • Re: (Score:3, Interesting)

      by FatdogHaiku ( 978357 )
      Interestingly, the U.S. Constitution was written with this passage about the census:

      Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons.

      So each non-free person (slave) counted as 3/5 of a person...
      That's Art
  • not quite 2/3 (Score:5, Insightful)

    by rla3rd ( 596810 ) on Thursday April 22, 2010 @06:13PM (#31947640)
    can't these people do simple math?

    2 / 3 = 0.66666666...
    106 / 236 = 0.660194175

    Whats the problem here? It didn't pass.
    • Re:not quite 2/3 (Score:4, Informative)

      by siwelwerd ( 869956 ) on Thursday April 22, 2010 @06:23PM (#31947848)
      Further, 137/206 is still less than 2/3. So they in fact needed 138 to pass. This is why I tell my students to never use decimal approximations; simply use the exact number.
      • Re: (Score:3, Interesting)

        by hldn ( 1085833 )

        i was taking this basic 'intro to' math class. i got the first exam back, and most of my answers were marked incorrect when i knew they were right. the supposedly correct answers were written on my test and some were relatively close, while some were not. perhaps you can guess where this is going.

        i asked the teacher what was up and as she was checking my test, i saw her answer sheet had her work shown on it and realized quickly what the problem was. yes, she had rounded all the fractions to two place dec

    • 106 / 236 = 0.660194175

      Evidently an article on math and/or writing skills is needed on slashdot as well...

    • by complete loony ( 663508 ) <Jeremy.Lakeman@gmai l . c om> on Thursday April 22, 2010 @06:27PM (#31947906)
      Personally I think their real problem is the number of seats they have. They should just drop 2 positions in their next election and this will never come up again.
    • > can't these people do simple math?

      One guy made a stupid mistake. It doesn't mean someone can't do math, it just means he got one math question wrong--the failure wasn't so much the bad math, it was (1) the failure of him to check his math a second time when the vote came out as close as it did, and (2) the fact that they didn't have someone else check it.

      It's okay to not notice an extra decimal place on a first approximation. It's not okay if it suddenly might matter because you're within a vote of n

    • Re: (Score:2, Interesting)

      by mysidia ( 191772 )

      "2/3 majority" does not mean 0.666666666666666666666666667 of the voters.

      It means that 3 times the number of supporters must be at least twice the total number of voters.

    • by igxqrrl ( 749937 ) on Thursday April 22, 2010 @09:41PM (#31950034)

      can't these people do simple math? 2 / 3 = 0.66666666... 106 / 236 = 0.660194175 Whats the problem here? It didn't pass.

      Me and my Pentium beg to differ.My pentium begs to differ.

  • This doesn't seem that hard.

    206 * (2/3) = 412/3 = 137 + 1/3

    I'd side with the commenter that more than 136 votes are needed. Now, whether or not you truncate the decimal or round it, I'm not sure. In this case it doesn't matter though, it comes to 137 either way.

    Obviously you can't have 1/3 of a vote.

    • by pluther ( 647209 )

      So, really, they need 138 votes.

      137 137 1/3, therefore it wouldn't pass.

    • Re: (Score:3, Informative)

      by ferrocene ( 203243 )

      137 votes does not give you 2/3rds. It is less than 2/3rds. If the law requires 2/3rds, in what situation would 137/206 be sufficient?

      Put it this way - put the equation into C++ and compile and see how it comes out.

      if( 137/206 >= 2/3 )

      You need 138 for that equation to be true.

    • 137 votes is not "at least 2/3" of 206, 138 votes is "at least 2/3". 137 votes is still less than 2/3. This isn't difficult.

    • Now, whether or not you truncate the decimal or round it, I'm not sure. In this case it doesn't matter though, it comes to 137 either way.

      Nope, it doesn't. The correct answer is 138 votes to pass. As you noted, 206 times 2/3 is 137 plus one third. 137 votes is less than two thirds of 206; it therefore doesn't pass either.

      (If the supermajority calculation is confusing, consider a conceptually-easier simple majority (1/2) case. In the hypothetical case of 101 voters, a pass is 51 votes - being the first integer greater than 50.5 - not 50.)

      Why is this hard?

  • (136 / 206 == 0.660194174757281553398058252427181) is less than (2/3 == 0.66666666666666666666666666repeating)

    The vote does not pass.

  • hmmm (Score:2, Funny)

    by Anonymous Coward

    Did the Town Clerk and Town Accountant ever work for Verizon?

  • by WahCheng ( 1543195 ) on Thursday April 22, 2010 @06:16PM (#31947692)
    Clearly 70 times 2 is greater than 136 Therefore there is NOT a 2/3 majority. The matematics of politics, however, is not like the math we all know and love....
  • In[1]:= 136/206 >= 2/3

    Out[1]= False

  • A few seconds with a calculator shows that 136/206 is less than two thirds. For the relentlessly pedantic, it works out to .66019... which is clearly less than .66666666...
  • 136/206 = 0.66019 137/206 = 0.66505 138/206 = 0.66990 2/3 = 0.66667 Clearly the measure does not pass with 2/3 of the vote. Even if you round off to 2 digits you would get 136/206 = 0.66, and 2/3 = 0.67, so it still doesn't pass.
  • by jarrodlikesmath ( 1795858 ) on Thursday April 22, 2010 @06:19PM (#31947734)
    is irrational
  • What is this, "Are you smarter than a 5th grader?" Wasn't it in Mass. where they

    136/206~0.6602, less than 2/3. The measure did not pass.

    I haven't kept up on Mass. politics, but hopefully the AG they're going to ask isn't Martha Coakley, who thought a glorified Lite-Brite was a bomb [].
  • by dpbsmith ( 263124 ) on Thursday April 22, 2010 @06:20PM (#31947768) Homepage

    The ratio of 2/3 to 1/3 is 2:1. In order for a measure to pass by a two-thirds vote, the majority must have more than twice as many as the minority. 136 is less than two times 70, so the vote does not pass.

    • Actually having twice as much is sufficient. They don't need more than twice. If it had been 140 to 70, it passes. And failing this kind of math, simple election and voting math, should be sufficient to remove both individuals from their jobs.
      • Re: (Score:3, Interesting)

        They could never have (with their current number of seats) 140 to 70. It would have to become 140 to 66, to match their 206 total.

    • Too easy perhaps (Score:3, Informative)

      by stomv ( 80392 )

      the majority must have more than twice as many as the minority

      No. A majority vote requires 50% + epsilon to pass. However, a 2/3 vote typically requires that the majority must have at least twice as many as the minority, not "more than twice as many". Which is to say, if there are three people voting, only two voting yea passes the bill, not three.

  • The Town Clerk and Town Accountant believe that since .66 * 206 is less than 136, the vote passes.

    Wow. Haven't RTFA because it would scare me, but if the summary is even halfway correct then we may as well say 2/3 is just 0.6. They'd only need 123 votes to pass...

  • ok - 206 = total of yeas and nays.

    But .66?? who thinks like that... unless they are lazy and using a calculator. Even .666 puts it over 137, but "666" is a bad number ;-)

    A grade school kid would quickly come up with "137 or more" without even getting into decimal places...

    (206 X 2) / 3

    Doing the long division, by hand yields 137 with 1 left over (e.g. 137 1/3) no arguing over decimal points. OTOH, this is law we are talking about, not math.
  • by PolygamousRanchKid ( 1290638 ) on Thursday April 22, 2010 @06:22PM (#31947814)

    The engineer pulls out his calculator, types in the results, and gives the answer.

    The mathematician goes to the whiteboard, and writes a proof for the answer.

    The politician whispers, "What do you want the answer to be . . . ?"

  • A 2/3 vote is easy to see if it passes. You must have at least 2x the number of 'yes' votes than 'no' votes. 136 to 70 fails because 136 is less than 140. It is as simple as that.

    Also, this news is almost a year old. April 30, 2009 is the date on the article.

  • by Paul Rose ( 771894 ) on Thursday April 22, 2010 @06:25PM (#31947864)
    >>the results were 136 for, 70 against. The vote required 2/3 approval to

    the question: is 136 / 206 >= 2 / 3 ??
    is the same as: is 3*136 >= 2 * 206 (multiple each side by 206 * 3)
    or: is 408 >= 412
    or: DID NOT PASS
  • by Jah-Wren Ryel ( 80510 ) on Thursday April 22, 2010 @06:26PM (#31947874)

    Date on the article is April 30th, 2009.

    So, does anyone know if basic math skills prevailed?

    • by Anthony Rosequist ( 1110043 ) on Thursday April 22, 2010 @07:45PM (#31948858)
      Yup []

      Voters did approve one of four petitioned zoning articles, one that would require cottage colonies to be in operation for at least three years before they can be turned into condominium ownership. Zoning articles require a two-thirds majority and the first vote was close, counted as 139 in favor and 64 opposed. A recount was held that was tallied at 136-70 and declared to be passed by Town Clerk Cynthia Slade, utilizing a multiplier of 0.66 to determine two-thirds, the figure the town has always used. Unfortunately, this vote was so close that the inaccurate fraction made the difference, and several months later the attorney general’s office negated the approval as not meeting the two-thirds threshold.

  • When there are more than 163 votes in total, going from 2 to 3 decimal places in the representation of 2/3 increases the number of votes required by one.

    Perhaps choosing a repeating decimal in your definition of a majority is not very smart.

  • 2/3 of 206 is 137.33333333... the town clerk and accountant fail at math. 66% is NOT the same as 2/3; in fact 2/3 would be 67% if you rounded it.
  • Why use decimals at all?

    Since I have yet to see a third of a person, that means you need 138 votes.

  • Geez...50 comments and NONE of them make mention that Truro, Mass not too long ago lost their Postmaster. Not to mention the entire contingent of postal employees is ineligible to vote! []

  • by Minwee ( 522556 ) <> on Thursday April 22, 2010 @07:01PM (#31948366) Homepage
    As Barbie says, "Math is hard, let's go shopping!"
  • by Skapare ( 16644 ) on Thursday April 22, 2010 @08:18PM (#31949246) Homepage

    ... gets 137.333...(repeating). So 137 votes is not even enough. 136 is clearly not.

  • by NewToNix ( 668737 ) on Thursday April 22, 2010 @08:29PM (#31949376) Journal
    "one zoning amendment which the voters passed in April - to require cottage colonies to operate as such for three years before conversion to condos is permitted - was reversed on a vote count challenge by a recent decision of the Mass. Attorney General's office. "

    From: []

  • by GNUALMAFUERTE ( 697061 ) <> on Thursday April 22, 2010 @09:11PM (#31949766)

    Updated 20100422 23:55 by timothy: Oops! This story is a year old (rounding up), which I didn't spot quickly enough. Hope they've got it all worked out in the meantime.

    Generating news isn't simple. You have to investigate, contact the sources, write the article, correct it, publish it in a readable way, etc, etc.

    Agregating news isn't that hard. All you have to do is check the source, the date and place of the article, if it's serious and still relevant, write a small summary (or cut and past it from the article) and submit. Not that hard at all. Google news does a better job than Slashdot at it. A damn perl script does a better job than 20+ slashdot editors. Even Fark is doing better than slashdot. If you post some old copypasta on the randomness and caos that is /b/, it'll be spotted instantly. The 13 year old kids at /b/ do a better job than slashdot's team of editors.

    I usually don't complain about article quality, dups, etc. I believe it's better to just let it go and move on. I say "hey, anyone can make a mistake". But it just gets worse everyday. We trust slashdot. We just spent a lot of our time discussing this issue, and trying to provide meaningful answers. It turned out to be an issue that happened almost a year ago. That is worse than reading slashdot on April 1st (at least you KNOW it's all bullshit on April fools day).

    Even taking all the stupid trolls into account, this community is much more valuable than the site that is hosting it. Yes, we can be a bunch of assholes sometimes, but I believe this is still true: Slashdot's community is la creme de la creme of the Internet. Just tell me of any other place where you can get a high profile open source developer, a NASA researcher that has written code for the Shuttle, a guy from Star Trek, a lawyer that understands copyright law, one of the founders of Apple, the Father of quake, an employee from almost every single technology corporation in the world, plus a huge crowd of engineers, coders, technology enthusiasts, writers, philosophers, sysadmins, doctors, lawyers, politicians, and generally smart people. The Slashdot community is amazing. Unique. I can't think of any other place with such diversity and such a high concentration of people that matters. Don't get me wrong, I'm not saying that /. readers are some kind of superior race. I'm just saying that the distribution of people in /. isn't average. We certainly have less cab drivers and more world changers than any other community out there. What worries me, is that the the site hosting that community is not up to the task. I love Slashdot. I've been in here for a long long time, and I have no intention of leaving. This is an off topic comment, and it'll certainly be flagged as such ... But I just felt like sharing this lines with you. What can we do to improve this place? it is, after all, like a second home to many of us.

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