June 1, 2009

If You Had 2/3 Of A Brain

Jim Correia tweeted this story earlier about trouble figuring out how to fulfill a 2/3 majority vote:

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums. [...]

The exact count of the vote - 136 to 70 -had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

Brazil said she used the calculation of .66 multiplied by 206 to obtain the number.

But using .6666 - a more accurate version of two-thirds - the affirmative vote needed to be 137 instead of 136, according to an anonymous caller to town hall and to the Times.

Slade said that she called several of her colleagues to see how they calculate a two-thirds vote, and the answer varied widely. In Provincetown, Town Clerk Doug Johnstone uses .66. But Johnstone said he'd never had a close vote where it might matter.

Whoa. There are so many things wrong here.

They're using 0.66 as an approximation of 2/3 in a calculation where the measured data has a precision of three digits? Sloppy.

Even if you don't know about precision, there is no need to approximate 2/3 this roughly on a calculator; you just divide 2 by 3 and the calculator does the approximation for you.

Here's what my calculator says is the value for 2/3: 0.6666666667. That's an approximation good enough for any town vote. It is rounded up at the end there because nobody's calculator can represent 2/3 in decimal without rounding.

Which brings us to another question: why did they round their approximation down? 0.6 repeating rounds at two decimal places up to 0.67, but they chose to round down to 0.66. It turns out that 206 * 0.67 = 138.02. Whoops, looks like they might be two votes shy! 0.67 isn't a very good approximation of 2/3; it's just slightly better than 0.66.

Why deal with approximations at all? 206 * 2/3 = 137 1/3. (I'm sure you can do the math with the fractions.) That extra 1/3 tells me that 137 votes is not enough to exceed 2/3 of 206. We have to ask "what does it mean for a vote to be required to pass by a 2/3 majority?"

In government matters, rules decide the way things work. If they have some bylaw that says the "yea" vote must equal or exceed the total number of votes multiplied by 0.66 and then subsequently truncated, then that's the law. But it seems silly to have a law that essentially defines 0.66 as 2/3, and I don't believe we do.

I think the law probably intends that at least 2/3 of the voters must vote for the measure. That is very easy to test.

If the yea votes divided by the total vote is greater than 2/3, then it passed! It's calculated just like it sounds. The ratio must be greater than 2/3.

Is 136 more than 2/3 of 206?

136 / 206 = 0.660194175

0.660194175 is indisputably less than 2/3. So the 2/3 requirement has not been achieved. What happens if you get one more vote?

137 / 206 = 0.665048544

Whoops! 137 votes is also less than 2/3! it's close, but it's actually about 0.00161812298 votes less than 2/3. I say "about" because everything is an approximation when you're dealing with a repeating decimal like 2/3. Well, not everything. I can tell you with absolute certainty that 137 is less than 2/3.

138 / 206 = 0.669902913

Hooray! 138 is definitely more than 2/3 of 206. It's arithmetically defensible that with 138 votes you have met or exceeded a 2/3 majority.

"But James" you say. 137 and even 136 are close enough!!! Look, either a number of votes is over 2/3 of the total or it isn't. That isn't up for argument. And you need at least 138 to exceed 2/3. Let's try a f'rinstance.

Let's say the law says you need a 2/3 majority. We'll use the Truro accountant's method. But, oops - only four people show up to vote! Two vote yea and two vote nay. So, how many votes did you need for your 2/3 majority? Four * 0.66 = 2.64. Continuing the trend of rounding down, that means we need two votes. I think you can see that's wrong. 50% definitely does not satisfy a 2/3 majority. It's a silly example, but it's also silly to approximate 2/3 with 0.66, and to always round down.

To be fair, they were obviously taken by surprise because they had adopted a lousy method for calculating 2/3 of the vote. Perhaps the law calls for a really flawed method for calculating 2/3 of a vote. I doubt it, but stranger things have happened. Regardless of that, shouldn't a town accountant be able to tell you whether a number is at least two thirds of some other number? That's middle school math, the last time I checked.

Posted by James at June 1, 2009 9:41 PM
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Wow. Just, wow. That was stunning. My favorite part of that was "But using .6666 - a more accurate version of two-thirds..." That's true. But, as you pointed out, 'times 2 divided by three' is a MUCH more accurate 'version' of two-thirds. holy crap. Is that really so difficult? Do we NEED versions of numbers?

Beyond that, I have *never* before seen two-thirds approximated as .66, .6666, or even .6666666666666!
The last digit in the approximation is always rounded up in any representation of two-thirds. Ye gods! We're all doomed!

Posted by: briwei at June 1, 2009 11:49 PM

Wow. Just wow. How about we get even simpler. Even if we agree to go to 2 sig figs and use decimals rather than fractions 0.66 is not the correct apporximation of 2/3, 0.67 is. If they had use this they'd have the correct # required (138).

What's stunning is that this is a story at all. Someone really had trouble figuring this out? They could have oh I don't know, called a math teacher, asked a 5th grader, shot themselves in the head thereby reducing the # required by 1 and ensuring they're removed from the gene pool.

Posted by: B.O.B. (bob) at June 2, 2009 7:56 AM

I don't know whether this underscores people's problems with fractions, with number sense, or if it is just an indictment of math skills in Truro, MA.

I can't find any subsequent news story about whether the state AG's office straightened the situation out.

Posted by: James at June 2, 2009 8:21 AM

As I said above, "0.660194175 is indisputably less than 2/3."

As you can see, that number is larger than the approximation they're using for 2/3, which is the crux of the problem.

Posted by: James at June 2, 2009 8:25 AM

I must be missing something. This seemed fairly easy to me. Is this a trick question?

Add 136 and 70 to get the total votes: 206

Divide 206 by 3: 68.66666666

Multiply 68.66666666 by 2: 137.3333333

Votes requred to attain a 2/3 majority: 137

Posted by: Patti M. at June 2, 2009 8:48 AM

It looks like you're 1/3 of a vote shy of having 2/3 of 206 votes.

Not until 138 do you pass the 2/3 threshold. Now, that may be a matter of some rule. Perhaps they don't look at it as a threshold. Perhaps they just do the (quite correct) calculation you did and then round. That's reasonable.

But 136 would still not be enough votes. They're either off by 1 or 2, and it appears it's because in 2009, a town in Massachusetts can't calculate 2/3 of 206, even with the help of their accountant.

Posted by: James at June 2, 2009 9:21 AM

I missed that the accountant got it wrong. That's even more stunning. Although with all the people getting in trouble for not reporting taxes correctly I guess I shouldn't be. Maybe I should just use their rounding rules when I do my taxes.

Posted by: B.O.B. (bob) at June 2, 2009 10:24 AM

Maybe they're using The Math that Karl Rove has.

Posted by: Patti M. at June 3, 2009 8:56 AM

As a resident of next to Truro, I am laughing.

As a person with a math degree I have a headache... from banging it against my desk.

Posted by: Sara at June 4, 2009 10:07 PM

My wife the humanities teacher gave the link to her math colleagues to use in teaching a real world example of why math matters.

Posted by: briwei at June 5, 2009 11:27 AM

They also serve who only stand as a negative example.

Posted by: James at June 5, 2009 11:46 AM

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