# How do you measure two-thirds?

From an article by Mary Ann Bragg which appeared on CapeCodeOnline and was also printed in this month’s College Mathematics Journal:

TRURO — Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

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 new limit requires that those properties be in operation for three years before being converted to condominiums.

The idea behind the zoning amendment is to slow the pace of condominium development in Truro and preserve more affordable accommodations for tourists, according to citizens proposing the warrant article.

Currently Truro does not allow condominiums complexes to be built outright in its zoning bylaws. Instead, property owners must build a cottage colony, cabins, motel or hotel first and then covert it to condominiums through a special permit.

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.

But is it?  Is 136 a sufficient number of votes to be considered two-thirds of the total 206 votes?  Let’s check:

If you use the fact that $\frac{2}{3} \approx 0.66$ and then proceed to multiply 206 by 0.66 you get 135.96.  There were 136 votes in favor which is  more than 135.96 so that means it passes, right?  If you think so, then you’d be WRONG!!!

The main problem is the rounding.  In fact, $\frac{2}{3} = 0.666666\ldots$ or using repeated decimal notation, $\frac{2}{3} = 0.\bar{6}$.  When you round, you are actually creating an error that, in this case, makes a pretty significant difference.

Think of it another way, lets compare 136 / 206 to 2 / 3.  First, just do it by decimal approximation:

$\frac{136}{206} \approx 0.660194174757 < 0.6666666667 \approx \frac{2}{3}$

My calculator cannot exactly represent either of these fractions but its accurate to 12 decimal places and I can clearly see that 136/206 < 2/3 so the vote should not pass.

Do you remember another way you can compare fractions?  Find a common denominator and convert each fraction, then compare.

$\frac{136}{206} \cdot \frac{3}{3} = \frac{408}{618}$

$\frac{2}{3} \cdot \frac{206}{206} = \frac{412}{618}$

So, here we see that, again,

$\frac{136}{206} = \frac{408}{618} < \frac{412}{618} = \frac{2}{3}$

This second method of checking is even better than the first because there are no approximations involved.  We’ve confirmed, absolutely, that 136 votes out of a total of 206 does NOT constitute two-thirds.

Fortunately, a good citizen made an anonymous call in Truro, MA, to clear this up.  What perplexes me is that they decided they needed to let the State Attorney General’s office decide on the correct count. The mathematical explanation wasn’t good enough. Can you say quantitative illiteracy?

## 4 thoughts on “How do you measure two-thirds?”

1. What I do get is that if you simply multiply 206 by 2 and divide by 3, you get ~137.3 which easily implies that you need 138 out of 206 votes for the thing to pass. And that’s using a cheap marketing calculator I have lying around.

The accountant, Trudy Brazil, should seriously be considering a new position as I fear all her accounting calculations could be called into question now. Even 3 places (.666) violates her decision. I’d be more willing to accept a 137 vote as passing as it’s closer to the actual value (even though it’s not).

2. Or, if you divide 206 by 3 and multiply by 2 you get 137.3, so, no the vote didn’t pass. Perhaps the person doing the calculating was wanting it to pass a little to much.

3. @Trae @N’Ida

Both your methods are still relying on rounding figures, so even though they are correct in the ruling they have the same problem as the accountants method; you dont get 137.3, you get 137.33333…
The use of a common denominator is far more convincing.

4. Burt says:

The vote was 136 to 70. If the vote is split 2 to 1, then 136 must be twice 70. But 136 is less than 140 so it must be less than two-thirds of the vote, and 70 must be more than one-third of the vote. You don’t need a calculator for that.