Rounding error on Professional Entry Test Sample Questions (English)

Чему только учат в школа в США (да и в Израиле)? Люди искренне верят, что $\frac{1}{3}$ можно представить конечной десятичной дробью.

Примечание: В качестве десятичного разделителя была использована точка (а не запятая), как это принято в англоязычных странах, так как текст написан был мной в оригинале на английском.

UPDATE 25-08-2010:О вреде округленияEND OF UPDATE

There is more below.
Ниже есть продолжение.


Here http://www.civilservice.la.gov/Examining/StaffingInfo/TestInformation/testinformation.asp can be found link to the Professional Entry Test (PET) Sample Questions. Here is direct link.

It is look like the Psychometric Entrance Test. The questions are more straight-forward. Inference is pretty shock for me, what does it mean: "Probably, but not necessarily true"? How it can be determined? Formal logic operates only with True\False, so these type of the answer can be objectively checked. Fuzzy or another exotic logic was not used in the interpretation, it was formal logic with twist.

But the real shock was not here. I found mistake in the question 19. The correct answer is 5.None of the above. Here is the question and proposed answer

19. A state park is budgeted at an amount 9 times the amount budgeted for a nearby city park. If the combined yearly budget of both parks is $$$1,000,000, what is the average monthly budget of the city park?

1. $$$8,111.00
2. $$$8,222.22
3. $$$8,333.33
4. $$$8,444.44
5. none of the above.

The correct answer is number 3. Let x = the annual city park budget and 9 x = the annual state park budget. Therefore, 10 x = 1,000,000 and x = $$$100,000. $$$100,000 divided by 12 = $$$8,333.33, the average monthly city park budget.

The error is here "$$$100,000 divided by 12 = $$$8,333.33", it is false. $\frac{100,000}{12}=8,333+\frac13=8,333.(3)$. The last number is read as 8333 and three in period, that means infinite repetition of the 3. It is not equal to 8,333.33, this is numerically (mathematically) different number. So, the correct answer is none of the above, because again 8,333.33 is not equal to 8,333.(3).

If you will say, well we should round it to the cents, I will say you two things:
a) Rounding is not mention in the question;
b) I struggle with exact same problem at my work. I should make some calculation with the money on the computer application I write. There are many tiny details here, but for our discussion I have to use much higher precision for intermediate calculation, otherwise, the rounding error will be significant. Let consider it with simple example. If you calculate average salary per month up to cents, let say it was 20,456.67 cent and then multiply it to let say 9 year=9*12=108 month, than you will have error up to dollar. Consider numerical example:

20,456.67*108=2,209,320.36

But the "true" number was 20,456.(6)=20,456+2/3=61,370/3, so

$20,456.(6)*108=\frac{61,370}{3}*108=\frac{61,370*108}{3}=61,370*\frac{108}{3}=$
$=61370*36=2,209,320.00$.

so we have error not in 1 cent, but in 2,209,320.36-2,209,320.00=0.36 or 36 cents!