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Two countries, Lativia and Manavia, have different business

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Two countries, Lativia and Manavia, have different business [#permalink] New post 11 Jul 2006, 02:26
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Two countries, Lativia and Manavia, have different business cycles: in Lativia, two downturn years are always separated by exactly 10 expansion years, while in Manavia, two downturn years are always separated by 18 expansion years. In each country, any calendar year is either a downturn year or an expansion year. 1904 was a downturn year for Manavia and 1905 was a downturn year for Latavia. In which decade will the two countries' economies contract in the same year?

(A) 2011-2020 (B) 2021-2030 (C) 2031-2040 (D) 2041-2050 (E) 2051-2060

Last edited by kevincan on 12 Jul 2006, 03:53, edited 2 times in total.
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 [#permalink] New post 11 Jul 2006, 07:25
Nobody likes countries with strange names?
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Re: PS: Lativia and Manavia [#permalink] New post 11 Jul 2006, 08:02
kevincan wrote:
Two countries, Lativia and Manavia, have different business cycles: in Lativia, two years of economic downturn are always separated by exactly 10 years of expansion, while in Manavia, two downturn years are separated by 18 years of expansion. In 1904, Manavia?s economy contracted, while Latavia´s economy did so one year later. In which decade will the two economies contract in the same year?

(A) 2011-2020 (B) 2021-2030 (C) 2031-2040 (D) 2041-2050 (E) 2051-2060

On the real exam I ll go with C on this question
I struggled about 2 min but has no clue how to solve it))And C is my bet blind shoot answer choice.
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 [#permalink] New post 11 Jul 2006, 08:34
This question is very tricky (or I am stupid ...)

First, observe that he text doesn't state wheter Manavia's economy contracts in 1903 or in 1905.

Manavia's business cycle is either 03-04, 05-22, 23-24, 25-42 ... (2003-2004, 2005-2022) or 04-05, 06-23, 24-25 ...( 2004-2005, 2006-2023).

Lativia's business cycle is 05-06, 7-16, 17-18 ....
It is easier to write this as follows: 6, 16, 18, 28, 30, 40, 42, 52, 54, 64, 66, 76, 78, 88, 90, 100, 102, 112, 114, 124, 126, 136, 138, 148, 150, 160 ...
(the number shows us, when the expansion or contraction phase ends).

Now: Compare both business cycles.
You can see that both economies contracts in 2025 for the first time.

Hence, B is the correct answer (I think).
(Note: If Manavia's economy contracts in 1903 and 1904 there is no date between 2010 and 2060 where both economies contract)

Aah ... ok.
Paddy is right. 2037

Last edited by game over on 12 Jul 2006, 15:48, edited 2 times in total.
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Re: PS: Lativia and Manavia [#permalink] New post 11 Jul 2006, 08:54
kevincan wrote:
Two countries, Lativia and Manavia, have different business cycles: in Lativia, two years of economic downturn are always separated by exactly 10 years of expansion, while in Manavia, two downturn years are separated by 18 years of expansion. In 1904, Manavia’s economy contracted, while Latavia´s economy did so one year later. In which decade will the two economies contract in the same year?

(A) 2011-2020 (B) 2021-2030 (C) 2031-2040 (D) 2041-2050 (E) 2051-2060


your question is little unclear. when the lativia's economic downturn ended? in 1904 (begaining or ending) or 1905 or 1906?
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 [#permalink] New post 12 Jul 2006, 03:54
I've changed the wording to make it clearer. Thanks for your feedback!
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 [#permalink] New post 12 Jul 2006, 15:31
C.

I got the series of downturn years for Lativia and for Manavia, comparing the lists, got 2037 as the common year.

Is there any faster method? This one took a long time...
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 [#permalink] New post 12 Jul 2006, 16:03
I get 2048= 1904+144=104+18*8=1905+143=1905+13*11

Any ideas on how we can get this quickly (less than 2-3 minutes)?

Clue: Forget about the actual years!
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 [#permalink] New post 12 Jul 2006, 22:48
Another way is the LCM of the 10 and 18 that is 60

So only after a gap of 60 years there economies will contract.

Hence only after 2026 will there economies contract. The decade after 2036 is 2031-2040

hence C.

It is a hard question considering that you should not chose 2021-2030.... because the event will only happen after that.
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 [#permalink] New post 12 Jul 2006, 23:51
Any ideas on how we can get this quickly (less than 2-3 minutes)?

Clue: Forget about the actual years!
In Lativia, contraction years happen once every 11 years, whereas in Manavia, they occur every 19 years

We have to find positive integers m and n such that

1905+11m=1904+19n
1+11m=19n

i.e. Find smallest positive integer value of n such that 19n divided by 11 yields a remainder of 1.

Look at positive multiples of 19, subtract multiples of 11 to find remainder when divided by 11

n=1......19............... remainder 8
n=2......8+19=27..... remainder 27-22=5
n=3......5+19=24..... remainder 24-22=2
n=4......2+19=21..... remainder 10
n=5......29............... remainder 7
n=6......26............... remainder 4
n=7......23................ remainder 1 19*7=133

So n is 7 and the year in which both economies contract is 1904+133=2037

Check: 1905+11*12=2037
  [#permalink] 12 Jul 2006, 23:51
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