Dear Harley1980,
This is a great question, and I am happy to help.

There is some number of Progressives in the country as a whole. Call that number P.

In each province, the number of traditionalists is P/10 = 0.1*P

In five provinces, the number of traditionalist would be that times five, 0.5*P. In other words, the number of traditionalist is half the number of progressives.

Let T be the total number of traditionalists. Then P = 2T, and the total population in the country is 3T. T is 1/3 of this. OA = (B).

Does all this make sense?
Mike
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Hi All,

This question can be solved by TESTing VALUES.

We're given a series of facts to work with:
1) There are 5 provinces
2) Each province includes a certain number of Progressives and a certain number of Traditionalists
3) Each province has the SAME number of Traditionalists; the number of Traditionalists in each province = 1/10 the TOTAL number of Progressives in the ENTIRE COUNTRY.

We're asked 'what fraction of the COUNTRY is TRADITIONALISTS?'

If we say that each province has 10 Traditionalists in it, then we can calculate the TOTAL number of Progressives in the ENTIRE COUNTRY....

10 = (1/10)(ALL Progressives)
100 = All Progressives

Since there are 5 provinces, each with 10 Traditionalists in it, there are (5)(10) = 50 Total Traditionalists

50 Traditionalists + 100 Progressives = 150 Total people.

Since we're asked for the fraction that is Traditionalists.... 50/150 = 1/3

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Hi Sash143,

When TESTing VALUES on this question, you can use ANY number of Traditionalists (I simply chose 10 because it made all of the numbers 'round numbers'). The prompt tells us that the number of Traditionalists is the SAME in all 5 provinces, so you MUST multiply your Value by 5 (to find the TOTAL number of Traditionalists). Whatever number you Test ALSO impacts the TOTAL number of Progressives in the country (you have to multiply that Value by 10 to get the Total number of Progressives).

The prompts asks us to consider the WHOLE COUNTRY, so we don't actually have to count up the number of Traditionalists in each province (we already know THAT Total).

GMAT assassins aren't born, they're made,
Rich
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This can definitely be done using the abstract method, but here I'm just sharing the number picking technique for this question:

Say there are 100 Progressives in the country.
This means that each province has 10 Traditionalists, so there are 50 Traditionalists in the whole country,
Hence the total population of the country: 100+50=150
The number of traditionalists amounts to 50/150 i.e. 1/3 of the total population.

Let there be 2P and 1T per province. (Total P = 10, each province has 1/10th of 10 = 1T)
Therefore, fraction of traditionalists = \(\frac{(1*5)}{(2*5)+(1*5)}\)
\(=\frac{5}{15} = \frac{1}{3}\)