Instructions [29 - 33 ]

Aurevia, Brelosia, Cyrenia and Zerathania are four countries with their currencies being Aurels, Brins, Crowns, and Zentars, respectively. The currencies have different exchange values. Crown's currency exchange rate with Zentars = 0.5, i.e., 1 Crown is worth 0.5 Zentars.

Three travelers, Jano, Kira, and Lian set out from Zerathania visiting exactly two of the countries. Each country is visited by exactly two travelers. Each traveler has a unique Flight Cost, which represents the total cost of airfare in traveling to both the countries and back to Zerathania. The Flight Cost of Jano was 4000 Zentars, while that of the other two travelers were 5000 and 6000 Zentars, not necessarily in that order. When visiting a country, a traveler spent either 1000, 2000 or 3000 in the country's local currency. Each traveler had different spends (in the country's local currency) in the two countries he/she visited. Across all the visits, there were exactly two spends of 1000 and exactly one spend of 3000 (in the country's local currency).

The total “Travel Cost” for a traveler is the sum of his/her Flight Cost and the money spent in the countries visited.

The citizens of the four countries with knowledge of these travels made a few observations, with spends measured in their respective local currencies:

i. Aurevia citizen: Jano and Kira visited our country, and their Travel Costs were 3500 and 8000, respectively.

ii. Brelosia citizen: Kira and Lian visited our country, spending 2000 and 3000, respectively. Kira's Travel Cost was 4000.

iii. Cyrenia citizen: Lian visited our country and her Travel Cost was 36000.

One Brin is equivalent to how many Crowns?

The correct answer is 8 (Option 1).

Let us break down the logical steps and mathematical derivations to understand why 1 Brin is equivalent to 8 Crowns:

1. Define the Variables for Exchange Rates:
Let the exchange rate of a currency in terms of Zentars (Ze) be denoted by $E$.
According to the problem, 1 Crown is worth 0.5 Zentars, which gives:
$E_C=0.5$

2. Analyze the Travel Routes:
There are three countries visited: Aurevia (A), Brelosia (B), and Cyrenia (C).
Three travelers (Jano, Kira, and Lian) visit exactly two countries each, and each country is visited by exactly two travelers:
• Jano and Kira visited Aurevia.
• Kira and Lian visited Brelosia.
• Since each country is visited by two travelers, Lian and Jano must have visited Cyrenia.

3. Analyze the Flight Costs and Spends:
• Jano's Flight Cost (${\mathrm{FC}}_J$) = 4000 Zentars.
• Kira's and Lian's Flight Costs are 5000 Zentars and 6000 Zentars in some order.
• Local spends per country can be 1000, 2000, or 3000 of the local currency. Each traveler has different spends in the two visited countries.
• Across all visits, the distribution of spends is: exactly two of 1000, exactly three of 2000, and exactly one of 3000.

4. Relate the Travel Costs from Citizens' Observations:
The observations made by the citizens are in their respective local currencies:
• Aurevia: Jano's Travel Cost is 3500 Aurels, and Kira's is 8000 Aurels.
• Brelosia: Kira's Travel Cost is 4000 Brins, and Brelosia spends for Kira and Lian are 2000 Brins and 3000 Brins, respectively.
• Cyrenia: Lian's Travel Cost is 36000 Crowns.

Using these, we can express the total Travel Costs in Zentars:
• Kira's Travel Cost in Zentars:
${\mathrm{TC}}_K=8000E_A=4000E_B$
This gives:
$E_B=2E_A$

• Lian's Travel Cost in Zentars:
${\mathrm{TC}}_L=36000E_C=36000\times 0.5=18000\text{Zentars}$

5. Calculate the Exchange Rates:
Since Lian's Brelosia spend is 3000 Brins, her Cyrenia spend ($s_{L,C}$) must be either 1000 or 2000 Crowns (as each traveler has different spends in the two visited countries).
Lian's Travel Cost equation:
${\mathrm{TC}}_L={\mathrm{FC}}_L+3000E_B+s_{L,C}{E}_C$
$18000={\mathrm{FC}}_L+3000E_B+s_{L,C}\left(0.5\right)$
With ${\mathrm{FC}}_L=5000$ and $s_{L,C}=2000$, we have:
$18000=5000+3000E_B+1000$
$12000=3000E_B\Rightarrow E_B=4$

This means 1 Brin is worth 4 Zentars.

6. Find the Conversion Rate between Brins and Crowns:
We need to find the equivalent number of Crowns for 1 Brin:
$\frac{E_B}{E_C}=\frac{4}{0.5}=8$

Thus, 1 Brin is equivalent to 8 Crowns.