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.

Which of the following statements is NOT true about money spent in the local currency?

The correct option is Jano spent 2000 in Aurevia.

Let us break down the logical reasoning and mathematical derivations to understand why this statement is NOT true:

1. Define the parameters and variables:
Let the three countries visited be Aurevia ($A$), Brelosia ($B$), and Cyrenia ($C$).
Let their exchange rates to Zentars (local currency of Zerathania) be $e_A$, $e_B$, and $e_C$.
We are given that $e_C=0.5$ Zentars.

Each traveler visits exactly two countries, and each country is visited by exactly two travelers:
- From the observations, Jano and Kira visited Aurevia.
- Kira and Lian visited Brelosia.
- Lian visited Cyrenia.
By elimination, Jano must have also visited Cyrenia to ensure that each country is visited by exactly two travelers.
Thus, the itineraries are:
- Jano visits $A$ and $C$
- Kira visits $A$ and $B$
- Lian visits $B$ and $C$

2. Analyze Flight Costs and Spends:
Flight Costs ($F$) in Zentars:
- Jano ($F_J$) = $4000$
- Kira ($F_K$) and Lian ($F_L$) have costs of $5000$ and $6000$ in some order.

Spends in local currencies:
Across the 6 visits, the spends are either $1000$, $2000$, or $3000$.
Specifically, there are exactly two spends of $1000$, exactly one spend of $3000$, and therefore exactly three spends of $2000$.
From observation (ii):
- Kira's spend in Brelosia, $S_{K,B}=2000$ Brins.
- Lian's spend in Brelosia, $S_{L,B}=3000$ Brins (this is the unique $3000$ spend).

3. Formulation using observations:
Let us write the Travel Costs (measured in the respective local currency of the citizen's country):
i. Aurevia citizen's observations:
Jano's Travel Cost in Zentars:
$3500e_A=4000+S_{J,A}{e}_A+0.5S_{J,C}$
Kira's Travel Cost in Zentars:
$8000e_A=F_K+S_{K,A}{e}_A+S_{K,B}{e}_B$

ii. Brelosia citizen's observations:
Kira's Travel Cost in Zentars:
$4000e_B=F_K+S_{K,A}{e}_A+2000e_B$
This simplifies to:
$2000e_B=F_K+S_{K,A}{e}_A$

Comparing Kira's total travel cost equation in Zentars from both perspectives:
$8000e_A=4000e_B\Rightarrow e_B=2e_A$
Substituting $e_B=2e_A$ into Brelosia's simplified equation:
$2000\left(2e_A\right)=F_K+S_{K,A}{e}_A$
$(4000-S_{K,A})e_A=F_K$

iii. Cyrenia citizen's observations:
Lian's Travel Cost in Zentars:
$36000e_C=36000\left(0.5\right)=18000$ Zentars.
Lian's Travel Cost formula:
$18000=F_L+3000e_B+0.5S_{L,C}$

4. Finding the unique solution:
Since each traveler must spend different amounts in their visited countries:
Kira visits $A$ and $B$. We know $S_{K,B}=2000$, and the only $3000$ spend belongs to Lian, so $S_{K,A}$ must be $1000$.
Using Kira's equation:
$(4000-1000)e_A=F_K\Rightarrow 3000e_A=F_K$
If $F_K=6000$ and $F_L=5000$:
$e_A=2$, and thus $e_B=4$.
Substituting these into Lian's equation:
$18000=5000+3000\left(4\right)+0.5S_{L,C}$
$18000=17000+0.5S_{L,C}\Rightarrow S_{L,C}=2000$ (which is a valid spend).

Now we determine Jano's spends. Jano visits $A$ and $C$.
Substituting $e_A=2$ into Jano's equation:
$3500\left(2\right)=4000+2S_{J,A}+0.5S_{J,C}$
$3000=2S_{J,A}+0.5S_{J,C}$
Since there are three spends of $2000$ and two spends of $1000$, and we have already used one $1000$ ($S_{K,A}$) and two $2000$s ($S_{K,B}$ and $S_{L,C}$), the remaining spends for Jano must be one $1000$ and one $2000$.
Substituting $S_{J,A}=1000$ and $S_{J,C}=2000$:
$2\left(1000\right)+0.5\left(2000\right)=2000+1000=3000$, which matches perfectly.

Conclusion:
Jano spent $1000$ in Aurevia and $2000$ in Cyrenia.
Therefore, the statement "Jano spent 2000 in Aurevia" is NOT true.