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.

How many Zentars did Lian spend in the two countries he visited?

The correct answer is 13000.

To find the total amount in Zentars that Lian spent in the two countries visited, we can systematically set up the relations between the travelers' flight costs, spends, and exchange rates.

1. Understanding the Settings and Variables:
Let the exchange rates of the country currencies to Zentars (Z) be:
- 1 Aurel (Aurevia) = $e_A$ Zentars
- 1 Brin (Brelosia) = $e_B$ Zentars
- 1 Crown (Cyrenia) = $e_C$ Zentars = 0.5 Zentars (given)

There are three travelers, each visiting exactly two countries, and each of the three visited countries (Aurevia, Brelosia, and Cyrenia) is visited by exactly two travelers:
- Aurevia is visited by Jano (J) and Kira (K).
- Brelosia is visited by Kira (K) and Lian (L).
- Cyrenia is visited by Lian (L) and Jano (J).

2. Flight Costs and Spends:
- Jano's flight cost: $F_J=4000$ Zentars.
- Kira's and Lian's flight costs: $\{F_K,F_L\}=\{5000,6000\}$ Zentars.
- Local currency spends: 6 spends in total, consisting of exactly two of 1000, exactly one of 3000, and therefore exactly three of 2000.
- Each traveler has different local spends in their two visited countries.

3. Analyzing the Citizens' Observations:
Spends and travel costs are measured in local currencies where indicated:
- From Brelosia: Kira and Lian spent 2000 and 3000 Brins, respectively. So, $S_{K,B}=2000$ and $S_{L,B}=3000$ Brins. Kira's Travel Cost was 4000 Brins. Since Kira's Travel Cost in Zentars is $4000e_B$:
$4000e_B=F_K+S_{K,A}{e}_A+2000e_B$
Simplifying this gives:
$2000e_B=F_K+S_{K,A}{e}_A$ (Equation 1)

- From Aurevia: Jano's and Kira's Travel Costs were 3500 and 8000 Aurels, respectively. Thus:
For Jano: $3500e_A=4000+S_{J,A}{e}_A+0.5S_{J,C}$
Rearranging Jano's equation:
$(3500-S_{J,A})e_A=4000+0.5S_{J,C}$ (Equation 2)

For Kira: $8000e_A=F_K+S_{K,A}{e}_A+2000e_B$
Substituting Equation 1 into Kira's equation:
$8000e_A=F_K+S_{K,A}{e}_A+(F_K+S_{K,A}{e}_A)$
$8000e_A=2F_K+2S_{K,A}{e}_A$
Dividing by 2:
$F_K=(4000-S_{K,A})e_A$ (Equation 3)

- From Cyrenia: Lian's Travel Cost was 36000 Crowns. In Zentars, this is:
$36000\times 0.5=18000$ Zentars.
So, $F_L+3000e_B+0.5S_{L,C}=18000$ (Equation 4)

4. Solving for the Spends:
Since Kira's spends in Brelosia ($S_{K,B}$) is 2000, and local spends in both countries must be different, Kira's spend in Aurevia ($S_{K,A}$) cannot be 2000. Out of the remaining spends available, $S_{K,A}$ must be 1000.
Substituting $S_{K,A}=1000$ into Equation 3:
$F_K=3000e_A$
Since $F_K\in \{5000,6000\}$, let us test $F_K=6000$. This yields:
$e_A=2$
Then from Equation 1:
$2000e_B=6000+1000\left(2\right)=8000\Rightarrow e_B=4$
Since $F_K=6000$, Lian's flight cost must be $F_L=5000$ Zentars.

To check consistency for Jano's spends in Equation 2:
With $e_A=2$, we have:
$(3500-S_{J,A})\times 2=4000+0.5S_{J,C}$
Using $S_{J,A}=1000$ and $S_{J,C}=2000$ (which are distinct and valid spends):
$2500\times 2=4000+1000=5000$, which is consistent!

The remaining spend for Lian in Cyrenia must be $S_{L,C}=2000$ Crowns.
Let us verify Equation 4 for Lian:
$5000+3000\left(4\right)+0.5\left(2000\right)=5000+12000+1000=18000$, which perfectly holds.

5. Calculating Lian's Total Spends in Zentars:
Lian's spends in Brelosia and Cyrenia converted to Zentars are:
- Spend in Brelosia: $3000\text{Brins}\times 4\text{Zentars/Brin}=12000$ Zentars.
- Spend in Cyrenia: $2000\text{Crowns}\times 0.5\text{Zentars/Crown}=1000$ Zentars.

Total spends = $12000+1000=13000$ Zentars.