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.

What is the sum of Travel Costs for all travelers in Zentars?

The correct answer is 41000.

Let us represent the countries as follows:
• Aurevia ($A$) with local currency Aurels (exchange rate: 1 Aurel = $a$ Zentars)
• Brelosia ($B$) with local currency Brins (exchange rate: 1 Brin = $b$ Zentars)
• Cyrenia ($C$) with local currency Crowns (exchange rate: 1 Crown = 0.5 Zentars, so $c=0.5$)
• Zerathania ($Z$) with local currency Zentars (the base currency)

Based on the observations, we can determine the country visits for the three travelers:
• Jano visited Aurevia ($A$) and Cyrenia ($C$).
• Kira visited Aurevia ($A$) and Brelosia ($B$).
• Lian visited Brelosia ($B$) and Cyrenia ($C$).

The flight costs in Zentars are:
• Jano's Flight Cost ($F_J$) = 4000 Zentars.
• The flight costs of Kira ($F_K$) and Lian ($F_L$) are 5000 and 6000 Zentars in some order.

Let the spend of traveler $T$ in country $X$ be represented as $S_{T,X}$. According to the rules, each traveler spent either 1000, 2000, or 3000 in local currency, with different spends in their two visited countries. Across all visits, there are exactly two spends of 1000, one spend of 3000, and three spends of 2000.

From the Brelosia citizen's observation, we have:
• Spend of Kira in Brelosia ($S_{K,B}$) = 2000 Brins.
• Spend of Lian in Brelosia ($S_{L,B}$) = 3000 Brins (which is the unique 3000 spend).
• Kira's Travel Cost in Brelosia currency was 4000 Brins.

From the Cyrenia citizen's observation, Lian's Travel Cost in Crowns was 36000. Since 1 Crown = 0.5 Zentars:

$T{C}_L=36000\times 0.5=18000\text{ Zentars}$

From the Aurevia citizen's observation:
• Jano's Travel Cost in Aurels was 3500. So, $T{C}_J=3500a$.
• Kira's Travel Cost in Aurels was 8000. So, $T{C}_K=8000a$.

Since Kira's Travel Cost in Brins was 4000, we also have:

$T{C}_K=4000b$

Equating the two expressions for Kira's Travel Cost:

$8000a=4000b\Rightarrow b=2a$

We also express Kira's Travel Cost as:

$T{C}_K=F_K+S_{K,A}\cdot a+S_{K,B}\cdot b$

Substituting $T{C}_K=4000b$ and $S_{K,B}=2000$:

$4000b=F_K+S_{K,A}\cdot a+2000b$

$2000b=F_K+S_{K,A}\cdot a$

Using $b=2a$:

$4000a=F_K+S_{K,A}\cdot a\Rightarrow F_K=(4000-S_{K,A})a$

Since Kira must have different spends in Aurevia and Brelosia, and $S_{K,B}=2000$, it must be that $S_{K,A}\ne 2000$. Since the only possible spends are 1000, 2000, or 3000, and 3000 is already taken by Lian, we must have:

$S_{K,A}=1000$

Substituting this back gives:

$F_K=3000a$

Since $F_K$ must be either 5000 or 6000:
If $F_K=6000$, then $a=2$ and $F_L=5000$.
If $a=2$, then $b=4$.

Let us verify this with Lian's total cost:

$T{C}_L=F_L+S_{L,B}\cdot b+S_{L,C}\cdot c$

$18000=5000+3000\left(4\right)+S_{L,C}\left(0.5\right)$

$18000=17000+0.5S_{L,C}\Rightarrow S_{L,C}=2000$

Next, let us check Jano's total cost:

$T{C}_J=F_J+S_{J,A}\cdot a+S_{J,C}\cdot c$

Substituting $T{C}_J=3500a=7000$ and $F_J=4000$:

$7000=4000+2S_{J,A}+0.5S_{J,C}$

$2S_{J,A}+0.5S_{J,C}=3000$

To satisfy the unique spend distributions (exactly two 1000s, three 2000s, and one 3000) and ensure Jano's spends are different, we set $S_{J,A}=1000$ and $S_{J,C}=2000$. This fits perfectly:

$2\left(1000\right)+0.5\left(2000\right)=2000+1000=3000$

With all parameters determined, the Travel Costs for each traveler in Zentars are:
• Jano: $T{C}_J=7000$ Zentars
• Kira: $T{C}_K=16000$ Zentars
• Lian: $T{C}_L=18000$ Zentars

The sum of Travel Costs for all travelers in Zentars is:

$7000+16000+18000=41000$