Three countries — Pumpland (P), Xiland (X) and Cheeseland (C) — trade among themselves and with the (other countries in) Rest of World (ROW). All trade volumes are given in IC (international currency). The following terminology is used:

• Trade balance = Exports - Imports

• Total trade = Exports + Imports

• Normalized trade balance = Trade balance / Total trade, expressed in percentage terms

The following information is known.

1. The normalized trade balances of P, X and C are 0%, 10%, and -20%, respectively.

2. 40% of exports of X are to P. 22% of imports of P are from X.

3. 90% of exports of C are to P; 4% are to ROW.

4. 12% of exports of ROW are to X, 40% are to P.

5. The export volumes of P, in IC, to X and C are 600 and 1200, respectively. P is the only country that exports to C.

Which among the countries P, X, and C has/have the least total trade?

The correct option is Both X and C.

Let us find the total trade (Exports + Imports) for each of the three countries: Pumpland (P), Xiland (X), and Cheeseland (C) by translating the given statements into mathematical equations.

1. Basic Definitions and Relations:
Let $E_i$ and $I_i$ denote the total exports and total imports of country $i$ ($i\in \{P,X,C\}$) in International Currency (IC).
Let $E_{ij}$ denote the exports from country $i$ to country $j$, which is also equal to the imports of country $j$ from country $i$ ($I_{ji}$).
The normalized trade balance (NTB) is defined as:
$\mathrm{NTB}=\frac{\mathrm{Exports}-\mathrm{Imports}}{\mathrm{Exports}+\mathrm{Imports}}\times 100\%$

From Information 1, we obtain the relations for the normalized trade balances:
• For Pumpland (P): ${\mathrm{NTB}}_P=0\%\Rightarrow E_P=I_P$
• For Xiland (X): ${\mathrm{NTB}}_X=10\%\Rightarrow \frac{E_X-I_X}{E_X+I_X}$
• For Cheeseland (C): ${\mathrm{NTB}}_C=-20\%\Rightarrow \frac{E_C-I_C}{E_C+I_C}$

2. Analysing Cheeseland (C):
From Information 5, Pumpland (P) is the only country that exports to C. Thus, all of C's imports come from P:
$I_C=E_{PC}=1200$

Using the relation for C's trade balance:
$I_C=1.5E_C\Rightarrow 1200=1.5E_C\Rightarrow E_C=800$

Thus, the total trade of Cheeseland (C) is:
${\mathrm{TT}}_C=E_C+I_C=800+1200=2000$

3. Distribution of C's Exports:
From Information 3:
• Exports of C to P: $E_{CP}=0.9\times E_C=0.9\times 800=720$
• Exports of C to ROW: $E_{C,\mathrm{ROW}}=0.04\times E_C=0.04\times 800=32$
• The remaining exports of C must go to Xiland (X):
$E_{CX}=E_C-E_{CP}-E_{C,\mathrm{ROW}}=800-720-32=48$

4. Finding the remaining trade values:
Let $E_{\mathrm{ROW}}$ be the total export volume of the Rest of World (ROW).
From Information 4:
• Exports of ROW to X: $E_{\mathrm{ROW},X}=0.12E_{\mathrm{ROW}}$
• Exports of ROW to P: $E_{\mathrm{ROW},P}=0.4E_{\mathrm{ROW}}$

Now we calculate the total imports of X ($I_X$). X imports from P, C, and ROW:
$I_X=E_{PX}+E_{CX}+E_{\mathrm{ROW},X}=600+48+0.12E_{\mathrm{ROW}}=648+0.12E_{\mathrm{ROW}}$

From the trade balance relation for X:
$E_X=\frac{11}{9}$

From Information 2, we know:
• $E_{XP}=0.4E_X$
• $E_{XP}=0.22I_P\Rightarrow I_P=\frac{20}{11}$

Substitute $E_{XP}=0.4E_X$ into the equation for $I_P$:
$I_P=\frac{20}{11}$

Substitute $E_X=\frac{11}{9}$ into the expression for $I_P$:
$I_P=\frac{8}{11}$

Pumpland (P) imports from X, C, and ROW:
$I_P=E_{XP}+E_{CP}+E_{\mathrm{ROW},P}$
Substitute $E_{XP}=0.22I_P$, $E_{CP}=720$, and $E_{\mathrm{ROW},P}=0.4E_{\mathrm{ROW}}$:
$I_P=0.22I_P+720+0.4E_{\mathrm{ROW}}\Rightarrow 0.78I_P=720+0.4E_{\mathrm{ROW}}$

Now we substitute $I_P=\frac{20}{9}$ and $I_X=648+0.12E_{\mathrm{ROW}}$:
$0.78(\frac{20}{9}$
$\frac{15.6}{9}$
$15.6(72+\frac{0.12}{9}$
$1123.2+0.208E_{\mathrm{ROW}}=720+0.4E_{\mathrm{ROW}}$
$1123.2-720=(0.4-0.208)E_{\mathrm{ROW}}$
$403.2=0.192E_{\mathrm{ROW}}\Rightarrow E_{\mathrm{ROW}}=2100$

5. Calculating Total Trade Volumes:
Using $E_{\mathrm{ROW}}=2100$:
• Imports of X: $I_X=648+0.12\times 2100=648+252=900$
• Exports of X: $E_X=\frac{11}{9}$
• Total trade of X: ${\mathrm{TT}}_X=E_X+I_X=1100+900=2000$

• Imports of P: $I_P=\frac{20}{9}$
• Since $E_P=I_P$, we have $E_P=2000$
• Total trade of P: ${\mathrm{TT}}_P=E_P+I_P=2000+2000=4000$

Conclusion:
The total trade volumes are:
• Pumpland (P) = 4000
• Xiland (X) = 2000
• Cheeseland (C) = 2000
Thus, both X and C have the least total trade (2000 each).