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.

What is the trade balance of ROW?

The correct option/answer is 200.

Here is the step-by-step derivation to find the trade balance of the Rest of World (ROW).

Step 1: Set up definitions and interpret normalized trade balances
Let _A denote the total exports of country A, and I_A denote the total imports of country A.
The trade balance of country A is given by:
$$T_B\left(A\right)=E_A-I_A$$
For any closed system of trade, the sum of all trade balances must equal zero:
$$T_B\left(P\right)+T_B\left(X\right)+T_B\left(C\right)+T_B\left(\mathrm{ROW}\right)=0$$

From the normalized trade balances:
1) For Pumpland (P): The normalized trade balance is 0%, which means exports equal imports:
$$E_P=I_P\Rightarrow T_B\left(P\right)=0$$
2) For Xiland (X): The normalized trade balance is 10%:
$$\frac{E_X-I_X}{E_X+I_X}=0.10\Rightarrow E_X-I_X=0.10E_X+0.10I_X\Rightarrow E_X=\frac{11}{9}{I}_X$$
3) For Cheeseland (C): The normalized trade balance is -20%:
$$\frac{E_C-I_C}{E_C+I_C}=-0.20\Rightarrow E_C-I_C=-0.20E_C-0.20I_C\Rightarrow I_C=1.5E_C$$

Step 2: Determine trade values for Cheeseland (C)
We are given that Pumpland is the only country that exports to Cheeseland. Therefore, all imports of Cheeseland come from Pumpland:
$$I_C=E_{P\&to;C}=1200$$
Using Cheeseland's import-export relationship:
$$E_C=\frac{I_C}{1.5}=\frac{1200}{1.5}=800$$
This gives the trade balance of Cheeseland:
$$T_B\left(C\right)=E_C-I_C=800-1200=-400$$

We can also compute Cheeseland's exports to other regions:
• Exports to Pumpland: $E_{C\&to;P}=0.90\times 800=720$
• Exports to ROW: $E_{C\&to;\mathrm{ROW}}=0.04\times 800=32$
• Exports to Xiland: $E_{C\&to;X}=E_C-E_{C\&to;P}-E_{C\&to;\mathrm{ROW}}=800-720-32=48$

Step 3: Analyze trade flows for Pumpland (P) and Xiland (X)
We are given:
• 40% of Xiland's exports go to Pumpland: $E_{X\&to;P}=0.40E_X$
• 22% of Pumpland's imports are from Xiland: $E_{X\&to;P}=0.22I_P$
Equating these two expressions:
$$0.40E_X=0.22I_P\Rightarrow E_X=0.55I_P$$
Since $E_X=\frac{11}{9}{I}_X$, we have:
$$I_X=\frac{9}{11}{E}_X=\frac{9}{11}\left(0.55I_P\right)=0.45I_P$$

Step 4: Solve the equations using imports of X and P
The imports of Xiland (X) come from Pumpland (P), Cheeseland (C), and ROW:
$$I_X=E_{P\&to;X}+E_{C\&to;X}+E_{\mathrm{ROW}\&to;X}$$
Substituting the given values:
$$I_X=600+48+0.12E_{\mathrm{ROW}}=648+0.12E_{\mathrm{ROW}}$$
Since $I_X=0.45I_P$, we write Equation 1:
$$0.45I_P-0.12E_{\mathrm{ROW}}=648$$

Next, the imports of Pumpland (P) come from Xiland (X), Cheeseland (C), and ROW:
$$I_P=E_{X\&to;P}+E_{C\&to;P}+E_{\mathrm{ROW}\&to;P}$$
Substituting the known expressions:
$$I_P=0.22I_P+720+0.40E_{\mathrm{ROW}}$$
Rearranging gives Equation 2:
$$0.78I_P-0.40E_{\mathrm{ROW}}=720$$

Now, let's solve the system of linear equations:
Multiply Equation 1 by 3.333 (or $\frac{10}{3}$) to align the coefficients of $E_{\mathrm{ROW}}$:
$$1.50I_P-0.40E_{\mathrm{ROW}}=2160$$
Subtract Equation 2 from this new equation:
$$(1.50-0.78)I_P=2160-720$$
$$0.72I_P=1440\Rightarrow I_P=2000$$

Using $I_P=2000$, we can compute Xiland's imports and exports:
$$I_X=0.45\times 2000=900$$
$$E_X=0.55\times 2000=1100$$
The trade balance of Xiland is:
$$T_B\left(X\right)=E_X-I_X=1100-900=200$$

Step 5: Compute the trade balance of ROW
Since the sum of the trade balances of all trading entities must be zero:
$$T_B\left(P\right)+T_B\left(X\right)+T_B\left(C\right)+T_B\left(\mathrm{ROW}\right)=0$$
Substitute the values we calculated:
$$0+200+(-400)+T_B\left(\mathrm{ROW}\right)=0$$
$$-200+T_B\left(\mathrm{ROW}\right)=0\Rightarrow T_B\left(\mathrm{ROW}\right)=200$$
Thus, the trade balance of the Rest of World (ROW) is 200.