Lost in wording with matrices (1 Viewer)

leehuan · Apr 6, 2016

Just start me off please. No need to complete the question.

$Assume 3 countries $A, B, $ and $C$ trade with one another and no-one else, that a common currency is used and that each country's total income comes from trade with the others or sales to itself and nothing else.$

$\\ A$ spends $\frac{3}{4},\frac{1}{8},\frac{1}{8}$ of its income on goods from $A, B, C$ respectively.$\\B$ spends $\frac{1}{5},\frac{3}{5},\frac{1}{5}$ of its income on goods from $A, B, C$ respectively.$\\ C$ spends $\frac{1}{4},\frac{1}{4},\frac{1}{2}$ of its income on goods from $A, B, C$ respectively.$$

$\\$Find the (relative) income of each country.$$

He · Apr 6, 2016

Haha, just reminded me I have to finish those for tomorrow

InteGrand · Apr 6, 2016

leehuan said:
Just start me off please. No need to complete the question.

$Assume 3 countries $A, B, $ and $C$ trade with one another and no-one else, that a common currency is used and that each country's total income comes from trade with the others or sales to itself and nothing else.$

$\\ A$ spends $\frac{3}{4},\frac{1}{8},\frac{1}{8}$ of its income on goods from $A, B, C$ respectively.$\\B$ spends $\frac{1}{5},\frac{3}{5},\frac{1}{5}$ of its income on goods from $A, B, C$ respectively.$\\ C$ spends $\frac{1}{4},\frac{1}{4},\frac{1}{2}$ of its income on goods from $A, B, C$ respectively.$$

$\\$Find the (relative) income of each country.$$

Since we are only after relative incomes, we can set the income of A to be 1 (in unit of currency), then find B and C's incomes; these'll be ratios of that of A. Let the income of B be b and C be c (these are relative to A's income).

From info regarding things bought from A, we have:

3/4 + (1/5)b + (1/4)c = 1 (LHS is the proportion of each country's income spent on A, multiplied by their total income, added up. This adds up to the total income of A, which we have said is 1, hence the RHS).

From info regarding things bought from B, we have:

1/8 + (3/5)b + (1/4)c = b.

From info regarding things bought from C, we have:

1/8 + (1/5)b + (1/2)c = c.

We thus have a linear system, which you can try and solve. (Rearrange each equation so that only things with b and c are on the LHS, and only on the LHS).

Lost in wording with matrices (1 Viewer)

leehuan

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He

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InteGrand

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