An investor plans to allocate exactly $12{,}000 between two investment options. Government bonds offer a 4% annual return, while corporate...

1. TRANSLATE the problem information

• Given information:
  • Total investment: $12,000
  • \(\mathrm{x}\) = amount in bonds (thousands of dollars)
  • \(\mathrm{y}\) = amount in stocks (thousands of dollars)
  • Bond return rate: 4% (annual)
  • Stock return rate: 9% (annual)

• What this tells us: We need an equation showing how the investment amounts relate to the total.

2. INFER what equation we actually need

• The question asks for "amounts invested" - this means we need the allocation constraint
• The return rates (4%, 9%) would be relevant for calculating profits, but not for the basic allocation
• We need: Amount in bonds + Amount in stocks = Total investment

3. Set up the constraint equation

• Since x and y are in thousands of dollars: \(\mathrm{x + y = 12}\)
• This represents: Investment in bonds + Investment in stocks = Total investment
• Note: $12,000 = 12 thousand dollars (matching our variable units)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see the 4% and 9% return rates and assume they must use them in the equation.

They think: "The problem mentions returns, so the equation must involve \(\mathrm{0.04x + 0.09y = something}\)." This leads them to incorrectly focus on returns rather than the allocation constraint.

This may lead them to select Choice A (\(\mathrm{0.04x + 0.09y = 12}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students get confused about units and scaling, trying to incorporate both percentages and dollar amounts.

They might think: "4% means 4, and we have thousands, so maybe \(\mathrm{4x + 9y = 1200}\)?" This shows confusion about how percentages, units, and scaling work together.

This may lead them to select Choice C (\(\mathrm{4x + 9y = 1200}\)).

The Bottom Line:

This problem tests whether students can distinguish between different types of constraints. The key insight is recognizing that "amounts invested" refers to the allocation constraint (how money is divided), not the return constraint (how much profit is made).