Question:A financial analyst is managing a portfolio with a total initial value of $50{,}000, split between a stock fund and...

1. TRANSLATE the problem information

• Given information:
  • Total initial portfolio value: $50,000
  • Stock fund increased by 6% after one year
  • Bond fund decreased by 10% after one year
  • Total portfolio value after one year: $50,800
  • Need to find: Original value of stock fund

• What this tells us: We have two unknowns (original stock value and original bond value) and can create equations from the constraints.

2. INFER the approach

• Since we have two unknowns, we need two equations
• First equation comes from initial total value
• Second equation comes from final total value after percentage changes
• Substitution method will work well since the first equation easily isolates one variable

3. TRANSLATE the constraints into equations

Let \(\mathrm{s}\) = original stock fund value, \(\mathrm{b}\) = original bond fund value

• Equation 1: \(\mathrm{s + b = 50,000}\)
• Equation 2: \(\mathrm{1.06s + 0.90b = 50,800}\)
  (6% increase means new value = \(\mathrm{1.06 \times original}\))
  (10% decrease means new value = \(\mathrm{0.90 \times original}\))

4. SIMPLIFY using substitution

• From equation 1: \(\mathrm{b = 50,000 - s}\)
• Substitute into equation 2:
  \(\mathrm{1.06s + 0.90(50,000 - s) = 50,800}\)
• Distribute: \(\mathrm{1.06s + 45,000 - 0.90s = 50,800}\)
• Combine like terms: \(\mathrm{0.16s = 5,800}\)
• Solve: \(\mathrm{s = 36,250}\) (use calculator)

Answer: C. $36,250

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to correctly convert percentage changes into mathematical expressions. They might use 0.06 instead of 1.06 for the stock increase, or 0.10 instead of 0.90 for the bond decrease.

For example, writing the second equation as: \(\mathrm{0.06s + 0.10b = 50,800}\) instead of \(\mathrm{1.06s + 0.90b = 50,800}\). This fundamental translation error makes the entire system incorrect and leads to confusion when the math doesn't work out to any answer choice.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during the substitution and simplification process, particularly when distributing \(\mathrm{0.90(50,000 - s)}\) or when combining the coefficients of s.

A common mistake is: \(\mathrm{1.06s + 45,000 - 0.90s = 50,800}\) becomes \(\mathrm{0.16s = 95,800}\) instead of \(\mathrm{0.16s = 5,800}\). This leads to \(\mathrm{s = 598,750}\), which is nowhere near any answer choice and causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can correctly translate percentage language into mathematical operations. The key insight is that "increased by 6%" means the new value is 106% of the original (multiply by 1.06), not just add 6% of the original.