The value of a certain stock portfolio, P, is 1,200% of the combined value of a mutual fund M and...

1. TRANSLATE the problem information

• Given information:
  • P = 1,200% of (M + B), which means \(\mathrm{P = 12(M + B)}\)
  • M = 48% of B, which means \(\mathrm{M = 0.48B}\)
  • All values are positive
• What we need to find: What percent of M is P? (This means we need \(\mathrm{P/M}\) as a percent)

2. INFER the approach

• We have two equations but three variables, so we need to eliminate one variable
• Since we want to find P in terms of M, we should eliminate B
• From \(\mathrm{M = 0.48B}\), we can solve for B and substitute

3. SIMPLIFY to eliminate B

From \(\mathrm{M = 0.48B}\), solve for B:

\(\mathrm{B = M/0.48}\)

4. SIMPLIFY by substitution

Substitute \(\mathrm{B = M/0.48}\) into \(\mathrm{P = 12(M + B)}\):

\(\mathrm{P = 12(M + M/0.48)}\)

\(\mathrm{P = 12M(1 + 1/0.48)}\)

5. SIMPLIFY the fraction arithmetic

Calculate \(\mathrm{1/0.48}\):

\(\mathrm{1/0.48 = 1/(48/100) = 100/48 = 25/12}\)

So:

\(\mathrm{P = 12M(1 + 25/12)}\)

\(\mathrm{P = 12M(12/12 + 25/12)}\)

\(\mathrm{P = 12M(37/12)}\)

\(\mathrm{P = 37M}\)

6. TRANSLATE back to percentage

Since \(\mathrm{P = 37M}\), this means \(\mathrm{P/M = 37 = 3,700\%}\)

Answer: D. 3,700%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle with converting "1,200% of (M + B)" correctly, sometimes interpreting it as \(\mathrm{P = 1.2(M + B)}\) instead of \(\mathrm{P = 12(M + B)}\).

This leads to \(\mathrm{P = 1.2M(37/12) = 3.7M}\), giving 370% instead of 3,700%, causing confusion when this doesn't match any answer choice. This leads to guessing or selecting Choice C (1,800%) as the "closest" large percentage.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make arithmetic errors when calculating \(\mathrm{1/0.48}\) or when simplifying the final expression.

Common mistakes include calculating \(\mathrm{1/0.48 \approx 2.1}\) (instead of \(\mathrm{25/12}\)), leading to \(\mathrm{P \approx 12M(3.1) \approx 37.2M}\), which might cause them to round incorrectly and select Choice D (3,700%) through luck rather than proper calculation.

The Bottom Line:

This problem tests whether students can handle multi-step percentage relationships systematically. The key challenge is maintaining precision through the algebraic manipulations while correctly interpreting percentages greater than 100%.