A biologist prepares nutrient solutions for a cell culture experiment using two stock solutions, Solution P and Solution Q. The...

1. TRANSLATE the problem information

• Given information:
  • First batch: 3 liters Solution P + 2 liters Solution Q = 0.85 kg nutrient total
  • Second batch: 1 liter Solution P + 4 liters Solution Q = 0.95 kg nutrient total
  • Need to find: concentration of nutrient in Solution Q (kg/L)

2. TRANSLATE to mathematical equations

• Let \(\mathrm{P}\) = concentration in Solution P (kg/L) and \(\mathrm{Q}\) = concentration in Solution Q (kg/L)
• First batch equation: \(\mathrm{3P + 2Q = 0.85}\)
• Second batch equation: \(\mathrm{P + 4Q = 0.95}\)

3. INFER the solution approach

• This is a system of two linear equations with two unknowns
• Since we only need Q, we can use either elimination or substitution
• Substitution might be cleaner since the second equation has P with coefficient 1

4. SIMPLIFY using substitution method

• From equation 2: \(\mathrm{P = 0.95 - 4Q}\)
• Substitute into equation 1:
  \(\mathrm{3(0.95 - 4Q) + 2Q = 0.85}\)
• Distribute: \(\mathrm{2.85 - 12Q + 2Q = 0.85}\)
• Combine like terms: \(\mathrm{2.85 - 10Q = 0.85}\)
• Isolate Q: \(\mathrm{-10Q = 0.85 - 2.85 = -2.00}\)
• Solve: \(\mathrm{Q = 0.20}\)

Answer: D (0.20)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the equations by confusing what each variable represents or mixing up the coefficients from each batch.

For example, they might write the first equation as \(\mathrm{2P + 3Q = 0.85}\) instead of \(\mathrm{3P + 2Q = 0.85}\), swapping the volume coefficients. This leads to solving the wrong system entirely and typically results in an answer not among the choices, causing them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make algebraic errors during the solution process.

A common mistake is sign errors when distributing or combining like terms, such as getting \(\mathrm{2.85 + 10Q = 0.85}\) instead of \(\mathrm{2.85 - 10Q = 0.85}\). This could lead them to calculate \(\mathrm{Q = -0.20}\), and then they might select Choice A (0.15) thinking they made a sign error and should take the absolute value.

The Bottom Line:

This problem tests whether students can translate a real-world mixing scenario into a mathematical system and then execute the algebra correctly. The context makes the setup more challenging than a pure algebraic system, and the multi-step algebra provides opportunities for calculation errors.