Machine A and Machine B work together to complete a production order. When Machine A works for 2 hours and...

1. TRANSLATE the problem information into mathematical equations

• Given information:
  • Machine A works 2 hours, Machine B works 5 hours → produce 23 units total
  • Machine A works 4 hours, Machine B works 3 hours → produce 21 units total

• Let \(\mathrm{a}\) = Machine A's rate (units per hour) and \(\mathrm{b}\) = Machine B's rate (units per hour)

• This gives us the system:
  • \(\mathrm{2a + 5b = 23}\)
  • \(\mathrm{4a + 3b = 21}\)

2. INFER the solution approach

• This is a system of two equations with two unknowns
• We can use elimination method since the coefficients allow for clean elimination
• Target: eliminate variable \(\mathrm{a}\) to solve for \(\mathrm{b}\) directly

3. SIMPLIFY using elimination method

• Multiply the first equation by -2 to get: \(\mathrm{-4a - 10b = -46}\)
• The second equation remains: \(\mathrm{4a + 3b = 21}\)
• Add the equations: \(\mathrm{(-4a - 10b) + (4a + 3b) = -46 + 21}\)
• This eliminates \(\mathrm{a}\): \(\mathrm{-7b = -25}\)
• Solve for \(\mathrm{b}\): \(\mathrm{b = \frac{25}{7}}\)

4. APPLY CONSTRAINTS to verify the answer

• Machine B's rate is \(\mathrm{\frac{25}{7}}\) units per hour
• Since this represents a production rate, it should be positive ✓
• The fraction \(\mathrm{\frac{25}{7}}\) is already in lowest terms since \(\mathrm{gcd(25,7) = 1}\) ✓

Answer: \(\mathrm{\frac{25}{7}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct equations from the word problem, often confusing which machine works for how many hours in each scenario.

They might write equations like \(\mathrm{2a + 3b = 23}\) or mix up the coefficients, leading to completely wrong systems. This fundamental error makes any subsequent work meaningless and typically leads to confusion and guessing among answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the system correctly but make arithmetic errors during elimination, such as incorrectly multiplying the first equation by -2 or making sign errors when adding equations.

For example, they might get \(\mathrm{-7b = 25}\) instead of \(\mathrm{-7b = -25}\), leading to \(\mathrm{b = -\frac{25}{7}}\). This may cause them to second-guess their work or select an incorrect answer if negative options are available.

The Bottom Line:

This problem requires careful translation of two different scenarios into a consistent system of equations. The key insight is recognizing that both scenarios involve the same two machines with the same rates - only the working times change.