A swimming pool can be filled by pump A alone in 4 hours and by pump B alone in 6...

1. TRANSLATE the pump information into rates

• Given information:
  • Pump A alone fills the pool in 4 hours
  • Pump B alone fills the pool in 6 hours

• What this tells us:
  • Pump A's rate: \(\frac{1}{4}\) of the pool per hour
  • Pump B's rate: \(\frac{1}{6}\) of the pool per hour

2. INFER how much work each pump does

• Since rate × time = work done:
  • Pump A running for a hours fills: \(\mathrm{a} \times \frac{1}{4} = \frac{\mathrm{a}}{4}\) of the pool
  • Pump B running for b hours fills: \(\mathrm{b} \times \frac{1}{6} = \frac{\mathrm{b}}{6}\) of the pool

3. TRANSLATE the final condition into an equation

• The problem states they fill exactly \(\frac{3}{4}\) of the pool together
• Total work = \(\frac{\mathrm{a}}{4} + \frac{\mathrm{b}}{6} = \frac{3}{4}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the given times (4 hours, 6 hours) with rates and mistakenly think these should appear directly in the equation.

They might reason: "Pump A takes 4 hours and runs for a hours, so that's 4a. Pump B takes 6 hours and runs for b hours, so that's 6b." This leads them to incorrectly set up \(4\mathrm{a} + 6\mathrm{b} = \frac{3}{4}\).

This may lead them to select Choice C (\(4\mathrm{a} + 6\mathrm{b} = \frac{3}{4}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that rates are \(\frac{1}{4}\) and \(\frac{1}{6}\), but then flip the denominators when setting up fractions, thinking "pump A works for a hours out of 4 total hours needed."

This confusion about what \(\frac{\mathrm{a}}{4}\) actually represents might lead them to construct incorrect equations like \(\frac{\mathrm{a}}{3} + \frac{\mathrm{b}}{2} = \frac{3}{4}\).

This may lead them to select Choice B (\(\frac{\mathrm{a}}{3} + \frac{\mathrm{b}}{2} = \frac{3}{4}\)).

The Bottom Line:

The key challenge is correctly TRANSLATING "fills the pool in X hours" into a rate (\(\frac{1}{\mathrm{X}}\) pools per hour), then understanding that multiplying this rate by actual running time gives the fraction of work completed. Students who focus on the given times (4 and 6) without converting to rates will struggle.