A water tank is filled using two pumps, Pump A and Pump B. In one operation, Pump A runs for...

1. TRANSLATE the problem information

• Given information:
  • Pump A runs for 3 hours
  • Pump B runs for 12 hours
  • Together they deliver 4,680 liters total
  • Equation: \(3\mathrm{r} + 12\mathrm{s} = 4{,}680\)

• What this tells us: The numbers 3 and 12 are time periods, and 4,680 is a total volume

2. INFER the approach using dimensional analysis

• For the equation to make sense, both sides must have the same units
• Right side: \(4{,}680\) liters (volume)
• Left side: \(3\mathrm{r} + 12\mathrm{s}\) must also equal liters
• Strategy: Figure out what units \(\mathrm{r}\) must have

3. INFER what r represents

• The term \(3\mathrm{r}\) must be in liters (to add up to the total)
• 3 represents hours (time Pump A operates)
• For \((\text{hours}) \times \mathrm{r} = \text{liters}\), \(\mathrm{r}\) must be in liters/hour
• Therefore: \(\mathrm{r}\) = flow rate of Pump A in liters per hour

Answer: A. The average flow rate, in liters per hour, of Pump A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students focus on the variables \(\mathrm{r}\) and \(\mathrm{s}\) without connecting them to the real-world context of pumps and time periods.

They might think "\(\mathrm{r}\) is just some number" without recognizing that 3 and 12 represent specific time durations. This leads them to misinterpret what \(\mathrm{r}\) could represent, potentially selecting Choice C (The total volume, in liters, pumped by Pump A) because they're thinking about total amounts rather than rates.

Second Most Common Error:

Poor INFER reasoning: Students don't apply dimensional analysis to check unit consistency.

Without checking units, they might assume \(\mathrm{r}\) represents total volume since the right side of the equation is in liters. They miss that for the equation to be dimensionally correct, \(\mathrm{r}\) must be a rate (liters per hour) not a total volume. This leads to confusion about whether to choose answer choices about rates versus totals.

The Bottom Line:

Success requires connecting the mathematical equation to the physical situation - understanding that time coefficients multiplied by rate variables give total volumes.