A landscaper uses a hose that puts 88x ounces of water in a bucket in 5y minutes. Which expression represents...

1. TRANSLATE the problem information

• Given information:
  • Hose puts \(88\mathrm{x}\) ounces of water in \(5\mathrm{y}\) minutes
  • Need to find ounces of water in \(9\mathrm{y}\) minutes at the same rate
• What this tells us: We have an amount and time, so we can find a rate

2. INFER the approach

• Since we know amount and time for one scenario, we can find the rate of water flow
• Once we have the rate, we can find the amount for any time period using: Amount = Rate × Time
• The key insight: rates stay constant, so we can apply the same rate to different time periods

3. SIMPLIFY to find the rate

• \(\mathrm{Rate} = \mathrm{Amount} \div \mathrm{Time}\)
• \(\mathrm{Rate} = 88\mathrm{x}\text{ ounces} \div 5\mathrm{y}\text{ minutes} = \frac{88\mathrm{x}}{5\mathrm{y}}\text{ ounces per minute}\)

4. SIMPLIFY to find the new amount

• \(\mathrm{Amount} = \mathrm{Rate} \times \mathrm{New Time}\)
• \(\mathrm{Amount} = \frac{88\mathrm{x}}{5\mathrm{y}} \times 9\mathrm{y}\)
• \(\mathrm{Amount} = \frac{88\mathrm{x} \times 9\mathrm{y}}{5\mathrm{y}} = \frac{792\mathrm{xy}}{5\mathrm{y}} = \frac{792\mathrm{x}}{5}\text{ ounces}\)

Answer: D. \(\frac{792\mathrm{x}}{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that they need to find the rate first before calculating the amount for the new time period. Instead, they might try to set up incorrect proportions or perform operations directly on the given numbers without understanding the underlying rate concept.

This confusion often leads to random manipulation of the numbers and variables, causing them to select incorrect answer choices or abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to multiply \(\frac{88\mathrm{x}}{5\mathrm{y}} \times 9\mathrm{y}\) but make algebraic errors during simplification. They might forget to cancel the y terms or incorrectly multiply \(88 \times 9 = 792\).

This may lead them to select Choice A (\(\frac{8\mathrm{x}}{240}\)) or other incorrect expressions due to calculation mistakes.

The Bottom Line:

This problem challenges students to recognize that rate problems require finding the unit rate first, then applying it to new conditions. The presence of variables makes the arithmetic more abstract, requiring careful algebraic manipulation alongside conceptual understanding of rates.