Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the...

1. TRANSLATE the problem information

• Given information:
  • First 10 hours: \(\$8/\mathrm{hour}\)
  • Remaining hours: \(\$10/\mathrm{hour}\)
  • Saves \(90\%\) of weekly earnings
  • Need to save at least \(\$270\)

• What this tells us: We need to find minimum hours at the higher rate.

2. TRANSLATE the setup

• Calculate fixed earnings: \(10 \times \$8 = \$80\)
• Let \(\mathrm{x}\) = additional hours at \(\$10/\mathrm{hour}\)
• Total weekly earnings = \(\$80 + \$10\mathrm{x}\)
• Amount saved = \(0.9(\$80 + \$10\mathrm{x})\)

3. INFER the mathematical approach

• Since we need "at least \(\$270\)," this becomes an inequality problem
• Set up: \(0.9(\$80 + \$10\mathrm{x}) \geq \$270\)

4. SIMPLIFY to solve the inequality

• Distribute: \(0.9 \times \$80 + 0.9 \times \$10\mathrm{x} \geq \$270\)
• Calculate: \(\$72 + \$9\mathrm{x} \geq \$270\)
• Subtract \(\$72\): \(\$9\mathrm{x} \geq \$198\)
• Divide by 9: \(\mathrm{x} \geq 22\)

Answer: C. 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students forget about the \(90\%\) savings rate and work with total earnings instead of saved earnings.

They set up: \(\$80 + \$10\mathrm{x} \geq \$270\), leading to \(\mathrm{x} \geq 19\). Since 19 isn't an option, they might round up to the nearest choice and select Choice D (16) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors, particularly when calculating \(0.9 \times \$80 = \$72\) or when dividing \(\$198 \div 9\).

Common mistakes include getting \(\$70\) instead of \(\$72\), or miscalculating the final division. This leads to slightly wrong values that don't match any answer choice, causing confusion and potentially selecting Choice D (16) as the closest "reasonable" answer.

The Bottom Line:

This problem requires careful attention to the savings percentage - it's not asking for total earnings needed, but for earnings that result in at least \(\$270\) in savings. Students who miss this key detail will set up the wrong inequality entirely.