In an article about exercise, it is estimated that a 160text{-pound} adult uses 200 calories for every 30 minutes of...

1. TRANSLATE the problem information

• Given information:
  • 160-pound adult: 200 calories per 30 minutes hiking
  • 160-pound adult: 150 calories per 30 minutes bicycling
  • Adult completed 1 hour of bicycling
  • Target: 1,900 total calories from both activities

• What we need to find: Hours of hiking required

2. INFER the solution approach

• We need to set up an equation relating calories from hiking and bicycling
• Since rates are given per 30-minute periods, let's work with that time unit initially
• Let \(\mathrm{h}\) = number of 30-minute hiking periods needed
• Let \(\mathrm{b}\) = number of 30-minute bicycling periods completed

3. TRANSLATE the constraint equation

• Total calories equation: \(\mathrm{200h + 150b = 1,900}\)
• Since the adult bicycled for 1 hour = 2 periods of 30 minutes: \(\mathrm{b = 2}\)

4. SIMPLIFY the equation

• Substitute \(\mathrm{b = 2}\): \(\mathrm{200h + 150(2) = 1,900}\)
• Calculate: \(\mathrm{200h + 300 = 1,900}\)
• Subtract 300: \(\mathrm{200h = 1,600}\)
• Divide by 200: \(\mathrm{h = 8}\)

5. INFER the final conversion

• \(\mathrm{h = 8}\) represents 30-minute periods of hiking
• Convert to hours: 8 periods ÷ 2 periods per hour = 4 hours

Answer: D. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may forget that the adult has already completed 1 hour (2 periods) of bicycling and set up the equation as just \(\mathrm{200h = 1,900}\), ignoring the bicycling component entirely.

This leads them to: \(\mathrm{200h = 1,900}\), so \(\mathrm{h = 9.5}\), and they select Choice A (9.5) without realizing this represents 30-minute periods, not hours.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{200h + 150b = 1,900}\) but substitute \(\mathrm{b = 1}\) instead of \(\mathrm{b = 2}\), thinking "1 hour of bicycling" means 1 period rather than 2 periods of 30 minutes.

This gives them \(\mathrm{200h + 150 = 1,900}\), leading to \(\mathrm{h = 8.75}\), and they select Choice B (8.75).

Third Most Common Error:

Poor unit conversion: Students solve correctly to get \(\mathrm{h = 8}\) thirty-minute periods but then subtract 2 instead of dividing by 2 to convert to hours, thinking they need to "account for" the bicycling time.

This leads them to calculate \(\mathrm{8 - 2 = 6}\) hours and select Choice C (6).

The Bottom Line:

This problem challenges students to carefully track units throughout their solution and properly account for given information when setting up their equation. The combination of rate problems with unit conversion creates multiple opportunities for error.