In a study skills program, students earn points for different learning activities. Reading assignments earn 150 points for every 45...

1. TRANSLATE the problem information

• Given information:
  • Reading assignments: \(\mathrm{150}\) points for every \(\mathrm{45}\) minutes
  • Practice problems: \(\mathrm{200}\) points for every \(\mathrm{30}\) minutes
  • Student already completed \(\mathrm{90}\) minutes of practice problems
  • Goal: \(\mathrm{1,800}\) total points from both activities
• What we need to find: How many hours of reading assignments needed

2. INFER the approach

• Since we have a total goal and already know one component, we can work backwards
• Strategy: Find points already earned, subtract from total, then calculate reading time needed
• First step: Convert all rates to points per hour for easier calculation

3. SIMPLIFY the rates to points per hour

• Reading rate: \(\mathrm{150}\) points \(\mathrm{\div}\) \(\mathrm{45}\) minutes = \(\mathrm{150 \div (45/60)}\) hours
  \(\mathrm{= 150 \div 0.75}\)
  \(\mathrm{= 200}\) points per hour
• Practice problems rate: \(\mathrm{200}\) points \(\mathrm{\div}\) \(\mathrm{30}\) minutes = \(\mathrm{200 \div (30/60)}\) hours
  \(\mathrm{= 200 \div 0.5}\)
  \(\mathrm{= 400}\) points per hour

4. SIMPLIFY to find points already earned

• Practice time completed: \(\mathrm{90}\) minutes = \(\mathrm{90/60 = 1.5}\) hours
• Points from practice: \(\mathrm{1.5}\) hours \(\mathrm{\times}\) \(\mathrm{400}\) points/hour
  \(\mathrm{= 600}\) points

5. INFER remaining points needed from reading

• Total goal: \(\mathrm{1,800}\) points
• Already earned: \(\mathrm{600}\) points
• Still needed from reading: \(\mathrm{1,800 - 600 = 1,200}\) points

6. SIMPLIFY to find reading time required

• Points needed: \(\mathrm{1,200}\) points
• Reading rate: \(\mathrm{200}\) points per hour
• Reading time needed: \(\mathrm{1,200 \div 200 = 6}\) hours

Answer: D (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle with converting mixed units (minutes in the rates vs. hours in the answer). They might try to work directly with the given rates without converting to a consistent time unit first.

For example, they might calculate: \(\mathrm{90}\) minutes of practice = \(\mathrm{90/30 = 3}\) "blocks" of practice = \(\mathrm{3 \times 200 = 600}\) points (this part is correct), but then get confused trying to work with reading in 45-minute blocks while the answer asks for hours. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the strategy but make arithmetic errors in the unit conversion, especially when converting 45 minutes to 0.75 hours or calculating \(\mathrm{150 \div 0.75 = 200}\).

They might incorrectly calculate the reading rate as \(\mathrm{150/45 = 3.33}\) points per minute, then multiply by 60 to get 200 points per hour, but make errors in this multi-step process. This may lead them to select Choice C (5) if they underestimate the reading rate.

The Bottom Line:

This problem tests your ability to work with rates in different time units while managing multi-step backwards calculations. Success depends on converting everything to the same time unit early in the solution.