The total number of problems P solved by a student can be represented by the expression \(\mathrm{h(h + 22)}\), where...

1. TRANSLATE the problem information

• Given information:
  • Total problems solved: \(\mathrm{P = h(h + 22)}\)
  • Time spent studying: \(\mathrm{h}\) hours
• What we need to find: Rate of solving problems (problems per hour)

2. INFER the mathematical relationship

• Since we're looking for rate (problems per hour), we need to use:
  \(\mathrm{Rate = \frac{Total\ problems}{Time}}\)
• This means: \(\mathrm{Rate = \frac{P}{h}}\)

3. SIMPLIFY the algebraic expression

• Substitute the given expression:
  \(\mathrm{Rate = \frac{h(h + 22)}{h}}\)
• Divide each term in the numerator by h:
  \(\mathrm{Rate = \frac{h}{h} + \frac{22h}{h}}\)
  \(\mathrm{= 1 + 22}\)
  \(\mathrm{= (h + 22)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the given expression to the rate formula, instead thinking that since h represents hours, it must be the rate.

This misconception leads them to select Choice A (\(\mathrm{h}\)) without considering what "rate" actually means in this context.

Second Most Common Error:

Conceptual confusion about rates: Students might think the rate is constant and focus only on the "22" in the expression, not understanding that the rate changes with time in this problem.

This leads them to select Choice B (22), missing that the student's rate actually increases as study time increases.

The Bottom Line:

This problem challenges students to distinguish between total quantity, time, and rate while working with algebraic expressions rather than simple numbers. The key insight is recognizing that rate problems follow the same Total = Rate × Time relationship even when expressed algebraically.