A library's collection starts with 800 books. The librarian orders 30 new books each week to add to the collection....

1. TRANSLATE the problem information

• Given information:
  • Starting amount: 800 books
  • Rate of increase: 30 books per week
  • Target amount: 1,700 books
  • Find: number of weeks (w)

2. INFER the mathematical approach

• This is a linear growth situation where we're adding a constant amount each time period
• We need a linear equation in the form: starting amount + (rate × time) = final amount
• Strategy: Set up equation and solve for the unknown time variable

3. TRANSLATE into mathematical notation

Set up the equation:
\(\mathrm{800 + 30w = 1{,}700}\)

4. SIMPLIFY by solving the equation

• Subtract 800 from both sides:
  \(\mathrm{30w = 1{,}700 - 800}\)
  \(\mathrm{30w = 900}\)

• Divide both sides by 30:
  \(\mathrm{w = 900 ÷ 30 = 30}\)

Answer: B (30 weeks)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the equation incorrectly by forgetting to include the initial 800 books, writing just \(\mathrm{30w = 1{,}700}\) instead of \(\mathrm{800 + 30w = 1{,}700}\).

When they solve \(\mathrm{30w = 1{,}700}\), they get w = 56.67 weeks, which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students might misinterpret the problem as asking "how many books will be added" rather than "how many weeks," leading them to calculate \(\mathrm{1{,}700 - 800 = 900}\) and select this as their answer.

Since 900 isn't among the choices, this causes them to get stuck and guess.

The Bottom Line:

Success on this problem depends on correctly translating the English description into a mathematical model. Students must recognize that the final amount equals the starting amount plus the accumulated additions over time.