Sarah is currently x years old. In 3 years, she will be twice as old as she was 5 years...

1. TRANSLATE the problem information

• Given information:
  • Sarah is currently \(\mathrm{x}\) years old
  • In 3 years, she will be \(\mathrm{x + 3}\) years old
  • 5 years ago, she was \(\mathrm{x - 5}\) years old
  • The key condition: "In 3 years, she will be twice as old as she was 5 years ago"

• What this tells us: We need to set up an equation where (age in 3 years) = 2 × (age 5 years ago)

2. TRANSLATE the condition into an equation

• The statement "In 3 years, she will be twice as old as she was 5 years ago" becomes:
  \(\mathrm{x + 3 = 2(x - 5)}\)

3. SIMPLIFY to solve for x

• Apply distributive property: \(\mathrm{x + 3 = 2x - 10}\)
• Move all x terms to one side: \(\mathrm{x + 3 = 2x - 10}\)
• Rearrange: \(\mathrm{3 + 10 = 2x - x}\)
• Combine: \(\mathrm{13 = x}\)

So Sarah is currently 13 years old.

4. SIMPLIFY to find the final answer

• We need \(\mathrm{2x + 7}\), not just \(\mathrm{x}\)
• Substitute \(\mathrm{x = 13}\):
  \(\mathrm{2x + 7 = 2(13) + 7}\)
  \(\mathrm{= 26 + 7}\)
  \(\mathrm{= 33}\)

Answer: 33

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students struggle with converting the time relationships into algebraic expressions. They might write something like "\(\mathrm{x + 3 = 2x - 5}\)" instead of "\(\mathrm{x + 3 = 2(x - 5)}\)", forgetting that "twice as old as she was 5 years ago" means 2 times the quantity \(\mathrm{(x - 5)}\), not \(\mathrm{2x - 5}\).

This leads to the wrong equation: \(\mathrm{x + 3 = 2x - 5}\), which gives \(\mathrm{x = 8}\). Then \(\mathrm{2x + 7 = 2(8) + 7 = 23}\), causing confusion since this isn't among typical answer choices and leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the correct equation but make algebraic errors when solving. For example, they might incorrectly combine terms or make sign errors, getting \(\mathrm{x = 7}\) instead of \(\mathrm{x = 13}\). This leads to \(\mathrm{2x + 7 = 2(7) + 7 = 21}\), again causing them to second-guess their work.

The Bottom Line:

This problem tests your ability to carefully translate time-based relationships into algebra. The phrase "twice as old as she was 5 years ago" specifically means \(\mathrm{2 \times (x - 5)}\), and getting this translation right is crucial for the entire solution.