Question:The real number r satisfies \(4(\mathrm{r} - 3) = 16\).What is the value of r + 7?Enter your answer as...

1. TRANSLATE the problem requirements

• Given information:
  • Equation: \(4(\mathrm{r} - 3) = 16\)
  • Need to find: \(\mathrm{r} + 7\) (not just r)

• What this tells us: We must first solve for r, then use that value to calculate \(\mathrm{r} + 7\)

2. SIMPLIFY to solve for r

• Start with: \(4(\mathrm{r} - 3) = 16\)
• Divide both sides by 4: \(\mathrm{r} - 3 = 4\)
• Add 3 to both sides: \(\mathrm{r} = 7\)

3. SIMPLIFY to evaluate the requested expression

• Substitute \(\mathrm{r} = 7\) into \(\mathrm{r} + 7\):
• \(\mathrm{r} + 7 = 7 + 7 = 14\)

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students solve correctly for \(\mathrm{r} = 7\) but then enter 7 as their final answer, forgetting that the problem asks for \(\mathrm{r} + 7\), not r.

The problem clearly states 'What is the value of \(\mathrm{r} + 7\)?' but students often focus so intently on solving the equation that they miss this final step. This leads them to submit 7 instead of the correct answer 14.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors during the equation-solving process, such as incorrectly dividing 16 by 4 or making sign errors when adding 3 to both sides.

For example, if they miscalculate \(\mathrm{r} - 3 = 4\) and get \(\mathrm{r} = 2\) instead of \(\mathrm{r} = 7\), they would then compute \(\mathrm{r} + 7 = 2 + 7 = 9\), leading to an incorrect final answer.

The Bottom Line:

This problem tests both systematic equation solving and careful attention to what the question actually asks. The algebra is straightforward, but students must read precisely and complete both parts: solving for r AND evaluating \(\mathrm{r} + 7\).