13 is p% of 25. What is the value of p?
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
\(\mathrm{13}\) is \(\mathrm{p\%}\) of \(\mathrm{25}\). What is the value of \(\mathrm{p}\)?
1. TRANSLATE the problem information
- Given information:
- 13 is \(\mathrm{p\%}\) of 25
- Need to find the value of \(\mathrm{p}\)
- What this tells us: We have a percentage relationship where 13 represents the part and 25 represents the whole.
2. INFER the approach
- Since we know the part (13) and the whole (25), we can set up an equation to find the percentage
- The relationship "A is p% of B" translates to: \(\mathrm{A = \frac{p}{100} \times B}\)
3. TRANSLATE into mathematical notation
- Set up the equation: \(\mathrm{13 = \frac{p}{100} \times 25}\)
- This can also be written as: \(\mathrm{13 = \frac{25p}{100}}\)
4. SIMPLIFY to solve for p
- Simplify the fraction: \(\mathrm{13 = \frac{25p}{100} = \frac{p}{4}}\)
- Multiply both sides by 4: \(\mathrm{p = 13 \times 4 = 52}\)
Answer: 52
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the setup and write \(\mathrm{25 = \frac{p}{100} \times 13}\) instead of \(\mathrm{13 = \frac{p}{100} \times 25}\)
They incorrectly think "25 is p% of 13" instead of "13 is p% of 25." This fundamental misreading leads to \(\mathrm{p = \frac{25 \times 100}{13} \approx 192.3}\), which doesn't match any reasonable answer and causes confusion and guessing.
Second Most Common Error:
Incomplete SIMPLIFY execution: Students set up the correct equation \(\mathrm{13 = \frac{p}{100} \times 25}\) but make arithmetic errors in the solving process
They might incorrectly calculate \(\mathrm{\frac{13 \times 100}{25}}\) or forget to properly handle the fraction manipulation. This often leads to answers like 5.2 or 520, causing them to second-guess their work and potentially select an incorrect value through estimation.
The Bottom Line:
This problem tests whether students can correctly interpret percentage language and translate it into the proper mathematical relationship. The key insight is recognizing that when we say "A is p% of B," A is the result of taking p% of B, not the other way around.