The positive number a is 230% of the number b, and a is 60% of the number c. If c...

1. TRANSLATE the problem information

• Given relationships:
  • a is 230% of b → \(\mathrm{a = \frac{230}{100}b = 2.30b}\)
  • a is 60% of c → \(\mathrm{a = \frac{60}{100}c = 0.60c}\)
  • c is p% of b → \(\mathrm{c = \frac{p}{100}b}\)
• What we need to find: the value of p

2. INFER the solution strategy

• Key insight: Since both 2.30b and 0.60c equal a, we can set them equal to each other
• This will let us express c in terms of b, then use the third relationship to find p

3. SIMPLIFY by setting the expressions equal

• From step 1: \(\mathrm{2.30b = 0.60c}\)
• Solve for c:
  \(\mathrm{c = \frac{2.30b}{0.60}}\)
  \(\mathrm{c = \frac{230b}{60}}\)
  \(\mathrm{c = \frac{23b}{6}}\)

4. SIMPLIFY to find the percentage

• We know \(\mathrm{c = \frac{p}{100}b}\) and we found \(\mathrm{c = \frac{23}{6}b}\)
• Setting equal: \(\mathrm{\frac{p}{100}b = \frac{23}{6}b}\)
• Divide both sides by b: \(\mathrm{\frac{p}{100} = \frac{23}{6}}\)
• Multiply by 100:
  \(\mathrm{p = \frac{23}{6} \times 100}\)
  \(\mathrm{p = \frac{2300}{6}}\)
  \(\mathrm{p \approx 383.33}\) (use calculator)

Answer: D. 383

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the direction of percentage relationships, writing equations like "b is 230% of a" instead of "a is 230% of b."

This reversal leads to setting up \(\mathrm{b = 2.30a}\) instead of \(\mathrm{a = 2.30b}\), which produces completely different relationships and typically results in values around 138 rather than 383. This may lead them to select Choice A (138).

Second Most Common Error:

Poor INFER reasoning: Students fail to recognize they can set the two expressions for 'a' equal to each other, instead trying to work with three separate equations without connecting them systematically.

This leads to confusion about how to proceed, causing them to abandon systematic solution and guess among the choices.

The Bottom Line:

This problem tests your ability to manage multiple percentage relationships simultaneously. The key breakthrough is realizing that when two different expressions both equal the same variable, you can set those expressions equal to each other to eliminate that variable and solve for what you need.