The formula P = 2s + b_1 + b_2 gives the perimeter, P, of an isosceles trapezoid in terms of...
GMAT Advanced Math : (Adv_Math) Questions
The formula \(\mathrm{P = 2s + b_1 + b_2}\) gives the perimeter, \(\mathrm{P}\), of an isosceles trapezoid in terms of the length of its non-parallel sides, \(\mathrm{s}\), and the lengths of its bases, \(\mathrm{b_1}\) and \(\mathrm{b_2}\). Which of the following equations correctly expresses \(\mathrm{s}\) in terms of \(\mathrm{P}\), \(\mathrm{b_1}\), and \(\mathrm{b_2}\)?
1. TRANSLATE the problem requirement
- Given: Formula \(\mathrm{P = 2s + b_1 + b_2}\)
- Goal: Express s in terms of P, b₁, and b₂
- What this means: Isolate s on one side of the equation
2. SIMPLIFY through inverse operations
- Current equation: \(\mathrm{P = 2s + b_1 + b_2}\)
- I see that s is multiplied by 2, then b₁ and b₂ are added
- To isolate s, I need to "undo" these operations in reverse order
3. SIMPLIFY by removing the addition first
- Subtract b₁ and b₂ from both sides:
\(\mathrm{P - b_1 - b_2 = 2s + b_1 + b_2 - b_1 - b_2}\)
\(\mathrm{P - b_1 - b_2 = 2s}\)
4. SIMPLIFY by removing the multiplication
- Divide both sides by 2:
\(\mathrm{\frac{P - b_1 - b_2}{2} = \frac{2s}{2}}\)
\(\mathrm{\frac{P - b_1 - b_2}{2} = s}\)
Answer: D. \(\mathrm{s = \frac{P - b_1 - b_2}{2}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students incorrectly distribute the division operation.
Instead of dividing the entire expression \(\mathrm{(P - b_1 - b_2)}\) by 2, they might divide only P by 2, getting \(\mathrm{s = \frac{P}{2} - b_1 - b_2}\). This happens because they don't recognize that when dividing both sides by 2, everything on the left side must be divided by 2.
This may lead them to select Choice A \(\mathrm{(\frac{P}{2} - b_1 - b_2)}\)
Second Most Common Error:
Poor SIMPLIFY reasoning: Students use the wrong inverse operation for the bases.
Instead of subtracting b₁ and b₂ to "undo" their addition, they might mistakenly add them, getting \(\mathrm{P + b_1 + b_2 = 2s}\), then \(\mathrm{s = \frac{P + b_1 + b_2}{2}}\).
This may lead them to select Choice B \(\mathrm{(\frac{P + b_1 + b_2}{2})}\)
The Bottom Line:
This problem tests whether students can systematically apply inverse operations in the correct order. The key insight is working backwards through the operations: first undo addition/subtraction, then undo multiplication/division.