The area A of a trapezoid is given by \(\mathrm{A = \frac{1}{2} h (b + B)}\), where h, b, B,...
GMAT Advanced Math : (Adv_Math) Questions
The area A of a trapezoid is given by \(\mathrm{A = \frac{1}{2} h (b + B)}\), where h, b, B, and A are positive quantities, h is the height, and b and B are the lengths of the two parallel sides. Which equation correctly expresses B in terms of A, h, and b?
1. INFER the problem approach
- This is asking us to solve for B in terms of the other variables
- Strategy: Use algebraic manipulation to isolate B on one side of the equation
- We'll work systematically to 'undo' the operations affecting B
2. SIMPLIFY by eliminating the fraction first
Starting with: \(\mathrm{A = \frac{1}{2}h(b + B)}\)
- Multiply both sides by 2 to eliminate the fraction:
\(\mathrm{2A = h(b + B)}\)
3. SIMPLIFY by isolating the parentheses
- Divide both sides by h:
\(\mathrm{\frac{2A}{h} = b + B}\)
4. SIMPLIFY to isolate B completely
- Subtract b from both sides:
\(\mathrm{B = \frac{2A}{h} - b}\)
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make errors when manipulating the algebraic expressions, particularly with order of operations or fraction handling.
Some students might subtract b in the numerator before dividing by h, getting \(\mathrm{B = \frac{2A - b}{h}}\) instead of the correct \(\mathrm{B = \frac{2A}{h} - b}\). This leads them to select Choice A.
Second Most Common Error:
Poor SIMPLIFY execution with signs: Students might make sign errors during the final step.
Instead of subtracting b, they might accidentally add it, getting \(\mathrm{B = \frac{2A}{h} + b}\). This causes them to select Choice C.
The Bottom Line:
Success on this problem depends entirely on careful, step-by-step algebraic manipulation. The key insight is recognizing that you must work systematically to isolate B, treating each algebraic operation as a deliberate step toward your goal.