Two rectangular posters each have height h inches. The widths of the posters are 2h - 3 inches and 5h...
GMAT Advanced Math : (Adv_Math) Questions
Two rectangular posters each have height \(\mathrm{h}\) inches. The widths of the posters are \(2\mathrm{h} - 3\) inches and \(5\mathrm{h} + 1\) inches, respectively. What is the total area, in square inches, of the two posters in terms of \(\mathrm{h}\)?
1. TRANSLATE the problem information
- Given information:
- Two rectangular posters, both with height \(\mathrm{h}\) inches
- First poster width: \(\mathrm{2h - 3}\) inches
- Second poster width: \(\mathrm{5h + 1}\) inches
- Need: Total area in square inches
2. INFER the solution approach
- Since we need total area, we'll find each poster's area separately then add them
- Use the rectangle area formula: \(\mathrm{Area = height \times width}\)
3. SIMPLIFY to find each poster's area
For the first poster:
\(\mathrm{Area_1 = h \times (2h - 3)}\)
\(\mathrm{= h(2h - 3)}\)
\(\mathrm{= 2h^2 - 3h}\)
For the second poster:
\(\mathrm{Area_2 = h \times (5h + 1)}\)
\(\mathrm{= h(5h + 1)}\)
\(\mathrm{= 5h^2 + h}\)
4. SIMPLIFY to find the total area
\(\mathrm{Total\:area = Area_1 + Area_2}\)
\(\mathrm{= (2h^2 - 3h) + (5h^2 + h)}\)
\(\mathrm{= 2h^2 - 3h + 5h^2 + h}\)
\(\mathrm{= 7h^2 - 2h}\)
Answer: A (\(\mathrm{7h^2 - 2h}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students correctly set up both areas but make sign errors when combining like terms. They might calculate \(\mathrm{-3h + h = +2h}\) instead of \(\mathrm{-2h}\), getting \(\mathrm{7h^2 + 2h}\) as their final answer.
This leads them to select Choice B (\(\mathrm{7h^2 + 2h}\)).
Second Most Common Error:
Incomplete SIMPLIFY process: Students correctly find the individual areas but forget to combine the constant terms (the \(\mathrm{h}\) terms), only adding the \(\mathrm{h^2}\) coefficients. They might write \(\mathrm{2h^2 + 5h^2 = 7h^2}\) and stop there.
This causes them to select Choice D (\(\mathrm{7h^2}\)).
The Bottom Line:
This problem tests whether students can systematically work through multi-step algebraic manipulation while keeping track of signs and like terms. The key challenge is maintaining accuracy through the distribution and combining steps rather than rushing to a quick answer.