Square P has a side length of x inches. Square Q has a perimeter that is 176 inches greater than...
GMAT Advanced Math : (Adv_Math) Questions
Square P has a side length of \(\mathrm{x}\) inches. Square Q has a perimeter that is \(\mathrm{176}\) inches greater than the perimeter of square P. The function \(\mathrm{f}\) gives the area of square Q, in square inches. Which of the following defines \(\mathrm{f}\)?
1. TRANSLATE the problem information
- Given information:
- Square P has side length \(\mathrm{x}\) inches
- Square Q has perimeter 176 inches greater than square P's perimeter
- Need function f that gives area of square Q
2. INFER what we need to find the area
- To find the area of square Q, we need its side length
- To find the side length, we need to work with the perimeter information first
3. TRANSLATE and calculate square P's perimeter
- Square P side length = \(\mathrm{x}\) inches
- Square P perimeter = \(\mathrm{4x}\) inches (using perimeter formula)
4. TRANSLATE the relationship for square Q's perimeter
- Square Q perimeter = Square P perimeter + 176
- Square Q perimeter = \(\mathrm{4x + 176}\) inches
5. SIMPLIFY to find square Q's side length
- Since perimeter = 4 × side length
- Side length of Q = (perimeter of Q)/4
- Side length of Q = \(\mathrm{\frac{4x + 176}{4}}\) = \(\mathrm{x + 44}\) inches
6. SIMPLIFY to find square Q's area
- Area = (side length)²
- Area of Q = \(\mathrm{(x + 44)^2}\) square inches
- Therefore: \(\mathrm{f(x) = (x + 44)^2}\)
Answer: A. \(\mathrm{f(x) = (x + 44)^2}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "perimeter that is 176 inches greater" as meaning the side length is 176 inches greater, not the perimeter.
They might think: If the perimeter is 176 greater, then the side length is also 176 greater, so the side length of Q is \(\mathrm{x + 176}\).
This leads them to calculate area as \(\mathrm{(x + 176)^2}\), causing them to select Choice B (\(\mathrm{f(x) = (x + 176)^2}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that the perimeter of Q is \(\mathrm{4x + 176}\), but make arithmetic errors when dividing by 4 to find the side length.
They might incorrectly manipulate the expression or forget to divide the entire perimeter by 4, leading to confusion about which answer choice represents the correct side length.
This causes them to get stuck and guess among the remaining choices.
The Bottom Line:
Success requires carefully distinguishing between perimeter and side length, then systematically working through the relationship between them using the square's geometric properties.