Question:A rectangular garden has length 3/4x + 5/2 meters and width 1/2x + 1/2 meters, where x is a positive...
GMAT Advanced Math : (Adv_Math) Questions
- A rectangular garden has length \(\frac{3}{4}\mathrm{x} + \frac{5}{2}\) meters and width \(\frac{1}{2}\mathrm{x} + \frac{1}{2}\) meters, where \(\mathrm{x}\) is a positive real number.
- The area \(\mathrm{A(x)}\), in square meters, is the product of its length and width.
- When \(\mathrm{A(x)}\) is written in the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), where \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\) are constants, what is the value of \(\mathrm{b}\)?
1. TRANSLATE the problem information
- Given information:
- Length: \(\frac{3}{4}\mathrm{x} + \frac{5}{2}\) meters
- Width: \(\frac{1}{2}\mathrm{x} + \frac{1}{2}\) meters
- Area \(\mathrm{A(x)} = \text{length} \times \text{width}\)
- Need coefficient b when \(\mathrm{A(x)}\) is written as \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)
- What this tells us: We need to multiply the two binomials and find the coefficient of the x term.
2. SIMPLIFY by expanding the product
- Set up the multiplication: \(\mathrm{A(x)} = \left(\frac{3}{4}\mathrm{x} + \frac{5}{2}\right)\left(\frac{1}{2}\mathrm{x} + \frac{1}{2}\right)\)
- Use FOIL method:
- First terms: \(\left(\frac{3}{4}\mathrm{x}\right)\left(\frac{1}{2}\mathrm{x}\right) = \frac{3}{8}\mathrm{x}^2\)
- Outer terms: \(\left(\frac{3}{4}\mathrm{x}\right)\left(\frac{1}{2}\right) = \frac{3}{8}\mathrm{x}\)
- Inner terms: \(\left(\frac{5}{2}\right)\left(\frac{1}{2}\mathrm{x}\right) = \frac{5}{4}\mathrm{x}\)
- Last terms: \(\left(\frac{5}{2}\right)\left(\frac{1}{2}\right) = \frac{5}{4}\)
3. SIMPLIFY by combining like terms
- Combine the x terms: \(\frac{3}{8}\mathrm{x} + \frac{5}{4}\mathrm{x}\)
- Find common denominator: \(\frac{5}{4} = \frac{10}{8}\)
- Add: \(\frac{3}{8}\mathrm{x} + \frac{10}{8}\mathrm{x} = \frac{13}{8}\mathrm{x}\)
- Final form: \(\mathrm{A(x)} = \frac{3}{8}\mathrm{x}^2 + \frac{13}{8}\mathrm{x} + \frac{5}{4}\)
Answer: \(\frac{13}{8}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students correctly expand using FOIL but make arithmetic errors when adding the fractions \(\frac{3}{8} + \frac{5}{4}\). They might add numerators directly (3 + 5 = 8) and denominators directly (8 + 4 = 12), getting \(\frac{8}{12} = \frac{2}{3}\), or they might incorrectly convert \(\frac{5}{4}\) to eighths as \(\frac{5}{8}\) instead of \(\frac{10}{8}\).
This leads to confusion and incorrect final answers.
Second Most Common Error:
Incomplete SIMPLIFY process: Students successfully use FOIL to get the four terms but forget to combine the like terms (\(\frac{3}{8}\mathrm{x}\) and \(\frac{5}{4}\mathrm{x}\)). They might report one of the individual coefficients (\(\frac{3}{8}\) or \(\frac{5}{4}\)) instead of their sum.
This causes them to provide a partial answer rather than the complete coefficient.
The Bottom Line:
This problem tests precision in fraction arithmetic within algebraic expansion. Success requires both systematic binomial multiplication and careful fraction addition with different denominators.