Question:The function \(\mathrm{f(x) = \frac{5}{8}(16x - 24) + 6x}\) represents a linear relationship. When \(\mathrm{f(x)}\) is written in simplest f...
GMAT Algebra : (Alg) Questions
The function \(\mathrm{f(x) = \frac{5}{8}(16x - 24) + 6x}\) represents a linear relationship. When \(\mathrm{f(x)}\) is written in simplest form as \(\mathrm{f(x) = mx + b}\), where \(\mathrm{m}\) and \(\mathrm{b}\) are constants, what is the value of \(\mathrm{m}\)?
Enter your answer as an integer.
1. TRANSLATE the problem requirements
- Given: \(\mathrm{f(x) = \frac{5}{8}(16x - 24) + 6x}\)
- Find: The value of m when written as \(\mathrm{f(x) = mx + b}\)
- This means we need to simplify the expression and identify the coefficient of x
2. SIMPLIFY by distributing first
- Apply the distributive property to \(\mathrm{\frac{5}{8}(16x - 24)}\):
- \(\mathrm{\frac{5}{8} \times 16x = \frac{80x}{8} = 10x}\)
- \(\mathrm{\frac{5}{8} \times 24 = \frac{120}{8} = 15}\)
- So: \(\mathrm{f(x) = 10x - 15 + 6x}\)
3. SIMPLIFY by combining like terms
- Combine the x terms: \(\mathrm{10x + 6x = 16x}\)
- Keep the constant: \(\mathrm{-15}\)
- Result: \(\mathrm{f(x) = 16x - 15}\)
4. TRANSLATE to identify m
- In the form \(\mathrm{f(x) = mx + b}\), we have \(\mathrm{f(x) = 16x + (-15)}\)
- Therefore: \(\mathrm{m = 16, b = -15}\)
Answer: 16
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make arithmetic errors during distribution, particularly with fractions.
Common mistake: \(\mathrm{\frac{5}{8} \times 16x = 5x}\) (forgetting to multiply \(\mathrm{5 \times 16 = 80}\), then divide by 8)
Or: \(\mathrm{\frac{5}{8} \times 24 = 3}\) (incorrect fraction arithmetic)
This leads to wrong coefficients when combining terms, resulting in an incorrect value for m and confusion about the final answer.
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly simplify but forget what the question is asking for.
They might calculate \(\mathrm{f(x) = 16x - 15}\) correctly but then provide the wrong component as their answer (giving -15 instead of 16, or giving the entire expression).
This causes them to select an incorrect answer even though their algebra work was correct.
The Bottom Line:
This problem tests both algebraic manipulation skills and careful attention to what's being asked. Success requires precise fraction arithmetic and clear identification of which coefficient represents m.