During a 5-second time interval, the average acceleration a, in meters per second squared, of an object with an initial...
GMAT Advanced Math : (Adv_Math) Questions
During a 5-second time interval, the average acceleration \(\mathrm{a}\), in meters per second squared, of an object with an initial velocity of 12 meters per second is defined by the equation \(\mathrm{a = \frac{v_f - 12}{5}}\), where \(\mathrm{v_f}\) is the final velocity of the object in meters per second. If the equation is rewritten in the form \(\mathrm{v_f = xa + y}\), where \(\mathrm{x}\) and \(\mathrm{y}\) are constants, what is the value of \(\mathrm{x}\)?
1. TRANSLATE the problem information
- Given equation: \(\mathrm{a = \frac{v_f - 12}{5}}\)
- Target form: \(\mathrm{v_f = xa + y}\)
- Find: the value of x (coefficient of a)
2. INFER the approach
- We need to solve the given equation for \(\mathrm{v_f}\) to match the target form
- This requires isolating \(\mathrm{v_f}\) on one side of the equation
- Once in the form \(\mathrm{v_f =}\) (something)\(\mathrm{a +}\) (something), the coefficient of a will be our x value
3. SIMPLIFY through algebraic manipulation
Starting equation: \(\mathrm{a = \frac{v_f - 12}{5}}\)
Step 1: Eliminate the fraction by multiplying both sides by 5
- \(\mathrm{5a = v_f - 12}\)
Step 2: Isolate \(\mathrm{v_f}\) by adding 12 to both sides
- \(\mathrm{v_f = 5a + 12}\)
4. TRANSLATE the final form
- Our result: \(\mathrm{v_f = 5a + 12}\)
- Target form: \(\mathrm{v_f = xa + y}\)
- Comparing: \(\mathrm{x = 5, y = 12}\)
Answer: 5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize they need to solve for \(\mathrm{v_f}\) rather than trying to manipulate the equation in its current form. They might attempt to directly identify x and y from \(\mathrm{a = \frac{v_f - 12}{5}}\), leading to confusion about which variable should be isolated. This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students make algebraic manipulation errors, such as incorrectly handling the fraction or making sign errors when moving terms. For example, they might get \(\mathrm{v_f = 5a - 12}\) instead of \(\mathrm{v_f = 5a + 12}\), leading them to think \(\mathrm{x = 5}\) but miss the correct relationship entirely.
The Bottom Line:
This problem tests whether students can strategically recognize that equation rearrangement is needed and execute basic algebraic manipulation accurately. The key insight is understanding that "rewrite in the form" means solve for the specified variable.