Caleb used juice to make popsicles. The function \(\mathrm{f(x) = -5x + 30}\) approximates the volume, in fluid ounces, of...
GMAT Algebra : (Alg) Questions
Caleb used juice to make popsicles. The function \(\mathrm{f(x) = -5x + 30}\) approximates the volume, in fluid ounces, of juice Caleb had remaining after making x popsicles. Which statement is the best interpretation of the y-intercept of the graph of \(\mathrm{y = f(x)}\) in the xy-plane in this context?
Caleb used approximately \(5\) fluid ounces of juice for each popsicle.
Caleb had approximately \(5\) fluid ounces of juice when he began to make the popsicles.
Caleb had approximately \(30\) fluid ounces of juice when he began to make the popsicles.
Caleb used approximately \(30\) fluid ounces of juice for each popsicle.
1. TRANSLATE the function components
- Given: \(\mathrm{f(x) = -5x + 30}\)
- This represents fluid ounces remaining after making x popsicles
- Function is in form \(\mathrm{f(x) = mx + b}\) where:
- \(\mathrm{m = -5}\) (slope)
- \(\mathrm{b = 30}\) (y-intercept)
2. INFER what y-intercept means mathematically
- Y-intercept occurs when \(\mathrm{x = 0}\)
- Calculate: \(\mathrm{f(0) = -5(0) + 30 = 30}\)
- So the y-intercept point is \(\mathrm{(0, 30)}\)
3. TRANSLATE the mathematical result to context
- When \(\mathrm{x = 0}\): Caleb has made zero popsicles
- \(\mathrm{f(0) = 30}\): He has 30 fluid ounces remaining
- Since he hasn't made any popsicles yet, "30 ounces remaining" means he started with 30 ounces
Answer: C. Caleb had approximately 30 fluid ounces of juice when he began to make the popsicles.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the slope with the y-intercept or don't properly interpret what \(\mathrm{x = 0}\) means in context.
They might see that the coefficient -5 appears more prominently in the function and think this represents the y-intercept, or they might not realize that \(\mathrm{x = 0}\) represents the starting condition before any popsicles were made. Some students interpret "remaining juice" as something different from "started with" even when no popsicles have been made yet.
This may lead them to select Choice A (about 5 fluid ounces) or Choice D (about using 30 per popsicle).
Second Most Common Error:
Conceptual confusion about y-intercept: Students may correctly identify that the y-intercept is 30 but misunderstand what this value represents in the real-world context.
They might think the y-intercept represents the rate at which juice is used rather than the initial amount, or they might not connect that "remaining after 0 popsicles" is the same as "amount at the beginning."
This may lead them to select Choice D (Caleb used approximately 30 fluid ounces for each popsicle).
The Bottom Line:
This problem tests whether students can bridge between mathematical concepts (y-intercept) and real-world interpretation, specifically understanding that the starting condition \(\mathrm{(x = 0)}\) reveals the initial amount in a "remaining quantity" function.
Caleb used approximately \(5\) fluid ounces of juice for each popsicle.
Caleb had approximately \(5\) fluid ounces of juice when he began to make the popsicles.
Caleb had approximately \(30\) fluid ounces of juice when he began to make the popsicles.
Caleb used approximately \(30\) fluid ounces of juice for each popsicle.