The table shows the cost of buying pencils at the school bookstore. If the data were graphed as
, where
is the number of pencils and
is the cost, what would
, the slope of the line, represent? Cost of Pencils Number of Pencils Cost 3
2.45 12
6.30 the number of pencils that can be purchased for
3 the cost of purchasing three pencils the cost of purchasing one pencil Done?
The Correct Answer and Explanation is:
Correct Answer: The cost of purchasing one pencil
Explanation
The question is asking about the meaning of the slope (m) in the linear equation c=mp+bc = mp + b, where:
- pp is the number of pencils,
- cc is the total cost,
- mm is the slope, and
- bb is the y-intercept.
In this context, the slope mm represents the rate of change of cost with respect to the number of pencils — essentially, how much the cost increases for each additional pencil purchased.
To determine this, we calculate the slope using two data points from the table. Let’s use:
- (3 pencils, $1.05)
- (7 pencils, $2.45)
Using the slope formula: m=ΔcΔp=2.45−1.057−3=1.404=0.35m = \frac{\Delta c}{\Delta p} = \frac{2.45 – 1.05}{7 – 3} = \frac{1.40}{4} = 0.35
So, the slope m=0.35m = 0.35, which means each pencil costs $0.35. The units of the slope are dollars per pencil.
Now, let’s analyze what that slope represents among the answer choices:
- “The number of pencils that can be purchased for $1” → That would be the reciprocal (about 2.86), not the slope.
- “The number of pencils that can be purchased for $3” → Also a reciprocal-based value, not the slope.
- “The cost of purchasing three pencils” → That would be 3 × 0.35 = $1.05, which is a specific total, not the rate.
- ✅ “The cost of purchasing one pencil” → This matches exactly with the definition and calculation of slope: $0.35 per pencil.
Thus, the slope mm represents the cost of purchasing one pencil, making that the correct and most logical answer.
