Prealgebra Question of the Day

1. Slope represents climbing rate; using $\frac{2050}{3}$ instead of $\frac{1200}{2.5}$
2. Slope represents total elevation gained; using $\frac{2050-1200}{1.25}$ instead of $\frac{2050}{2.5}$
3. Slope represents climbing rate; using $\frac{2050}{3}$ instead of $\frac{2050-1200}{2.5}$ (correct answer)
4. Slope represents average elevation; using $\frac{2050-1200}{1.25}$ instead of $\frac{2050+1200}{2.5}$

Explanation: When you encounter linear relationships involving motion or change over time, always remember that slope represents the rate of change — in this case, how fast the climber gains elevation per hour. To find the correct slope (climbing rate), you need the change in elevation divided by the change in time. The climber starts at 1200 feet and reaches 2050 feet, so she climbs $2050 - 1200 = 850$ feet in 2.5 hours. Her climbing rate is $\frac{850}{2.5} = 340$ feet per hour. The mistake that gives 680 feet per hour comes from using $\frac{2050}{3}$. Someone might incorrectly think the denominator should be 3 (perhaps confusing 2.5 hours with something else) and use the final elevation 2050 instead of the elevation change. This gives $\frac{2050}{3} ≈ 683$ feet per hour, which rounds to about 680. Looking at the choices: Choice A incorrectly describes slope calculation methods. Choice B wrongly claims slope represents "total elevation gained" rather than rate, and describes an impossible calculation with 1.25 hours. Choice D incorrectly states slope represents "average elevation" and suggests adding elevations rather than finding their difference. Only choice C correctly identifies that slope represents climbing rate and pinpoints the exact mistake: using $\frac{2050}{3}$ instead of the proper formula $\frac{2050-1200}{2.5}$. Study tip: For any linear relationship involving change over time, always use $\text{slope} = \frac{\text{change in output}}{\text{change in input}}$. Don't forget to subtract initial values when finding the change.