Question

1. if 1 inch is equal to 150 miles, what is the scale factor?

options:

• \\(\frac{75}{1}\\)
• \\(\frac{1}{150}\\)
• \\(\frac{15}{1}\\)
• \\(\frac{1}{75}\\)

Explanation:

Step1: Recall scale factor definition

Scale factor is the ratio of the drawing length to the actual length. Here, drawing length (in inches) is 1, actual length (in miles) needs to be converted to inches? Wait, no—wait, actually, when the scale is 1 inch = 150 miles, we need to express the scale factor as (drawing unit)/(actual unit). But since both are linear, the scale factor is the ratio of the model (drawing) length to the real - world length.

First, we know that 1 mile = 63360 inches. But wait, maybe we don't need to convert units here? Wait, no—wait, the scale factor for a map is usually the ratio of the distance on the map (in the same unit as the actual distance) to the actual distance. Wait, no, let's think again. If 1 inch (map) represents 150 miles (actual), to find the scale factor, we can consider the ratio of the map distance to the actual distance. But we need to have the same units. Let's convert 150 miles to inches.

1 mile = 63360 inches, so 150 miles = 150×63360 inches = 9504000 inches. But that seems complicated. Wait, maybe the question is considering the scale factor as the ratio of the drawing length (1 inch) to the actual length (150 miles) in terms of the same linear measurement, but maybe the question is simplified. Wait, no—wait, the scale factor is (length on drawing)/(length in real life). If 1 inch (drawing) = 150 miles (real), then the scale factor is 1 inch / 150 miles. But since we are dealing with a scale factor, and if we consider the ratio of the two lengths (with the same unit), but maybe the question is using the ratio of the drawing length to the actual length without unit conversion (assuming that we are just taking the numerical ratio of the two quantities with their respective units, but that's not standard). Wait, no—maybe the question has a typo or is simplified. Wait, the options are fractions like 1/150, 75/1, etc. Let's think differently. The scale factor is the ratio of the model (drawing) to the actual. So if 1 inch (model) represents 150 miles (actual), the scale factor is 1 (inch) / 150 (miles). But since we are looking for the scale factor as a fraction, and the options are given as fractions without units, maybe the question is considering the ratio of the two quantities as numbers. So 1 (the number of inches) to 150 (the number of miles), so the scale factor is 1/150. Wait, that makes sense with the option $\frac{1}{150}$.

Step2: Analyze the options

• Option 1: $\frac{75}{1}$: This would mean the drawing is 75 times the actual length, which is not the case here.
• Option 2: $\frac{1}{150}$: If 1 inch (drawing) corresponds to 150 miles (actual), the ratio of drawing to actual is 1/150.
• Option 3: $\frac{15}{1}$: Drawing is 15 times actual, not correct.
• Option 4: $\frac{1}{75}$: 1 inch would correspond to 75 miles, which is not the case.

Answer:

$\frac{1}{150}$ (the second option, i.e., the option with $\frac{1}{150}$)