Question

oscar has a map of the campground where he is staying. the scale shows that 4 inches on the map represents an actual distance of 2 miles. oscar’s tent is at campsite 4a. (a) what number can you multiply each length on the map by to find the actual distance? (b) what is the actual distance from oscar’s campsite to the beach? the image shows a map with $2\frac{1}{2}$ in from 4a to beach, 3 in to general store, and scale 4 in. : 2 mi.

Explanation:

Part (a)

Step1: Find the scale factor

The scale is 4 inches on the map represents 2 miles. To find the number to multiply map length by, we calculate the ratio of actual distance to map distance. So, the scale factor is $\frac{2\ \text{miles}}{4\ \text{inches}}=\frac{1}{2}\ \text{miles per inch}$.

Step1: Identify the map distance to the beach

From the map, the distance from Oscar's campsite (4A) to the beach is $2\frac{1}{2}$ inches (or $\frac{5}{2}$ inches).

Step2: Multiply by the scale factor

Using the scale factor from part (a) ($\frac{1}{2}$ miles per inch), we multiply the map distance by the scale factor: $\frac{5}{2}\times\frac{1}{2}=\frac{5}{4} = 1.25$? Wait, no, wait. Wait, the scale is 4 inches = 2 miles, so 1 inch = $\frac{2}{4}=\frac{1}{2}$ mile. Wait, the distance to the beach is $2\frac{1}{2}$ inches. So $2\frac{1}{2}\times\frac{1}{2}=\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}$? Wait, no, maybe I misread the map. Wait, the vertical distance from 4A to the beach is $2\frac{1}{2}$ inches. Wait, the scale is 4 inches = 2 miles, so 1 inch = 0.5 miles. So $2\frac{1}{2}$ inches is $2.5\times0.5 = 1.25$? Wait, no, wait the first part: 4 inches = 2 miles, so the conversion factor is (2 miles)/(4 inches) = 0.5 miles per inch. So for the beach, the map distance is $2\frac{1}{2}$ inches (which is 2.5 inches). So actual distance is 2.5 * 0.5 = 1.25? Wait, but maybe the problem for part (b) is the distance to the beach, which is $2\frac{1}{2}$ inches. Wait, let's re - calculate.

Wait, 4 inches on map = 2 miles actual. So the scale factor (actual per map) is $\frac{2}{4}=\frac{1}{2}$ mile per inch. The distance from 4A to the beach on the map is $2\frac{1}{2}$ inches (or $\frac{5}{2}$ inches). So actual distance = $\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}=1.25$? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, 4 inches = 2 miles, so 1 inch = 0.5 miles. So 2.5 inches * 0.5 miles per inch = 1.25 miles? Wait, but let's check again.

Wait, the first part: the question (a) is what number to multiply each length on the map by to get actual distance. Since 4 inches map = 2 miles actual, then the multiplier is $\frac{2}{4}=\frac{1}{2}$. So for part (b), the distance from 4A to the beach is $2\frac{1}{2}$ inches. So we multiply $2\frac{1}{2}$ by $\frac{1}{2}$. $2\frac{1}{2}=\frac{5}{2}$, so $\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}=1.25$? Wait, but maybe the distance to the beach is $2\frac{1}{2}$ inches. Wait, but let's see the map: the vertical distance from 4A to the beach is $2\frac{1}{2}$ inches. So the actual distance is $2\frac{1}{2}\times\frac{1}{2}=\frac{5}{2}\times\frac{1}{2}=\frac{5}{4} = 1.25$ miles? Wait, no, maybe I messed up the scale. Wait, 4 inches = 2 miles, so 2 inches = 1 mile, 1 inch = 0.5 miles. So $2\frac{1}{2}$ inches is 2.5 inches. 2.5 * 0.5 = 1.25. So the actual distance is 1.25 miles, which is $\frac{5}{4}$ or 1.25.

Wait, but let's do it step by step:

Step1: Determine the map distance

From the map, the distance from Oscar's campsite (4A) to the beach is $2\frac{1}{2}$ inches. We can write $2\frac{1}{2}$ as an improper fraction: $2\frac{1}{2}=\frac{2\times2 + 1}{2}=\frac{5}{2}$ inches.

Step2: Apply the scale factor

We know from part (a) that the scale factor (the number to multiply map length by to get actual distance) is $\frac{1}{2}$ (miles per inch). So we multiply the map distance by this scale factor:

Actual distance = $\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}=1.25$ miles.

Answer:

$\frac{1}{2}$

Part (b)