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? (\frac{1}{2}) (b) what is the actual distance from oscar’s campsite to the beach? (square) mi (the map shows 2(\frac{1}{2}) in. from 4a to the beach, 3 in. to general store, campsites 4b, 4a, 4c, 4d, 4e, 4f, and scale 4 in. : 2 mi)

Explanation:

Part (a)

Step1: Find the scale factor

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

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

We use the scale factor from part (a) ($\frac{1}{2}\ \frac{miles}{inch}$). So, actual distance = map distance $\times$ scale factor. That is $\frac{5}{2}\ inches\times\frac{1}{2}\ \frac{miles}{inch}=\frac{5}{4}= 1.25$ miles? Wait, no, wait. Wait, the scale is 4 inches to 2 miles, so 1 inch is $\frac{2}{4}=\frac{1}{2}$ 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, wait, 4 inches is 2 miles, so 1 inch is $\frac{2}{4}=0.5$ miles. So $2.5$ inches times $0.5$ miles per inch. $2.5\times0.5 = 1.25$? Wait, but let's check again. Wait, the scale is 4 in : 2 mi, so the conversion factor is (2 mi)/(4 in) = 0.5 mi/in. The distance from 4A to the beach is $2\frac{1}{2}$ in, which is 2.5 in. So actual distance = 2.5 in * 0.5 mi/in = 1.25 mi? Wait, but maybe I misread the map. Wait, the map shows the distance from 4A to the beach is $2\frac{1}{2}$ inches? Wait, the vertical arrow is $2\frac{1}{2}$ in from 4A to the beach? Yes. So:

Step1: Convert the mixed number to improper fraction

$2\frac{1}{2}=\frac{5}{2}$ inches.

Step2: Multiply by the scale factor ($\frac{1}{2}$ mi per inch)

Actual distance = $\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}=1.25$? Wait, no, wait, 4 inches is 2 miles, so 1 inch is 2/4 = 0.5 miles. So 2.5 inches * 0.5 miles per inch = 1.25 miles. Wait, but let's do it with the scale. The scale is 4 in = 2 mi, so the ratio of actual to map is 2 mi / 4 in = 0.5 mi/in. So for a map distance of $d$ inches, actual distance is $d\times0.5$ miles. So for $d = 2.5$ inches, actual distance is $2.5\times0.5 = 1.25$ miles, which is $\frac{5}{4}$ or $1\frac{1}{4}$ miles. Wait, but maybe I made a mistake. Wait, let's check the first part again. The first part: what number to multiply each length on the map by to get actual distance. Since 4 in map = 2 mi actual, then 1 in map = 2/4 = 0.5 mi actual. So the number is 0.5, which is 1/2. So that's correct. Then for the beach, the distance from 4A to the beach is $2\frac{1}{2}$ inches. So multiply by 1/2: $2\frac{1}{2}\times\frac{1}{2}=\frac{5}{2}\times\frac{1}{2}=\frac{5}{4}=1.25$ miles. Wait, but 2.5 times 0.5 is 1.25. So the actual distance is 1.25 miles, or $\frac{5}{4}$ miles, or $1\frac{1}{4}$ miles.

Answer:

$\frac{1}{2}$

Part (b)