Question

chaya has a map that shows the lengths of some hiking trails. the scale is shown in the drawing. what number can you multiply each distance in the scale drawing by to find the actual distance? ? (image shows map with waterfall, parking, picnic area, ranger tower; distances 6 cm, 9 cm, 3 cm; scale 3 cm : 1.5 km)

Explanation:

Step1: Understand the scale

The scale given is \(3\) cm (on the map) corresponds to \(1.5\) km (actual distance). We need to find the multiplier \(k\) such that map distance \(\times k=\) actual distance.

Step2: Calculate the multiplier

We know that for \(3\) cm (map), actual distance is \(1.5\) km. So, to find \(k\), we can set up the equation \(3\times k = 1.5\) (in km, but we are just finding the multiplier, so we can solve for \(k\)).
Solving for \(k\), we divide both sides by \(3\): \(k=\frac{1.5}{3}\)
Calculating \(\frac{1.5}{3}=0.5\)? Wait, no, wait. Wait, actually, the scale is \(3\) cm to \(1.5\) km, so the multiplier is the actual distance per map distance. So \(1.5\) km per \(3\) cm. Let's convert that to per cm. So \(\frac{1.5\space km}{3\space cm}= 0.5\space km/cm\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's check again. Wait, the scale is \(3\) cm represents \(1.5\) km. So to find how much \(1\) cm represents, we do \(1.5\div3 = 0.5\) km per cm? Wait, no, wait, \(3\) cm is \(1.5\) km, so \(1\) cm is \(1.5\div3 = 0.5\) km? Wait, no, \(1.5\) km divided by \(3\) cm is \(0.5\) km per cm. But that seems small. Wait, maybe the scale is \(3\) cm to \(1.5\) km, so the multiplier is the ratio of actual to map. So actual distance = map distance \(\times\) (actual / map). So actual is \(1.5\) km, map is \(3\) cm. So \(1.5\) km / \(3\) cm = \(0.5\) km/cm. But that would mean that \(1\) cm on the map is \(0.5\) km in real life. Wait, but let's check with the given distances. For example, the distance from Parking to Ranger Tower is \(3\) cm. So actual distance would be \(3\times0.5 = 1.5\) km, which matches the scale. Similarly, the distance from Waterfall to Parking is \(6\) cm, so actual is \(6\times0.5 = 3\) km. And from Parking to Picnic Area is \(9\) cm, actual is \(9\times0.5 = 4.5\) km. Wait, but maybe the scale is given as \(3\) cm : \(1.5\) km, so the multiplier is \(1.5\div3 = 0.5\) km per cm, but the question is "what number can you multiply each distance in the scale drawing by to find the actual distance". So the number is \(0.5\) (in km per cm), but wait, maybe the units are such that we can think of it as a ratio. Wait, \(3\) cm corresponds to \(1.5\) km, so the multiplier is \(1.5\div3 = 0.5\) when the map distance is in cm and actual in km. Wait, but let's do the calculation again. Let's let \(x\) be the map distance (in cm) and \(y\) be the actual distance (in km). Then the scale is \(y = k\times x\), where \(k\) is the multiplier. We know that when \(x = 3\), \(y = 1.5\). So \(1.5 = k\times3\), so \(k=\frac{1.5}{3}=0.5\). Wait, but that would mean that each cm on the map is \(0.5\) km in real life. So the multiplier is \(0.5\) (km per cm). But let's check with the example. If the map distance is \(3\) cm, then \(3\times0.5 = 1.5\) km, which matches the scale. So the multiplier is \(0.5\)? Wait, no, wait, maybe I messed up the units. Wait, \(1.5\) km is \(1500\) meters, and \(3\) cm is \(0.03\) meters. Then the scale factor would be \(1500\div0.03 = 50000\). But that's a scale factor for unit conversion. But the question is asking for the number to multiply each distance in the scale drawing (which is in cm) to get actual distance (in km). So from the scale \(3\) cm : \(1.5\) km, we can write the ratio as \(\frac{1.5\space km}{3\space cm}=0.5\space km/cm\). So the number is \(0.5\) (when the map distance is in cm, multiplying by \(0.5\) gives km). Wait, but let's confirm. If the map distance is \(6\) cm, then \(6\times0.5 = 3\) km. If we check the scale, \(3\) cm is \(1.5\) km, so \(6\) cm is twice that, so \(3\) km, which is correct. Similarly, \(9\) cm is three times \(3\) cm, so \(4.5\) km, and \(9\times0.5 = 4.5\) km, which is correct. So the multiplier is \(0.5\) (km per cm). Wait, but maybe the question is expecting the multiplier in terms of km per cm, so the answer is \(0.5\). Wait, but let's do the calculation again. The scale is \(3\) cm represents \(1.5\) km. So to find the multiplier, we divide the actual distance by the map distance: \(1.5\div3 = 0.5\). So the number is \(0.5\).

Answer:

\(0.5\)