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? there is a scale drawing with a scale of 3 cm : 1.5 km, and some distances on the map: from waterfall to parking is 6 cm, parking to picnic area is 9 cm, parking to ranger tower is 3 cm.

Explanation:

Step1: Analyze the scale

The scale given is \(3\) cm : \(1.5\) km. We need to find the number (let's call it \(k\)) such that when we multiply the distance on the drawing (in cm) by \(k\), we get the actual distance (in km). So we can set up the equation \(3\times k = 1.5\).

Step2: Solve for \(k\)

To find \(k\), we divide both sides of the equation \(3k = 1.5\) by \(3\). So \(k=\frac{1.5}{3}= 0.5\) km per cm. Wait, no, actually, the scale is \(3\) cm represents \(1.5\) km, so to find the multiplier, we can calculate \(\frac{1.5\space km}{3\space cm}= 0.5\space km/cm\). But actually, if we consider the scale factor, for each cm on the drawing, the actual distance is \(0.5\) km? Wait, no, wait \(3\) cm is \(1.5\) km, so \(1\) cm is \(1.5\div3 = 0.5\) km? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \(3\) cm corresponds to \(1.5\) km, so to find the multiplier, we can find how much \(1\) cm corresponds to. So \(1.5\) km divided by \(3\) cm is \(0.5\) km per cm. But actually, the question is "what number can you multiply each distance in the scale drawing by to find the actual distance". So if the distance on the drawing is \(d\) (in cm), then actual distance \(D\) (in km) is \(D = d\times k\), where \(k\) is the multiplier. From the scale \(3\) cm \(= 1.5\) km, so \(k=\frac{1.5\space km}{3\space cm}= 0.5\space km/cm\). Wait, but let's check with the given scale. Wait, maybe the scale is \(3\) cm to \(1.5\) km, so the multiplier is \(1.5\div3 = 0.5\) km per cm? Wait, no, that would mean \(1\) cm is \(0.5\) km, \(2\) cm is \(1\) km, \(3\) cm is \(1.5\) km. Yes, that works. So the multiplier is \(0.5\) km per cm? Wait, but maybe the units are different. Wait, the question is about the number, so maybe the scale is in km and cm, but the multiplier is a number (with units, but the question says "what number", so probably the value. Wait, let's re - express. The scale is \(3\) cm : \(1.5\) km. So to find the multiplier, we can calculate \(\frac{1.5\space km}{3\space cm}=0.5\space km/cm\). But if we consider the scale factor, the multiplier is \(0.5\) (in km per cm). Wait, but maybe the question is expecting the multiplier as a number where the units are consistent. Wait, maybe the scale is \(3\) cm represents \(1.5\) km, so the conversion factor is \(1.5\div3 = 0.5\) km per cm. So when you multiply the distance in cm (on the drawing) by \(0.5\) (km/cm), you get the actual distance in km. Alternatively, maybe the scale is \(1\) cm : \(0.5\) km, so the multiplier is \(0.5\) (with units km per cm). But the question says "what number", so the numerical value is \(0.5\) (if we consider km per cm) or maybe \(500\) (if we convert to meters, since \(0.5\) km is \(500\) meters). Wait, no, the scale is \(3\) cm : \(1.5\) km. Let's do unit conversion. \(1.5\) km \( = 1500\) meters, \(3\) cm \( = 0.03\) meters. Then the scale factor is \(1500\div0.03 = 50000\). Wait, that's a big difference. Oh! I see my mistake. I confused km and cm. Let's convert everything to the same unit. Let's convert km to cm. \(1\) km \( = 100000\) cm, so \(1.5\) km \( = 1.5\times100000 = 150000\) cm. So the scale is \(3\) cm (drawing) : \(150000\) cm (actual). So the scale factor (multiplier) is \(150000\div3 = 50000\). Ah! That's the correct way. Because when dealing with scale drawings, the scale factor is usually a ratio of actual length to drawing length, and we need to have the same units. So let's redo this.

Step1: Convert actual distance to cm

The actual distance for \(3\) cm on the drawing is \(1.5\) km. Convert \(1.5\) km to cm. Since \(1\) km \( = 1000\) meters and \(1\) meter \( = 100\) cm, so \(1\) km \( = 1000\times100 = 100000\) cm. Then \(1.5\) km \( = 1.5\times100000 = 150000\) cm.

Step2: Find the scale factor (multiplier)

The scale is \(3\) cm (drawing) : \(150000\) cm (actual). So the multiplier (scale factor) is \(\frac{150000}{3}=50000\). So when we multiply the distance on the drawing (in cm) by \(50000\), we get the actual distance in cm. But the question says "find the actual distance", and the actual distance is in km? Wait, no, the scale is given as \(3\) cm : \(1.5\) km, so maybe we can work in km. Let's see, \(3\) cm corresponds to \(1.5\) km, so \(1\) cm corresponds to \(1.5\div3 = 0.5\) km? Wait, no, that would mean \(3\) cm is \(3\times0.5 = 1.5\) km, which matches. So if the distance on the drawing is \(d\) cm, then actual distance \(D\) km is \(D = d\times0.5\) km/cm? Wait, no, the units are cm (drawing) to km (actual). So the multiplier is \(0.5\) km per cm. But that seems odd. Wait, maybe the problem is that the scale is \(3\) cm represents \(1.5\) km, so the multiplier is \(1.5\div3 = 0.5\) km per cm. So if you have a distance of \(6\) cm on the drawing, actual distance is \(6\times0.5 = 3\) km. Let's check with the scale: \(3\) cm is \(1.5\) km, so \(6\) cm is \(3\) km, which is correct. \(9\) cm would be \(9\times0.5 = 4.5\) km, and \(3\) cm is \(1.5\) km. So that works. So the multiplier is \(0.5\) (with units km per cm) or \(500\) (with units meters per cm, since \(0.5\) km \( = 500\) meters) or \(50000\) (with units cm per cm). But the question says "what number can you multiply each distance in the scale drawing by to find the actual distance". The distance in the scale drawing is in cm, and the actual distance is in km. So the number is \(0.5\) (km per cm). Wait, but let's check the two approaches. If we use cm: drawing distance \(d\) cm, actual distance \(D\) cm is \(D = d\times50000\) (since \(3\) cm drawing is \(150000\) cm actual, so \(150000\div3 = 50000\)). Then to convert \(D\) cm to km, we divide by \(100000\), so \(D\) km \(=\frac{d\times50000}{100000}=d\times0.5\) km. So both methods are consistent. So the multiplier is \(0.5\) when the actual distance is in km, or \(50000\) when the actual distance is in cm. But the scale is given as \(3\) cm : \(1.5\) km, so the problem probably expects the multiplier in km per cm, so \(0.5\) km per cm, but as a number (without units, considering the context), it's \(0.5\)? Wait, no, that can't be. Wait, maybe I messed up the scale. Let's look at the image again. The scale is \(3\) cm : \(1.5\) km. So \(3\) cm on the map is \(1.5\) km in real life. So to find the multiplier, we can calculate how much \(1\) cm on the map is in real life. So \(1.5\) km divided by \(3\) cm is \(0.5\) km per cm. So if you have a distance of \(x\) cm on the map, the real distance is \(x\times0.5\) km. So the multiplier is \(0.5\) (in km per cm) or \(500\) (in meters per cm) or \(50000\) (in cm per cm). But the question is asking for the number to multiply each distance in the scale drawing (which is in cm) to get the actual distance (which is in km, as per the scale). So the number is \(0.5\) (km per cm). Wait, but let's verify with the given distances. For example, the distance from Waterfall to Parking is \(6\) cm. If we multiply by \(0.5\), we get \(6\times0.5 = 3\) km. The distance from Parking to Picnic Area is \(9\) cm, \(9\times0.5 = 4.5\) km. The distance from Parking to Ranger Tower is \(3\) cm, \(3\times0.5 = 1.5\) km. Which matches the scale (since \(3\) cm is \(1.5\) km). So that works. So the multiplier is \(0.5\). Wait, but earlier when I converted to cm, I got \(50000\), but that's because I was converting to cm. But the actual distance is in km, so the multiplier is \(0.5\) km per cm. So the answer is \(0.5\)? Wait, no, wait \(1.5\) km is \(1500\) meters, \(3\) cm is \(0.03\) meters. Then the scale factor is \(1500\div0.03 = 50000\) (meters per meter? No, centimeters to meters. Wait, no, the drawing is in cm, actual is in km. So the correct multiplier is \(0.5\) km per cm. So the number is \(0.5\).

Answer:

\(0.5\) (or \(50000\) if considering cm to cm, but based on the scale \(3\) cm : \(1.5\) km, the multiplier is \(0.5\) km per cm, so the number is \(0.5\))