TenMarks teaches you how to apply concepts of similar figures to find actual distances using maps.
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Learn about Maps, Models, and Actual Distance Actual Measurements Let’s learn about actual measurement when it comes to scaling. Let’s look at the question which says that on a map, the distance between City A and City B is 20 inches. So, the distance A to B on the map is 20 inches. That’s one fact given to us. It has also given to us that the scale on the map is 2 inches=4. Scale is 2 inches to 4.5 miles or 2 inches =4.5 mile, same thing. What is scale? Scale is the ratio of measurement on a drawing or a model to measurement of the actual object. So, if we all know that the ratio is 2 inches equals 4.5 miles, the scale ratio equals 2 inches/4.5 miles. That’s one fact given to us. Let’s create a little bit of extra space and look at the second fact. What we are also given is that the distance between A and B on the map equals 20 inches. We know that the distance on map equals 20 inches. Let’s say distance in reality, A to B actual. The actual distance, let’s say that is Y miles. The ratio between these two, again, it’s the distance on a map to distance actually, that’s also a scale. So, scale according to the numbers that we’ve been given is 20 inches/Y miles. That’s also a scale because remember the scale is the ratio of measurement on drawing which is 20 inches to measurement of actual object which we said is, let’s say it’s Y miles. So now, we know that this one scale and this is a second scale but these two scales are equal or equivalent or in proportion. They are one and the same. Let’s look at them which mean 2 inches/4.5 miles is the same as 20 inches/Y miles. If this is indeed true, then their cross product is 2 as well. So, if I remove the units for a minute, 2/4.5=20/Y. All I did was remove miles and inches form both sides. So, if we’ve got that then the cross products are the same. So, 2×y=4.5×20, so dividing both sides by 2, I get y=4.5×10, 20/2 is 10, equals 45. If y equals 45 and y was the actual distance between A and B in miles, then what we know is the actual distance between A and B is 45 miles. That’s the answer we were looking for. To quickly recap what we did here is we looked at the data we were given. If we know that the distance on a map is 20 inches and the scale is given to us 2:4.5 miles, then we can use the scale as one ratio, we know that the distance on the map is 20 inches. Let’s say the distance in reality is Y miles. We get a second ratio which is 20 inches=y miles. Since both of these are the scale, they are the same. Once I got two ratios with one variable, we can easily cross multiply to get y=45 or the distance in real life between A and B Cities is 45 miles.