Let’s pick this up right where the numbers left off — Page 281, Lesson 246

Let’s pick this up right where the numbers left off. We had a Representative Fraction of 5 over 1,296,400, and I showed you that we always want the numerator to be 1. So we divide both the top and the bottom by 5, and we get 1 over 259,280. That’s our final Representative Fraction. Now, I want to give you a really practical calculator tip, because in the exam you don’t want to be fumbling. The easiest way to enter this in one go is to evaluate the denominator first, then divide by the numerator. So for that example, you’d key in 7 times 1852 times 100, and then divide by 5. That gives you 259,280, which is the denominator. You then just write it down as 1 over 259,280. If instead you enter the numerator first, you get the RF expressed as a decimal. So 5 divided by the whole bracket, 7 times 1852 times 100, gives you 0.000003856. That’s a perfectly valid number, and if you want to use your x⁻¹ key to flip it, fine. But honestly, it’s quicker to do it the first way — denominator first. Now let’s talk about the types of scale question you’ll meet. There are two basic subdivisions: Simple Scale and Mercator Scale. Mercator Scale problems are dealt with in a later chapter, so for now we focus entirely on Simple Scale. And here’s the beautiful thing — every single Simple Scale problem is solved by substituting into one basic equation: RF = CL over ED. Let me define those letters for you. RF is the Representative Fraction, the scale of the chart. CL is the Chart Length, the distance measured on the chart itself. ED is the Earth Distance, the actual distance on the ground. So the equation says: the scale equals the chart length divided by the earth distance. Now, there are three ways this question can be posed. Sometimes you’re given the Chart Length and the Earth Distance, and asked to find the RF — that’s what we did in the previous example. Sometimes you’re given the scale, the RF, and the Earth Distance, and asked to find the Chart Length. And sometimes you’re given the RF and the Chart Length, and asked to find the Earth Distance. The method is always the same. You substitute what you know into that equation, apply any conversion factors necessary to get both CL and ED in the same units, rearrange if needed to make the unknown the subject of the equation, and then solve with your calculator. Let me walk you through Example 2 to show you exactly how that works. The question is: on a chart of scale 1 over 5,000,000, how many nautical miles on the Earth are represented by 1.7 centimetres? So we write down our equation: RF equals CL over ED. We know RF is 1 over 5,000,000. We know CL is 1.7 centimetres. And ED is what we’re trying to find. So we substitute: 1 over 5,000,000 equals 1.7 centimetres over ED. Now we rearrange to make ED the subject. We cross-multiply, so ED equals 1.7 times 5,000,000. But wait — that gives us an answer in centimetres, and the question asks for nautical miles. So we need to convert. Here’s the conversion. One nautical mile is 1852 metres. And one metre is 100 centimetres. So to get from centimetres to nautical miles, we divide by 1852 and then divide by 100. That’s why you see that denominator in the working: 1852 times 100. So ED equals 1.7 times 5,000,000, all divided by 1852 times 100. Work that out and you get the Earth distance in nautical miles. Let me just make sure you see the logic of that conversion. We measured 1.7 centimetres on the chart. The scale tells us that 1 centimetre on the chart represents 5,000,000 centimetres on the Earth. So 1.7 centimetres represents 1.7 times 5,000,000 centimetres on the ground. But we don’t want centimetres, we want nautical miles. So we convert: first from centimetres to metres by dividing by 100, then from metres to nautical miles by dividing by 1852. Put those together and you divide by 1852 times 100. That’s the whole of Simple Scale. One equation, three possible unknowns, and the only real trap is getting your units consistent. Get CL and ED into the same units before you substitute, and the arithmetic takes care of itself.