Let me start with the definition — Page 281, Lesson 244

I want to walk you through the concept of scale — this is one of the most fundamental ideas in navigation because a chart is useless unless you know how distances on it relate to real distances on the Earth. Let me start with the definition. Scale is the relationship between the length of a line drawn between two positions on a chart and the distance on the Earth between the same two points. In other words, if I measure a certain distance on the paper, what does that correspond to in the real world? There are three common methods of expressing scale, and I want to cover each one. First: Statement in words. This is exactly what it sounds like — you state the relationship in plain language. For example, the statement "One inch to ten nautical miles" means that a line one inch long on the chart represents a line ten nautical miles long on the Earth. This method is used by Jeppesen charts; for instance, the Jeppesen E(HI) 1/2 chart uses a scale of 1 inch to 20 nautical miles. Second: Graduated Scale Line. These are normally depicted at the bottom of the chart — you'll see them as a bar or ruler. They may be in nautical miles, statute miles, or kilometres. Here's an important connection: since one degree of latitude is equal to 60 nautical miles, the latitude scale on a chart effectively provides a graduated scale line. So if you look at the latitude markings along the side of a chart, you can use them to measure distances directly. There's a figure showing this — — which illustrates both a graduated scale line and a latitude scale. Third: Representative Fraction. This is the statement in words put into mathematical form. The equation is: Scale = Chart Length divided by Earth Distance By convention, the chart length is always 1. The dimensions above and below the line — the numerator and denominator — are the same unit, so they cancel each other out, making the fraction dimensionless. That's why it's called "representative" — it represents the relationship regardless of what units you use. For example, on the ICAO 1:500,000 topographical chart, 1 centimetre on the chart represents 500,000 centimetres on the Earth, and 1 inch on the chart represents 500,000 inches on the Earth. The ratio stays the same. Now, let me walk you through converting from a Statement in Words to a Representative Fraction. This is a practical skill you'll need. Suppose you have "One inch to ten nautical miles" and you want to convert it to a representative fraction. You substitute into the equation: Representative Fraction = 1 inch divided by 10 nautical miles But the problem is that the numerator is in inches and the denominator is in nautical miles — they're different units. You need them to be the same so they cancel. So first, multiply the denominator by 6080 to convert nautical miles to feet: 1 inch divided by (10 × 6080 feet) Then multiply by 12 to convert feet to inches: 1 inch divided by (10 × 6080 × 12 inches) Now both numerator and denominator are in inches, so you get: 1 divided by 729,600 So the representative fraction is 1:729,600. It's now dimensionless — it will work with centimetres or any other units as long as the ratios remain the same. Let me give you the conversion factors you'll find useful for scale problems: - 1 nautical mile = 6080 feet or 1852 metres - 1 metre = 3.28 feet - 1 inch = 2.54 centimetres - 5.4 nautical miles = 10 kilometres Now let's work through Example 1 from the book. On a particular chart, 5 centimetres represents 7 nautical miles. What is the scale? We use the equation: Representative Fraction = Chart Length divided by Earth Distance So that's 5 centimetres divided by 7 nautical miles. But we need the same units. We convert the 7 nautical miles to centimetres. First, multiply 7 by 1852 to convert nautical miles to metres, then multiply by 100 to convert metres to centimetres. So the denominator becomes 7 × 1852 × 100 centimetres. The fraction then simplifies to give you the representative fraction. That's the core of scale — the definition, the three methods of expression, and how to convert between them.