Dividing an area into unit squares

Dimensional Analysis is about converting units using ratios. Let’s keep this short and jump straight into how that works.

Say that you have 2 kilograms and you want to know how much that is in grams. Well, if one kilogram is a thousand grams, then two kilograms must be two thousand grams. Easy.

But what about something more complicated such as converting inches to cm? 50 inches are equivalent to 127 cm. How do we convert 40 inches to cm? Well, we use ratios.

Let’s work with a simpler example. Say you have a 100 inches, how do you convert that to cm? Well, since we can represent the conversion in a ratio, 50 in/127 cm, we can just multiply the fraction by 2/2, giving us 100 in/254 cm. So now we have a new conversion, which is that 100 inches are equal to 254 cm.

There’s also another way. We can divide the 100 inches into groups of 50 inches. By the definition of division, we can do that by dividing 100 by 50, which is 2. Meaning that we have 2 50 inches in a 100 inches. Since each 50 inch is equal to 127 cm, we can multiply 127 by 2, giving us 254 cm.

Going back to 40 inches, we can do the same thing. Divide 40 into groups of 50, which gives us 0.8. Meaning we have 0.8 50 inches in 40 inches. Since each 50 inch is equal to 127 cm, we can multiply 127 by .8, giving us 101.6 cm.

What if it’s the other way around? Let’s say we are converting 50 cm to inches. Well just like before, we can divide the 50 cm into groups of 127 cm, giving us around 0.39. Meaning, there are ~.39 127 cm in 50 cm. And since each complete 127cm gives us 50 inches, we can just multiply .39 by 50, giving us around 19.7 inches.

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