Can a light-year be represented as a ratio?

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Would it be valid to say that a light year is a ratio between the distance that light travels and the elapsing of one year?

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No—it's the product of a time and a speed. The speed of light is one lightyear per year. A lightyear is the distance travelled.

Suppose we define a walker-hour as the distance a typical person walks in one hour, and that a typical person walks at $3$ miles per hour. Then one walker-hour is three miles, and typical walking speed is one walker-hour per hour.

Lightyears are the same idea as walker-hours, but with a much bigger speed and distance.

You're right that the speed of light, a lightyear and a year are related—but the relationship is that you multiply the speed of light (about $300,000$ km/s) by the time ($1$ year) to get the distance ($1$ lightyear).

We can also talk about lighthours and lightseconds. A fun fact is that one nanolightsecond is almost exactly one foot (it's about $11.8$ inches).