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Usually such ratios have a specific name. For example, the ratio of a mass measurement (g) over a volume measurement (mL) is called density. The result is a number with the unit g/mL. Most cases that have different units will have a specific name if they're commonly used. Other examples include molarity, molality, joules, newtons, and ppm.
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What a ratio of two measurements with different units?
Time
A ratio of two measurements having different units?
If the two measurements are of the same type, such as length, we could use that ratio to convert from one measurements to another. An example is the ratio of 1.609 Km to 1 mile. Here, we can multiply some number of miles by 1.609 and find the kilometer distance. If the two measurements are of different types, that is often used as a definition of another quantity such as speed. Speed is defined as the ratio of the distance traveled to the amount of time it takes. These two examples are the most common uses when taking the ratio of two measurements, yet there may be a more generalized term or theorem here, but I've not heard of it yet.
What is a ratio of 2 measurements having different units?
It can be a conversion factor - though not necessarily. For example, 68 deg Fahrenheit = 20 deg Celsius. But there is no conversion factor for F-to-C: instead there is a linear equation.
What a scale written as a ratio in simplest form without units of measurements?
scale factor
Why is a ratio expressed without writing units?
In a ratio of measurements for the same kinds of units, units get canceled. For example, in a ratio of 3 meters / 8 meters, you can cancel the "meters" in the numerator and the denominator. An important ratio is pi, which is the ratio of the circumference of a circle to its diameter. If you measure the circumference in feet, and the diameter in feet, then divide circumference/diameter, then the result is the dimensionless quantity 3.14159265.... If you go back and measure both in meters, you get the same answer.In a ratio of two measurements, the units cancel, so it makes no difference whether you write the units, or not.
