Consider two lamps with the same maximum brightness but: the first lamp has a dial accepting discrete values ranging from 0 (off) to 360 (full brightness), and the second lamp has a dial accepting discrete values from 0 to 100.
Given any value for the first dial, how do I calculate the closest new value for the first dial such there exists a value for the second dial that results in exactly the same brightness?
For example, a value of 40 for the first lamp can be reduced to 36 so that the equivalent value of the second lamp is exactly 10.
Best Answer
The relationship between the first lamp $a$ and the second lamp $b$ is:
$$ a = \frac{360}{100}b = 3.6b $$
Clearly, the lowest whole-number multiple of 3.6 is 18 ($36 \times 5$). Thus, we must round to the nearest compatible interval of 18, which we can compute like so:
$$f_b(a) = 18[\frac{a}{18}]$$
where $[x]$ is the round function.
This function satisfies the aforementioned requirements:
$$f_b(40) = 36$$ $$\frac{f_b(40)}{360} = \frac{\frac{f_b(40)}{3.6}}{100}$$