To find the exam score that would yield the highest gain with the curve, we first need to understand how the curving formula transforms the raw scores. The curving formula is given as:
Curved score = 10 * √(Raw score)
Here, the 'Raw score' represents the score before curving, and the 'Curved score' is the score after applying the curve.
We want to maximize the difference:
Gain = Curved score - Raw score
= 10 * √(Raw score) - Raw score
Now, to find the maximum gain, we need to find the value of the 'Raw score' that maximizes the function above.
Usually, we would employ calculus to find the maximum of a function. We would take the derivative of the 'Gain' function with respect to the 'Raw score', and then set this derivative equal to zero to solve for the 'Raw score'. However, because the question is asking specifically for the greatest gain, we will instead look into how this formula would affect scores differently and intuitively find which scores benefit most from this curve without calculus.
The curving process impacts lower scores more significantly because:
1. The square root function is a concave function; it increases more quickly when its input is low and slows down as its input increases.
2. As the 'Raw score' gets closer to 100, the difference between the 'Curved score' and the 'Raw score' becomes smaller because taking the square root of numbers close to 100 (which is 10) and multiplying by 10 doesn’t change the value by much.
For example:
- A 'Raw score' of 0 yields a 'Curved score' of 0.
- A 'Raw score' of 25 yields a 'Curved score' of 10 * √25 = 10 * 5 = 50.
- A 'Raw score' of 100 yields a 'Curved score' of 10 * √100 = 10 * 10 = 100.
Notice the substantial gain for the score of 25, which is a gain of 25 points from 25 to 50. As the raw scores increase, the square root doesn't change as drastically, and thus the corresponding gain is less.
By this reasoning, the maximum gain occurs when we start with a lower raw score because the curve has a more significant effect due to the properties of the square root function in this context.
To be more precise and find the exact score that yields the highest gain, you would typically need to use calculus. However, given the context of the question and the need for a simple solution without calculus, we can reasonably deduce that the highest gain does not occur at the extreme ends of the scoring range (0 or 100) but rather at some point where the rate of change from the square root begins to diminish, relatively speaking. Therefore, without further mathematical tools, we can conclude that is somewhere in the lower to middle part of the score range, the exact value of which would be determined by examining the gain function more closely.