The raw scores on the physics group projects are 18, 29, 32, 35, 36, 49, 53, 64, 66. The teacher wants to rescale the scores using the linear formula G = kR + c, where G is the final grade and R is the raw score. She wants the highest score to scale to 100 and the median to scale to 80. What is the value of the product kc? Express your answer as a common fraction.
First, we need to find the values of k and c in the linear formula G = kR + c. Let's use the highest score and the median to set up two equations:
For the highest score:
100 = k(66) + c
For the median score:
80 = k(median) + c
To find the median score, we need to first sort the raw scores in ascending order:
18, 29, 32, 35, 36, 49, 53, 64, 66
The median is the middle score, which is 36.
Substituting 66 and 36 into the equations above, we get a system of two equations with two variables:
100 = 66k + c,
80 = 36k + c, solve for k, c
k=2 / 3
k * c =(2 / 3) * 56 =112 / 3 =37 + 1/3