Finding the median of a data set

A data set has 15 elements. The 15 elements in a second data set are obtained by multiplying each element in the first data set by 10. The 15 elements in a third data set are obtained by decreasing each element of the second data set by 20. The median of the third data set is 50. What is the median of the first data set?

I know the answer is 7 but I don't know how to set up this problem.

Re: Finding the median of a data set

Since the median may not have a nice algebraic expression (i.e., it may land between two numbers and we need to round), it is probably best to work backward from the given median value. You can express the transformations for each dataset algebraically to understand each of the datasets. Let them be denoted by the first capital letters, respectively. Then we start with A having 15 elements.

$\displaystyle B = 10A$

$\displaystyle C = B - 20 = 10A - 20 = 10(A - 2)$

Since the median of C = 50, we just follow the backward steps TKHunny pointed out. We can also use the algebra now,

$\displaystyle 10(Med(A) - 2) = Med(C) \Rightarrow Med(A) = \frac{Med(C)}{10} + 2 \Rightarrow Med(A) = 7$

Note, this approach is the same as TKHunny's save for the fact I used "10(A - 2)" instead of "10A - 20" to algebraically solve for Med(A). If we used the other form we would have had the same arithmetic as TKHunny's:

$\displaystyle 10Med(A) - 20 = Med(C) \Rightarrow Med(A) = \frac{Med(C) + 20}{10} \Rightarrow Med(A) = 7$