Set

niketdoshi123 wrote:Consider a set whose elements are {p, q, p + q, p-1, q+1}, where p<=q, and median is 3. In addition, it is given that each element is the mode of the set. Also, if p and q satisfy the equation
p^2 -(2+q)p + 2q = 0, then what is the average of these five terms?

a) 2
b) 2.4
c) 3
d) 4.2
e) 5

Let's begin with the equation: p^2 -(2+q)p + 2q = 0
Expand the middle part: p^2 - 2p - pq + 2q = 0
Factor in parts: p(p-2) - q(p-2) = 0
Simplify: (p-q)(p-2)=0
This means that p-q=0 or p-2=0

We have a problem with p-q=0. If it were the case that p-q=0, then p would equal q. However, the question tells us that "each element is the mode of the set"
In other words, every number in the set is different.
So, it cannot be the case that p-q=0, which means it must be true that p-2=0.

If p-2=0, then p=2.
Great, our set becomes a little clearer once we replace p with 2.
We get: {2, q, 2+q, 1, q+1}
Rearrange: {1, 2, q, q+1, q+2}

Aside: we know that q < q+1 < q+2 for all values of q.

Now, since we're told that the median is 3, it must be the case that q = 3, since this would be the only way for 3 to be the middle number.

If q=3, our set looks like this: {1, 2, 3, 4, 5}, which means the mean (average) = (1+2+3+4+5)/5 = 3

Answer = C

Cheers,
Brent