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niketdoshi123: Master | Next Rank: 500 Posts; Posts: 210; Joined: Thu Mar 08, 2012 11:24 pmFollowed by: 3 members; Thanked: 62 times

by niketdoshi123 » Thu Aug 09, 2012 4:15 am

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

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Source: Beat The GMAT — Problem Solving | Thu Aug 09, 2012 4:15 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Aug 09, 2012 7:27 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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adthedaddy: Master | Next Rank: 500 Posts; Posts: 167; Joined: Fri Mar 09, 2012 8:35 pmFollowed by: 3 members; Thanked: 39 times

by adthedaddy » Thu Aug 09, 2012 7:54 am

There is also an alternative way to resolve this.

Given Set contains {p,q,p+q,p-1,q+1}
Also, it is given that p<=q.
But as each element is the mode of the set (i.e. every element is different) p=q is not possible.
Hence, p<q.

Arranging the set from above details in ascending order, we get
{p-1,p,q,q+1,p+q}

Also, its given that median is 3.
From above, median is q.
Therefore, q=3

Substituting q=3 in the given eqn p^2 -(2+q)p + 2q = 0
we get p=2 or p=3

But p=3 not possible as p&q are distinct.
Thus, p=2.

Subsituting the above values in the set, we get it as {1,2,3,4,5}

Avg = (1+5)/2
= 3

[spoiler]Ans: Option C[/spoiler]

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