yumi2012 wrote:s={1,2,5,7,x}
If x is a positive integer, is the mean of set S greater than 4?
1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S
Target question:
Is the mean of set S greater than 4?
In other words,
is (1+2+5+7+x)/5 > 4?
Simplify:
Is 1+2+5+7+x > 20?
Simplify:
Is 15+x > 20?
Simplify:
Is x > 5?
Now that we've rephrased the target question as "
Is x > 5?", the question is much easier to handle.
Aside: If a set consists of an odd number of elements, the median will be the middle number.
Statement 1: The median of set S is greater than 2
So, the middle number (the median) is not 2. So, it must be either 5 or x (if x between 2 and 5).
What does this tell us about x? Here are two possibilities.
case a: x=
3, which gives us {1,2,
3,5,7}. In this case, the median is 3 and
x is not greater than 5.
case b: x=
6, which gives us {1,2,5,
6,7}. In this case, the median is 5 and
x is greater than 5.
Since we have conflicting answers to the rephrased target questions, statement 1 is NOT SUFFICIENT
Statement 2: The median of set S is equal to the mean of set S
We know that the mean = (x+15)/5,.
We can rewrite this as: the mean = (x/5) + (15/5) or the mean = (x/5) + 3
Important: Since x must be a positive integer, and since the other four numbers are positive integers, we know that the median must be a positive integer. If the median = mean, then the mean is also a positive integer.
For (x/5) + 3 to be a positive integer,
x must be divisible by 5.
Let's see what happens when x=5.
When x=5, the median=5 and the mean=4. Nope. x cannot equal 5.
Let's see what happens when x=10.
When x=10, the median=5 and the mean=5.
Great, that works.
Now, should we keep checking other values of x to see what happens? No, we don't need to.
Remember that our rephrased target question:
Is x > 5?
We already saw that
x can't equal 5, and we've shown that x must be a positive integer that's divisible by 5. Since
x could equal 10, we've already shown that
x must be greater than 5.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.
Answer =
B
Cheers,
Brent
