Aishwarya1204 wrote:Are all the numbers in a certain list of 15 numbers equal?
1) The sum of all the numbers in the list is 60
2) The sum of any 3 numbers in the list is 12
Answer is b
Target question:
Are all 15 numbers equal?
Statement 1: The sum of all the numbers in the list is 60.
There are several possible scenarios that satisfy this statement. Here are two.
Case a: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, in which case
all of the numbers are equal
Case b: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 7}, in which case
all of the numbers are not equal
Statement 1 is NOT SUFFICIENT
Statement 2: The sum of any 3 numbers in the list is 12.
This is a very powerful statement.
One approach is to ask, "Can I construct a list of 15 numbers,
in which the numbers are not all equal to 4 AND the sum of ANY 3 numbers is 12?"
If you try to create such a list, you'll find that it's impossible.
This, however, isn't a good proof. It's like "proving" there are no sasquatches by requiring others to find a sasquatch to prove you wrong.
Here's a more mathematical proof. It uses a technique called
Reductio ad absurdum (not required for the GMAT!)
First let's
assume that there is, indeed, a set
in which the numbers are not all equal to 4 and the sum of ANY 3 numbers is 12.
Let's go on to say that a, b, c, . .. n and o are in this magical set.
This means that a+b+
c must equal 12 (since they are 3 numbers in the set)
Also a+b+
d must equal 12.
And a+b+
e=12.
And a+b+
f=12.
.
.
.
And a+b+
o=12.
What does this tell us? It tells us that
c =
d =
e =
f = . . . =
n =
0
We know this because when we add each of these numbers to a and b, we keep getting the sum of 12.
So, far we've proven that c, d, e . . . n and o are all equal.
Using similar logic, we can show that the same applies to a and b. In other words, we can show that a=d=e=f-. . . and b=d=e=f=. . .
So, even though we began by
assuming that the numbers in the list are
not all equal, we were forced to conclude that the numbers
are all equal.
As such, our original assumption must be incorrect. In other words, there is no such set in which the numbers are not all equal to 4 AND the sum of ANY 3 numbers is 12.
Since such a list cannot exist, it
must be the case that
all 15 numbers are equal
So, statement 2 is SUFFICIENT
Answer =
B
Cheers,
Brent
