GmatKiss wrote:If square roots of numbers are considered positive and sqrt(a)+sqrt(b) = sqrt(c)+sqrt(d)+sqrt(e), then is a < c?
(1) c = d
(2) sqrt (b )+ sqrt (d )< sqrt ( e)
Nice question.
Aside: Since sqrt(a)+sqrt(b) = sqrt(c)+sqrt(d)+sqrt(e), we know that a, b, c, d and e are each greater than or equal to zero, otherwise we could not have a valid equation involving square roots of negative numbers.
Statement 1: c = d
When we replace d with c in the given equation, we get: sqrt(a)+sqrt(b) = sqrt(c)+sqrt(c)+sqrt(e)
We can simplify this as sqrt(a)+sqrt(b) = 2sqrt(c)+sqrt(e)
From here, can we determine whether or not a < c? No.
Given the equation, sqrt(a)+sqrt(b) = 2sqrt(c)+sqrt(e), we can come up with 2 conflicting cases:
case a: a=b=c=e=0 in which case a is
not less than c
case b: a=0, b=4, c=1 and e=0, in which case a
is less than c
Since we cannot determine whether or not a < c, statement 1 is not sufficient.
Statement 2: sqrt (b )+ sqrt (d )< sqrt ( e)
For statement 2, I'm going to rewrite the target question.
First, we know that if a < c then it must be true that sqrt(a)< sqrt(c) [as long as a and c are not negative]
So, we can rewrite the target question as:
Is sqrt(a)< sqrt(c)?
Better yet, we can use the fact that if x < y, then y - x > 0 to take the rewritten target question, (
Is sqrt(a)< sqrt(c)?) and rewrite it as
Is sqrt(c)- sqrt(a) > 0?
In other words,
Is sqrt(c)- sqrt(a) positive?
Now let's take the given equation:
sqrt(a) + sqrt(b) = sqrt(c) + sqrt(d)+ sqrt(e) and subtract sqrt(a), sqrt(d) and sqrt(e), from both sides to get:
sqrt(b) - sqrt(d)- sqrt(e) =
sqrt(c)- sqrt(a) [notice that the right-hand-side matches the new target question]
At this point, we'll take statement 2 and play around with it
We're given that sqrt(b)+ sqrt(d)< sqrt(e)
If we subtract sqrt(e) from both side, we get sqrt(b)+ sqrt(d) - sqrt(e) < 0
In other words, sqrt(b)+ sqrt(d) - sqrt(e) is negative
Important point: if sqrt(b)+ sqrt(d) - sqrt(e) is negative, we can further conclude that sqrt(b)
- sqrt(d) - sqrt(e) must be negative as well [since changing +sqrt(d) to -sqrt(d), will make the value of the expression even smaller].
At this point, we're almost done.
We just showed that
sqrt(b)- sqrt(d) - sqrt(e) must be negative
And we already know that
sqrt(b) - sqrt(d)- sqrt(e) =
sqrt(c)- sqrt(a)
From here, we can conclude that
sqrt(c)- sqrt(a)
must be negative.
Now, our new target question asks
Is sqrt(c)- sqrt(a) positive?
Well, we know that sqrt(c)- sqrt(a) is definitely negative, which means we can answer our new target question with certainty (the answer to the target question is "no")
As such, statement 2 is sufficient and the answer is
B
Cheers,
Brent
