One interesting inequality question, I came across recently, is this Data Sufficiency Question:
Is
?
1)
2)
The options are
(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
(B)Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
(C)Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
(D)Each statement alone is sufficient to answer the question.
(E)Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.
Solution :
I initially thought I would solve it using norms and then the idea of super ellipses came into my mind.
Here's how one can think about the above options
The option A) says that (x/z)^2 + (y/z)^2 >1
So basically, numbers x/z and y/z lie outside the unit circle i.e. X^2 +Y^2=1 (X and Y being the X- coordinate and Y-coordinate in Euclidean geometry)
Option B) says that (x/z) +(y/z) >1
Here the numbers x/z and y/z lie on the non origin side of the straight line X+Y=1 (X and Y being the X- coordinate and Y-coordinate in Euclidean geometry)
Now, neither of these two imply that the numbers lie outside the super ellipse (x/z)^4 + (y/z)^4 =1. This can be inferred from the diagram below. Clearly points above the line or outside the circle do not imply being outside the super ellipse. So option E. None are sufficient.
Is
?1)
2)
The options are
(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
(B)Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
(C)Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
(D)Each statement alone is sufficient to answer the question.
(E)Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.
Solution :
I initially thought I would solve it using norms and then the idea of super ellipses came into my mind.
Here's how one can think about the above options
The option A) says that (x/z)^2 + (y/z)^2 >1
So basically, numbers x/z and y/z lie outside the unit circle i.e. X^2 +Y^2=1 (X and Y being the X- coordinate and Y-coordinate in Euclidean geometry)
Option B) says that (x/z) +(y/z) >1
Here the numbers x/z and y/z lie on the non origin side of the straight line X+Y=1 (X and Y being the X- coordinate and Y-coordinate in Euclidean geometry)
Now, neither of these two imply that the numbers lie outside the super ellipse (x/z)^4 + (y/z)^4 =1. This can be inferred from the diagram below. Clearly points above the line or outside the circle do not imply being outside the super ellipse. So option E. None are sufficient.
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