Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
First, list out the six possibilities for the operation & acting on x and y:
Addition: x + y
Subtraction: x – y
Multiplication: xy
Division: x/y
Exponentiation: xy
Rooting (as shown in the problem): x1/y
Each of these six operations has an equal chance of being represented by the symbol &.
Now, we are asked “which of the following points in the xy-coordinate plane” does something with this symbol. Before we investigate exactly what that something is, we should recognize that this isn’t really a coordinate-plane problem. A point in the plane just represents a pair of numbers for x and y. So we are being asked to plug in the pairs of numbers given in the answer choices to see which one fits the condition we want.
This condition is that the expression x&y has a 50% chance of yielding the same number. Since there are six operations, this means that we want three (out of six) of the operations to yield the same number. We are also asked to ensure that the other three operations yield different numbers from this number and from each other.
So we now start testing:
A: (1, 1):
1 + 1 = 2
1 – 1 = 0
1*1 = 1
1/1 = 1
11 = 1
11/1 = 1
Four out of six are the same, so we can cross out A.
B: (1, 2):
1 + 2 = 3
1 – 2 = –1
1*2 = 2
1/2 = 0.5
12 = 1
11/2 = 1
Only two out of six are the same, so we can cross out B.
C: (2, 1):
2 + 1 = 3
2 – 1 = 1
2*1 = 2
2/1 = 2
21 = 2
21/1 = 2
Four out of six are the same, so we can cross out C.
D: (2, 2):
2 + 2 = 4
2 – 2 = 0
2*2 = 4
2/2 = 1
22 = 4
21/2 = square root of 2
Three out of six are the same, and the other three are all different. This is the condition we’re looking for. We should stop now, but if we really wanted to, we could check the last choice and find that it also failed to meet the condition.
The correct answer is D.
