GaneshMalkar wrote:If a,b,c and d are positive numbers, is a/b < c/d?
1. 0 < c-a/d-b
2. (ad/bc)^2 < ad/bc
OA - B
Target question:
Is a/b < c/d?
Statement 1: 0 < (c-a)/(d-b)
There are several sets of values that meet this condition. Here are two:
Case a: a = 1, b = 1, c = 3 and b = 2, in which case
a/b is less than c/d
Case b: a = 1, b = 1, c = 2 and b = 3, in which case
a/b is not less than c/d
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Aside: You asked whether we can approach statement 1 algebraically. This is difficult because we're dealing with an inequality, and when we deal with inequalities we must reverse the direction of the inequality whenever we multiply or divide both sides by a negative value.
So, knowing that 0 < (c-a)/(d-b), we know that either (c-a) and (d-b) are both positive, or (c-a) and (d-b) are both negative. So, we need to consider multiple cases where the inequality may or not be reversed each time.
At that point, it's easier to start checking different scenarios by plugging in values.
Statement 2: (ad/bc)^2 < ad/bc
IMPORTANT: Since a, b, c and d are all positive, we know that all of the products and quotients here will be positive. This is great because it means that we can multiply and divide and the direction of the inequality never changes.
Take (ad/bc)^2 < ad/bc
Divide both sides by (ad/bc) to get: ad/bc < 1 (we can do this because ad/bc MUST be positive)
Multiply both sides by c/d to get:
a/b < c/d (we can do this because c/d MUST be positive)
Now that we've demonstrated that
a/b < c/d, we can conclude that statement 2 is SUFFICIENT
Answer =
B
Cheers,
Brent
