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16 Sep 2014, 00:59
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Is the product of two positive numbers \(x\) and \(y\) greater than their sum?
(1) \(x\) and \(y\) are integers.
(2) \(y > x > 1\).
(1) \(x\) and \(y\) are integers.
(2) \(y > x > 1\).
16 Sep 2014, 00:59
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Official Solution:
Is the product of two positive numbers \(x\) and \(y\) greater than their sum?
Essentially, the question is asking: is \(xy > x + y\)?
(1) \(x\) and \(y\) are integers.
If \(x=y=1\), the answer is NO. However, if \(x=y=3\), the answer is YES. This is not sufficient.
(2) \(y > x > 1\).
If both \(x\) and \(y\) are very close to 1, the product \(xy\) will also be close to 1, whereas the sum \(x + y\) will exceed 2. In this scenario, we'd have a NO answer. However, if we take \(x=2\) and \(y=3\), the answer becomes YES. Thus, the information is not sufficient.
(1)+(2) Since from (1) we know that both \(x\) and \(y\) are integers, from (2) we can work out that the smallest \(x\) can be is 2 and the smallest \(y\) can be is 3. Even with these smallest values, \(xy > x + y\). As \(x\) and \(y\) get bigger, the difference between their product and their sum also gets bigger. So, \(xy\) has to be bigger than \(x+y\). Sufficient.
Answer: C
Is the product of two positive numbers \(x\) and \(y\) greater than their sum?
Essentially, the question is asking: is \(xy > x + y\)?
(1) \(x\) and \(y\) are integers.
If \(x=y=1\), the answer is NO. However, if \(x=y=3\), the answer is YES. This is not sufficient.
(2) \(y > x > 1\).
If both \(x\) and \(y\) are very close to 1, the product \(xy\) will also be close to 1, whereas the sum \(x + y\) will exceed 2. In this scenario, we'd have a NO answer. However, if we take \(x=2\) and \(y=3\), the answer becomes YES. Thus, the information is not sufficient.
(1)+(2) Since from (1) we know that both \(x\) and \(y\) are integers, from (2) we can work out that the smallest \(x\) can be is 2 and the smallest \(y\) can be is 3. Even with these smallest values, \(xy > x + y\). As \(x\) and \(y\) get bigger, the difference between their product and their sum also gets bigger. So, \(xy\) has to be bigger than \(x+y\). Sufficient.
Answer: C
19 May 2023, 03:54
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Statement (2) by itself is insufficient. If x and y are large, the answer to the question is "yes". If x=1.1 and y=1.2, the answer to the question is "no".
The correct answer is B, Statement (2) by itself is sufficient. If x=1.1 and y=1.2, then x*y=1.32; and x+y=1.3; so x*y>x+y, the answer to the question is "yes"
The correct answer is B, Statement (2) by itself is sufficient. If x=1.1 and y=1.2, then x*y=1.32; and x+y=1.3; so x*y>x+y, the answer to the question is "yes"
