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11 Aug 2021, 07:30
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If \(x\) and \(y\) are positive, is \(81^3 = 27^{y - x}\)?
(1) \(|y - x| =4\)
(2) \(2^y = 8^x\)
(1) \(|y - x| =4\)
(2) \(2^y = 8^x\)
11 Aug 2021, 22:44
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Give x and y are positive
To find: \(81^3 =\) \(27^{y - x}\) ?
i.e. Is \((3^4)^3 = (3^3)^{y-x}\)
i.e. Is \(3^{12}\) = \(3^{3y-3x}\)
i.e. Is 12 = 3y-3x
i.e. Is y-x = 4?
Statement (1) |y−x|=4
If y > x then y-x = 4
If y < x then y-x = -4
So, Statement (1) is insufficient by itself.
Statement (2) \(2^y=8^x\)
i.e. \(2^y = 2^{3x}\)
y = 3x
But we don't know if y-x = 4
e.g. if x=1 then y=3 and y-x = 2
if x=2 then y=6 and y-x = 4
So, Statement (2) is insufficient by itself.
Combining statement 1 and 2:
As x and y are positive and as per statement (2) y = 3x we get, y > x
So the statement 1: |y−x|=4 becomes y-x = 4
Hence, combining the two statements is sufficient.
Answer: C
To find: \(81^3 =\) \(27^{y - x}\) ?
i.e. Is \((3^4)^3 = (3^3)^{y-x}\)
i.e. Is \(3^{12}\) = \(3^{3y-3x}\)
i.e. Is 12 = 3y-3x
i.e. Is y-x = 4?
Statement (1) |y−x|=4
If y > x then y-x = 4
If y < x then y-x = -4
So, Statement (1) is insufficient by itself.
Statement (2) \(2^y=8^x\)
i.e. \(2^y = 2^{3x}\)
y = 3x
But we don't know if y-x = 4
e.g. if x=1 then y=3 and y-x = 2
if x=2 then y=6 and y-x = 4
So, Statement (2) is insufficient by itself.
Combining statement 1 and 2:
As x and y are positive and as per statement (2) y = 3x we get, y > x
So the statement 1: |y−x|=4 becomes y-x = 4
Hence, combining the two statements is sufficient.
Answer: C
