If x and y are greater than 0, is x = 1 ?

Fairly straightforward:

Statement 1:

Both of these are valid: \(\frac{x}{y}=\frac{7}{7}=1\) and \(\frac{x}{y}=\frac{100}{100}=1\) - Insufficient

Statement 2:

Both of these are valid: \(x*y=1*1=1\) and \(x*y=0.2*5=1\) - Insufficient

However, combining both statements, the only valid solution is: \(\frac{x}{y}=\frac{1}{1}=1=x*y=1*1\) where \(x=1\) and \(y=1\).

Don't forget those pesky real numbers! Nowhere does the stem insist that these be integers.

You are not wrong, but check there are multiple values that x can take ?
As per GMAC - All numbers are real numbers if not mentioned.

If X and Y are greater than 0, is x=1?
Question gives :
If X and Y are greater than 0.
Question asks : Is X = 1? --> Yes/No
(1) x/y = 1
Since y > 0 as per question stem, so x=y
x > 0 , so x can take any value - 1,2,3,etc
So 1 is not sufficient
(2) xy = 1
If x is equal to 1, then y is equal to 1
If x is equal to 2, then y is equal to 1/2
If x is equal to 3, then y is equal to 1/3
All are valid, but there is no unique value of x.
So (2) is not sufficient

Combining 1 and 2 - xy=1 and x=y
\(XY = 1\)
=> \(x^2 = 1\)
=> \(x = +1 or -1\)
But question tells x is greater than 1
So\(x = 1\)is the only value x can take.

OA is C

Why 2 isn't sufficient is that X or Y can also be fractions.

X = 1/2, Y = 2

XY = 1

ST 1
x/y=1 and this means x=y so x and y can be 1 , 2, 3 , 4 , .......
EX : 5/5=1 4/4=1 1000/1000=1
NOT sufficient

ST2
xy=1 and means x=1/y so x can be 1/5 so y = 5
EX" 1/5 * 5 = 1
also not sufficient

ST1 & ST2
x=y and xy=1
you can simply find that xx=1 this means x^2 = 1 so x= +1 or x= -1
and because y and x must be larger than 0 so x must be +1
sufficient


the answer is C

Nice question. I used to fall in this trap a lot.

S1: x/y = 1
hence x = y .

Not sufficient

S2: xy =1
No more info hence not sufficient

Now combining both

X^2 =1
or x = +1,-1

Here is the trap where i used to miss the question statement all together . Since x is positive. It cannot take the negative value.

Hence x = 1. Sufficient

Answer is D

Target question: Is x = 1 ?

Given: x and y are POSITIVE

Statement 1: x/y = 1
Let's TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 1. In this case, the answer to the target question is YES, x DOES equal 1
Case b: x = 2 and y = 2. In this case, the answer to the target question is NO, x does NOT equal 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: xy = 1
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = 1. In this case, the answer to the target question is YES, x DOES equal 1
Case b: x = 2 and y = 1/2. In this case, the answer to the target question is NO, x does NOT equal 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x/y = 1
Statement 2 tells us that xy = 1
Take x/y = 1 (statement 1) and multiply both sides by y to get: x = y
Now take xy = 1 (statement 2) and replace y with x to get: (x)(x) = 1
Simplify: x² = 1
This means EITHER x = 1, OR x = -1
However, since we're told that x is POSITIVE, it must be the case that x = 1
So, the answer to the target question is YES, x DOES equal 1
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Bunuel mcelroytutoring ScottTargetTestPrep

I understand this problem theoretically. Statement one says x and y are the same number. Statement two says that they are reciprocals of each other. Combined, the only number whose reciprocal is itself is one.

However, algebraically, I don't understand something. When I combine the two statements (both are equal to one), I see that "x/y=xy". This equation itself lends itself to multiple values of x; the "x" variables cancel each other out. This would lead me to believe that x can be multiple values (suggesting answer E), but I know that this is incorrect. Where am I going wrong here?

Setting x/y equal to xy is another way to solve this question. As you mentioned, x variables will cancel each other out (which is possible because we are given that x > 0) and we will be left with 1/y = y, or, equivalently, y^2 = 1. Now, y^2 = 1 implies that y = 1 or y = -1; but the latter is ruled out since we are given that y > 0. Thus, y = 1 and using either equation, we get that x = 1 as well.
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(1) \(if x = y = 1; yes\)
\(if x = y = 7; no\)

INSUFFICIENT

(2) \(xy = 1\)
if \(x = 0.5 y = 2; no\)
if \(x = y = 1; yes\)

INSUFFICIENT.

(1&2) Combined, x must be 1. SUFFICIENT.
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