Data Sufficiency

If x, y, and z are all distinct positive integers and the percent increase from x to y is equal to the percent increase from y to z, what is x? (1) y is prime (2) z = 9

Manhattan Prep (2015-05-17). GMAT Advanced Quant (Kindle Locations 4099-4101). Manhattan Prep Publishing. Kindle Edition.

OA A

bubbliiiiiiii wrote:If x, y, and z are all distinct positive integers and the percent increase from x to y is equal to the percent increase from y to z, what is x? (1) y is prime (2) z = 9

The percent increase from x to y is equal to the percent increase from y to z.
Implication:
x/y = y/z
xz = y², where x<y<z.

Statement 1:
If y=2, then xz = 2² = 4, implying that x=1 and z=4.
If y=3, then xz = 3² = 9, implying that x=1 and z=9.
If y=5, then xz = 5² = 25, implying that x=1 and z=25.
In every case, x=1.
SUFFICIENT.

Statement 2:
Since z=9, we get:
(x)(9) = y².
It's possible that x=1 and that y=3.
It's possible that x=4 and that y=36.
Since x can be different values, INSUFFICIENT.

The correct answer is A.

The percent increase from x to y is equal to the percent increase from y to z.
Implication:
x/y = y/z

Hi Mitch,

Thanks for responding.

Isn't percentage increase equation be the following?

(x-y)/x = (y-z)/y

bubbliiiiiiii wrote:

The percent increase from x to y is equal to the percent increase from y to z.
Implication:
x/y = y/z

Hi Mitch,

Thanks for responding.

Isn't percentage increase equation be the following?

(x-y)/x = (y-z)/y

If y is 10% greater than x, the, x/y = 10/11.
If z is 10% greater than y, then y/z = 10/11.
Since the fractions in red are equal, we get:
x/y = y/z.

If y is 30% greater than x, the, x/y = 10/13.
If z is 30% greater than y, then y/z = 10/13.
Since the fractions in red are equal, we get:
x/y = y/z.

Implication:
If the percent increase from x to y is equal to the percent increase from y to z, then x/y = y/z.

Algebraic proof:

Percent increase from x to y = (y-x)/x * 100.
Percent increase from y to z = (z-y)/y * 100.
Since the two percent increases are equal, we get:
(y-x)/x * 100 = (z-y)/y * 100
(y-x)/x = (z-y)/y
y/x - x/x = z/y - y/y
y/x - 1 = z/y - 1
y/x = z/y.
Thus:
x/y = y/z.

GMATGuruNY wrote:

bubbliiiiiiii wrote:If x, y, and z are all distinct positive integers and the percent increase from x to y is equal to the percent increase from y to z, what is x? (1) y is prime (2) z = 9

The percent increase from x to y is equal to the percent increase from y to z.
Implication:
x/y = y/z
xz = y², where x<y<z.

Statement 1:
If y=2, then xz = 2² = 4, implying that x=1 and z=4.
If y=3, then xz = 3² = 9, implying that x=1 and z=9.
If y=5, then xz = 5² = 25, implying that x=1 and z=25.
In every case, x=1.
SUFFICIENT.

Statement 2:
Since z=9, we get:
(x)(9) = y².
It's possible that x=1 and that y=3.
It's possible that x=4 and that y=36.
Since x can be different values, INSUFFICIENT.

The correct answer is A.

Mitch, seriously It is awesome question.
I reached at equation y2 = xz

But I answered C by mistake --- your explanation is awesome.

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If x, y, and z are all distinct positive integers and the percent increase from x to y is equal to the percent increase from y to z, what is x? (1) y is prime (2) z = 9

Transforming the original condition and the question by variable approach method we have percentage change =(After-Before)*100/Before, and thus (y-x)*100/x=(z-y)*100/y. multiplying both sides by 100 gives us (y/x)-1=(z/y)-1, y/x=z/y, y^2=xz. The question asks for the value of x=?, and we know that the percentage increases thus x<y<z. Thus, in case of 1) x = 1 if y=prime. Since this is a unique answer, it is sufficient. Therefore the answer is A.


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bubbliiiiiiii wrote: Isn't percentage increase equation be the following?

(x-y)/x = (y-z)/y

You've got the right idea, but remember that percent change is expressed as

100 * (New - Old)/Old

and we know that z > y > x.

So we have

100 * (y - x)/x = 100 * (z - y)/y, or

(y - x)/x = (z - y)/y, or

y*y - x*y = x*z - x*y, or

y*y = x*z