If x, y, and z are all distinct positive integers and the percent

Question

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

Bunuel · Accepted Answer

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?

The percent increase from x to y is equal to the percent increase from y to z: \(\frac{(y-x)}{x}*100 = \frac{(z-y)}{y}*100\) --> \(xz = y^2\) (\(0 < x < y < z\))

(1) y is prime:

y = 2, then \(xz = 4\), and since \(0 < x < y < z\) and all of them are positive integers, then x = 1 and z = 4.
y = 3, then \(xz = 9\), and since \(0 < x < y < z\) and all of them are positive integers, then x = 1 and z = 9.
y = 5, then \(xz = 25\), and since \(0 < x < y < z\) and all of them are positive integers, then x = 1 and z = 25.
y = 7, then \(xz = 49\), and since \(0 < x < y < z\) and all of them are positive integers, then x = 1 and z = 49.
...

We can see a pattern there, in all cases x = 1. Sufficient.

(2) z = 9 --> \(9x = y^2\). Case 1: x = 1 and y = 3; case 2: x = 4 and y = 6. Not sufficient.

Answer: A.

chetan2u · Answer

Hi..

I would simplify for myself by taking same % increase as multiplication by same number

Although Q becomes a bit easier with finding value of x, I'll answer it if say we were to find value of y as it would be a bit more challenging.
For finding x as the Q presently states, statement I is sufficient as shown below. x will always be 1.
A

Say we were to find value of y 
1) y is prime..
Starting with 1, I can find various combinations.
Say y is 2.... 1, 1*2, 2*2......1,2,4
Say y is 3.....1, 1*3, 3*3...... 1,3,9
Say y is 5......1,1*5,5*5.....1,5,25
Insufficient 
2) z=9
Possible values..
1,3,9.....1,1*3,3*3
4,6,9....4,4*(1.5),6*(1.5)
Two possible ways..
Insuff

Combined..
ONLY 1,3,9 remains.
Sufficient

C

gmat_for_life · Answer

Hello Bunuel,

I couldnt understand why (0<x<y<z). We could have selected the values of x,y and z as 4,2 and 1 or 9,3 and 1. The percentage changes for the above 2 cases would be negative but the changes would be equal. It was based on these 2 different sets that I ruled out option A as a possibility.

Could you please suggest where I might be going wring?

Regards,
Amit

chetan2u · Answer

hi anit,

the Q stem talks of % increase - "the percent increase from x to y", so we cannot take % decrease by taking x>y>z....

gmat_for_life · Answer

Oh right! My mistake! thanks a lot Chetan!

rohit8865 · Answer

simplifying the stem 
as percent increase from x to y is equal to the percent increase from y to z

(y-x)/x = (z-y)/y
y^2-xy = xz-xy
y^2 = xz-------(given in stem)
so x=??

(1) if y =3
then x=1,z=9 or x=9 ,z=1
Not suff

(2) no info of other integers
clearly insuff..

Combined
z=9 and y is prime then,

y^2= x*9
as 9 = 3^2 = Prime ^2, then for x,any value but 1 leaves y not anymore a prime number
rather x=1 leaves y =3
i.e Y^2(3^2) = x(1) * 9(z)
thus x=1,y=3 ,z=9
suff

Ans C

sahilkak007@gmail.com 
Please confirm OA

swapnilvar · Answer

Wrong;
Y= %increase* x= prime
Prime= no. itself * 1
Comparing both, since x and y are distinct, x can only be 1, hence statement 1 is sufficient

rekhabishop · Answer

Great approach, Bunuel I have learnt so much from you!! :shock:

shmba · Answer

(y-x)/x=(z-y)/y >> y^2=x*z 1. if y is prime, y^2 has only 3 factors 1,y,y^2. since x

energetics · Answer

+int x<y<z
Let a% = some % increase
a%*x = y 
a%*y = z

(1) y is prime  
Say y is 2, and a is 100%
If x = 1, then y is 2 and z is 4, this works.

Say y is 3, and a is 100%
x would have to be 1.5 for y to be 3, this is not possible due to int constraint.

Say y is 3 and a is 50%
Then x could be 2, and a 50% increase gives us y=3
BUT, 50% increase of 3 gives z=4.5, which is again not possible.

We can conclude that the only way to get the same percent increase and to maintain all int values is if x starts at 1. Sufficient. 
This makes sense because only 1 is a multiple of every number so essentially 1*y=y and then y^2 = z. This works for many cases but only if x=1.

(2) z = 9 
Say a is 200%, If x = 1 then y could be 3 and z could be 9
Say a is 50%, If x = 4 then y could be 6 and z could be 9
Not sufficient.

vijay282001 · Answer

Another way to solve A
let's say a is the percentage increase,
y = x(100+a)/100
since y is prime, so either x is 1 or (100 + a)/100 = 1,
(100 + a)/100 = 1 can't be true because it will yield a = 0 and x, y and z are unique, so x is equal to 1

Palladin · Answer

Question says percentage increase from  x  to  y  and from  y  to  z  is the same.
Few solutions say that, hence,  0<x<y<z .

I cannot completely agree.

If  x=4 ,  y=2  and  z=1 , where  0<z<y<x ,
Percentage increase from  x  to  y  =  (\frac{(2-4)}{4}) * 100  =  -50\% 
Percentage increase from  y  to  z  =  (\frac{(1-2)}{2}) * 100  =  -50\%

So,  0<x<y<z  may not always hold.
I can however conclusively say  x , y  and  z  are in ascending or descending order.

Percentage increase can be negative too, can it not?

kajaldaryani46 · Answer

————————- Could you please link similar Qs below for further practice? Bunuel Thank you!

Posted from my mobile device

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If x, y, and z are all distinct positive integers and the percent

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