Last visit was: 01 Sep 2026, 23:37 It is currently 01 Sep 2026, 23:37
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
   1   2   3   4 
805+ (Hard)|   Number Properties|   Roots|         
avatar
DhanyaAbhirami
avatar
Joined: 09 Mar 2024
Last visit: 05 Aug 2026
Posts: 59
Own Kudos:
47
 [1]
Given Kudos: 14
Posts: 59
Kudos: 47
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 04:16
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
(3/5)^(3/9x)<(3/5)^(2/4y)<(3/5)^(1/z)
1/z<1/2y<1/3x
z>2y>3x

x, y z must be from 1,4,6,8,9
x=1
y=4
z=9
Sum = 14
User avatar
Reon
User avatar
Joined: 16 Sep 2025
Last visit: 01 Sep 2026
Posts: 288
Own Kudos:
200
 [1]
Given Kudos: 18
Posts: 288
Kudos: 200
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 05:01
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
x,y,z are distinct single digit non-prime positive integers (1,4,6,8,9)
(27/125)^(1/9x) < (9/25)^(1/4y) < (3/5)^(1/z)
Find: x+y+z
Reduce all fractions to simplest forms
(3/5)^(3/9x) < (3/5)^(2/4y) < (3/5)^(1/z)
The base are all same and 0<3/5<1. So the function (3/5)^p decreases as p increases, hence 1/3x>1/2y>1/z
Lets choose values from (1,4,6,8,9)
Let, x=1...then next smallest is 4=y, try to put them in equation .... (1/3)>(1/8)>(1/9) ....True as we solve we need to choose highest value of x=9.... So no other possibilities.
x+y+z= 1+4+9= 14
User avatar
hritik04
User avatar
Joined: 11 Nov 2025
Last visit: 12 Jun 2026
Posts: 14
Own Kudos:
9
 [1]
Given Kudos: 25
Posts: 14
Kudos: 9
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 05:17
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?

A. 13
B. 14
C. 15
D. 16
E. 17

Gift
12 Days of Christmas Competition
This question is part of our holiday event
Win $40,000 in prizes: courses, tests, and more
Show more
Here, two concepts are used,
1. Exponents
2. When you deal with numbers between 0 and 1.

After creating a same base of 3/5, we can then use formula saying if bases are same power can be taken out. But because the base was a number between 0 and 1.
The greater would be the power number become smaller,so signs will get changed and the equation will become like this,

1/3x > 1/2y > 1/z

Now, after adjusting reciprocal we can write,

3x < 2y < z (because if 1/2>1/3 then 2<3)

Now, since only positive non prime numbers are llowed then we only have option {1,4,6,8,9}

By putting this we get z = 9, y = 4 and x = 1

Hence, option B is correct
User avatar
nousernamefine
User avatar
Joined: 14 Jan 2024
Last visit: 11 Apr 2026
Posts: 38
Own Kudos:
32
Given Kudos: 131
Location: India
Schools: Columbia '27 Wharton '27 Stanford '27
GPA: 4
Schools: Columbia '27 Wharton '27 Stanford '27
Posts: 38
Kudos: 32
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 05:27
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Its single digit no prime number. So only options= 1,4,6,8,9

Based on combination
1,4,6= 11
1,4,8=13
1,4,9=14
4,6,8=18
4,6,9=19
6,8,9=23

Only option 13= Option A
Bunuel
If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?

A. 13
B. 14
C. 15
D. 16
E. 17

Gift
12 Days of Christmas Competition
This question is part of our holiday event
Win $40,000 in prizes: courses, tests, and more
Show more
User avatar
Lizaza
User avatar
Joined: 16 Jan 2021
Last visit: 01 Aug 2026
Posts: 246
Own Kudos:
286
 [1]
Given Kudos: 7
GMAT 1: 710 Q47 V40
GMAT 1: 710 Q47 V40
Posts: 246
Kudos: 286
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 05:41
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I've attached my initial transformations as a photo for the ease of illustration, but the point is that the original equation can be eventually written as this:
\(0.6^{1/3x} < 0.6^{1/2y} < 0.6^{1/z}\)

Since 0.6 is less than 1, the higher the power, the less the number will become. Therefore, our powers need to be related in reverse to the final numbers, that is:
\(1/3x > 1/2y > 1/z\)

And since these variables are in the denominator of the fraction, they again bear an inverse relationship to the numerator, so we need to make one final inequality twist:
\(3x < 2y < z\)

Now let's deal with the numbers we actually have. Non-prime single-digits are: 1, 4, 6, 8, 9.
Obviously, since Z has to be one digit, \(3x = 1*3\), since if we take 4 three times, we'll land at 12.
Therefore, the smallest value Y can take is 4, and it means \(z > 2y = 8\), so \(z=9.\)

Hence, our final sum is \(S = x + y + z = 1 + 4 + 9 = 14\), and the answer is B.
Attachments

thumbnail_IMG_5938.jpg
thumbnail_IMG_5938.jpg [ 418.82 KiB | Viewed 355 times ]

User avatar
Mardee
User avatar
Joined: 22 Nov 2022
Last visit: 19 Aug 2026
Posts: 286
Own Kudos:
248
 [1]
Given Kudos: 21
Posts: 286
Kudos: 248
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 06:11
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
We need to find x + y + z

Given,
(27/125)^(1/9x) < (9/25)^(1/4x) < (3/5)^(1/x)

= (3/5)^(3/9x) < (3/5)^(2/4x) < (3/5)^(1/x)
= (3/5)^(1/3x) < (3/5)^(1/2x) < (3/5)^(1/x)

We know that,
0 < 3/5 < 1
=> inequality on exponents inverses
=> 1/3x > 1/2y > 1/z
=> 3x < 2y < z

It is given that only single distinct non prime integers are allowed = {1,4,6,8,9}
We try them out,
x = 1, y=4 , z= 8 (z=6 not possible since z >8 for y=4) => 3 < 8 < 9
WE see that other values with x = 1 is not possible

Now we try for x = 4,
=> 3x = 12
=> y = 8 (since 2y> 12)
But z > 16 not possible
=> higher values of x fail to meet the requirements

Thus the required values of x y and z = 1, 4 and 9

x+y+z = 1+4+9 = 14

B.14
User avatar
chattyyee
User avatar
Joined: 14 Jan 2025
Last visit: 16 Mar 2026
Posts: 29
Own Kudos:
11
 [1]
Given Kudos: 88
Location: India
Concentration: Technology, International Business
Schools: ISB '26
GPA: 3.7
WE:Information Technology (Consulting)
Products:
My Reviews
Schools: ISB '26
Posts: 29
Kudos: 11
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 06:25
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The first pattern recognition to be done here is see how each of them have powers of 3/5, so lets simplify that.
Each of them then leads to the underlying fraction to be the same, 3/5.
Post that, rewrite them by bringing the exponents into fractional powers themselves, such that it becomes 3/9x, 2/4y, 1/z. (1)
Now, its important to understand how signs work with fractions, and then exponents. In fractions, larger power = smaller number.
Hence, according to the inequality provided, the exponents must follow the following rule:
1/3x > 1/2y > 1/z from (1)
Next, lets simplify this to 3x < 2y < z. The logic behind this is all have to be positive numbers, hence we can take a reciprocal. Also since the reciprocal is a decreasing function, we flip the signs. ( Think of numbers 1,2,3 and if you took the reciprocal, how the inequality would change )
Let us now play with the values, identify that the values in place are {1,4,6,8,9} and z needs to be the largest, and then x needs to be smallest, and you get 1, 4, 9. ( You cannot have 6 for y as 2y=12 which is greater than 9 )
Hence, 14!
User avatar
Rahilgaur
User avatar
Joined: 24 Jun 2024
Last visit: 26 Jan 2026
Posts: 162
Own Kudos:
125
 [1]
Given Kudos: 47
GMAT Focus 1: 575 Q81 V82 DI72
Products:
My Reviews
GMAT Focus 1: 575 Q81 V82 DI72
Posts: 162
Kudos: 125
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 07:17
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?

A. 13
B. 14
C. 15
D. 16
E. 17

Gift
12 Days of Christmas Competition
This question is part of our holiday event
Win $40,000 in prizes: courses, tests, and more
Show more

Only option available for x, y, and z are 1,4,6, 8 and 9.

Compare all the parts by making base same that is 3/5..

The power will become 1/3x, 1/2y, 1/z . since 3/5 is less than 1 Greater the power -- lesser the total value will be hence we can directly compare powers instead -- 1/3x > 1/2y > 1 / z or 3x<2y<z

Upon trying multiple value from given options we arrive to x=1, y=4, z = 9 .... x+y+z =14 answer B
User avatar
cann2406
User avatar
Joined: 20 Feb 2025
Last visit: 01 Sep 2026
Posts: 15
Own Kudos:
12
 [1]
Given Kudos: 186
Posts: 15
Kudos: 12
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 07:23
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Convert each fraction into simpler form:
27/125 = (3/5)^3
9/25 = (3/5)^2

now powers will become:
1/3x,1/2y,1/z
Base <1:
1/3x > 1/2y > 1/z which implies, 3x<2y<z

Now, single digit non prime positive numbers are 1,4,6,8,9
When we put in above equation, we will get:
x=1,y=4,z=9

So, x+y+z = 1+4+9 = 14. Answer
User avatar
harsha12344556
User avatar
Joined: 24 May 2021
Last visit: 24 Jan 2026
Posts: 8
Own Kudos:
6
Given Kudos: 4
Location: India
Concentration: Finance, Entrepreneurship
GPA: 10
Posts: 8
Kudos: 6
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 07:46
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Firstly I have solved the all the roots into 3/5 form and received 1/3x,1/2y and 1/z So after comparing if it 3/5^1/3x <3/5^1/2y then 1/3X>1/2Y. So after solving this I have received only pair has 9,4,1 with distinct non prime positive integers and x+y+z is equal to 14
User avatar
forestmayank
User avatar
Joined: 05 Nov 2025
Last visit: 26 Aug 2026
Posts: 103
Own Kudos:
87
 [1]
Given Kudos: 6
Posts: 103
Kudos: 87
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 07:57
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
A root of a number can be written as a fractional power. Eg. Cube root of, 3_/5 can be written as 5^1/3.

Similarly,
9x_/27/125 can be written as (_/3/5)^3/9x, i.e. (_/3/5)^1/3x

Similarly, 4y_/9/25 would be (_/3/5)^2/4y, i.e. (_/3/5)^1/2y

And, z_/3/5 would be (_/3/5)^1/z

Since all the base of all values are equal, 1/3x > 2y > z
OR, 3x < 2y < z

Since the numbers are non-prime single digits, the options are 1,4,6,8,9
Substituting the values, the only options that fits best is z = 9, y = 4 and x = 1
Hence, x+y+z = 14
User avatar
akrutij
User avatar
Joined: 08 Jul 2025
Last visit: 18 Jan 2026
Posts: 12
Own Kudos:
10
 [1]
Given Kudos: 29
Posts: 12
Kudos: 10
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 08:06
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Given : x,y,z non prime integers
Simplifying the inequaities
27/125= (3/5) ^ 3/9x = (3/5)^1/3x
9/25=(3/5)^1/2y
& (3/5)^1/z

Given: (3/5)^1/3x < (3/5)^1/2y < (3/5)^1/z
Since 3/5 < 1 the powers will be reversed
1/3x> 1/2y> 1/z
= 3x<2y<z
if x = 1 next possible valyue for y = 4 and z = 9
x+y+z= 1+4+9= 14
User avatar
Zeus_
User avatar
Joined: 03 Jan 2024
Last visit: 01 Sep 2026
Posts: 33
Own Kudos:
22
Given Kudos: 7
Products:
GMAT Club Tests Forum Quiz
Posts: 33
Kudos: 22
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 08:23
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think the answer should be A since any value of x more than 1 would not hold the condition as true.
User avatar
givemethescore
User avatar
Joined: 29 Nov 2025
Last visit: 02 Jan 2026
Posts: 10
Own Kudos:
10
 [1]
Given Kudos: 11
Posts: 10
Kudos: 10
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 08:31
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
*: 3/5 <1, so as the exponent increases, the value of (3/5)^(1/3x) decreases. A large exponent gives a smaller result

**: when numerators equal, bigger denominator give smaller fraction

Consider: 3x < 2y < z, where x, y, z are all one-digit non-prime numbers.

Since 3x<z => 3x cannot exceed 9 => x has 3 options 1, 2, 3 => x=1 (eliminate 2 and 3, which are prime numbers)

Similarly, 3*1 < 2y < z => 2y cannot exceed 9 => y has 3 options 2, 3, 4 => y=4 (eliminate 2 and 3, which are prime number)

Therefore, 3*1 < 2*4 < z => z=9

Ultimately, x + y + z = 1 + 4 + 9 = 14
Attachments

IMG_5281.jpg
IMG_5281.jpg [ 1.42 MiB | Viewed 296 times ]

User avatar
Leebash
User avatar
Joined: 28 Sep 2025
Last visit: 08 Jul 2026
Posts: 9
Own Kudos:
5
 [1]
Given Kudos: 11
Posts: 9
Kudos: 5
 [1]
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 16 Dec 2025, 08:32
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Expand the equation and make all inside root values as of 3/5. Then we get 1/3x>1/2y>1/z inequality sign changes because 3/5 is less than 1 so exponent must be a greater value for the given Equation to be valid.
Again denominator values become finally 3x<2y<z ....
Check for distinct number which is single digit non prime positive we have[1,4,6,8,9]
The only possible value we can have is x=1, y=4, z=9
So x+y+z = 14
User avatar
sunshineeee
User avatar
Joined: 17 May 2020
Last visit: 30 Aug 2026
Posts: 174
Own Kudos:
53
Given Kudos: 266
Location: Indonesia
Schools: Stern '26 (WL) Tepper '26 (WL) BU'26 (A) Ross '26 Johnson '26
Products:
GMAT Club Tests Forum Quiz
Schools: Stern '26 (WL) Tepper '26 (WL) BU'26 (A) Ross '26 Johnson '26
Posts: 174
Kudos: 53
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 17 Dec 2025, 03:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
could you explain how to spot this?

"Since x, y, and z are distinct single-digit non-prime positive numbers"

why cant we use the prime positive numbers as the x,y,z?

Thanks
Bunuel


GMAT Club Official Explanation:



Notice that 27/125 = (3/5)^3 and 9/25 = (3/5)^2, so:

\(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\)

\(\sqrt[(3*3x)]{\frac{27}{125}} < \sqrt[(2*2y)]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\)

\(\sqrt[3x]{(\frac{27}{125})^{\frac{1}{3}}} < \sqrt[2y]{(\frac{9}{25} )^{\frac{1}{2}}} < \sqrt[z]{\frac{3}{5}}\)

\(\sqrt[3x]{\frac{3}{5}} < \sqrt[2y]{\frac{3}{5}} < \sqrt[z]{\frac{3}{5}}\)

Here’s the key idea: when you take a positive integer root of a number between 0 and 1, the result becomes larger than the original number, and the higher the root, the closer the value moves toward 1.

For example, \(\sqrt{\frac{1}{2}} < \sqrt[3]{\frac{1}{2}} < \sqrt[4]{\frac{1}{2}} < \sqrt[5]{\frac{1}{2}}...\)

Thus, \(\sqrt[3x]{\frac{3}{5}} < \sqrt[2y]{\frac{3}{5}} < \sqrt[z]{\frac{3}{5}}\) implies:

3x < 2y < z

Since x, y, and z are distinct single-digit non-prime positive numbers, they can be selected only from {1, 4, 6, 8, 9}.

Now, x cannot be 4 or greater, because in that case 3x becomes 12 or greater and cannot be less than z, which is at most 9. Thus, x = 1.

Similarly, y cannot be 6 or greater, so y = 4.

And finally, this forces z to be 9.

Thus: x = 1, y = 4, and z = 9.

(3 * 1) < (2 * 4) < 9

Therefore, x + y + z = 1 + 4 + 9 = 14.

Answer: B.
Show more
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 01 Sep 2026
Posts: 113,028
Own Kudos:
838,510
Given Kudos: 111,265
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,028
Kudos: 838,510
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 17 Dec 2025, 04:31
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sunshineeee
could you explain how to spot this?

"Since x, y, and z are distinct single-digit non-prime positive numbers"

why cant we use the prime positive numbers as the x,y,z?

Thanks

Show more

Please read the question:


If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?
User avatar
sunshineeee
User avatar
Joined: 17 May 2020
Last visit: 30 Aug 2026
Posts: 174
Own Kudos:
53
Given Kudos: 266
Location: Indonesia
Schools: Stern '26 (WL) Tepper '26 (WL) BU'26 (A) Ross '26 Johnson '26
Products:
GMAT Club Tests Forum Quiz
Schools: Stern '26 (WL) Tepper '26 (WL) BU'26 (A) Ross '26 Johnson '26
Posts: 174
Kudos: 53
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 17 Dec 2025, 07:45
Kudos
Add Kudos
Bookmarks
Bookmark this Post
thanks Bunuel!! Please read the question carefully, me!!
Bunuel


Please read the question:


If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?
Show more
User avatar
truedelulu
User avatar
Joined: 01 Sep 2025
Last visit: 22 Aug 2026
Posts: 81
Own Kudos:
70
Given Kudos: 18
Posts: 81
Kudos: 70
12 Days of Christmas GMAT Competition 2025 - Day 1: If x, y, and z 17 Dec 2025, 23:00
Kudos
Add Kudos
Bookmarks
Bookmark this Post
\(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\\
<=> ((\frac{3}{5})^3)^{\frac{1}{9x}} < ((\frac{3}{5})^2)^{\frac{1}{4y}} < (\frac{3}{5})^{\frac{1}{z}}\\
<=> (\frac{3}{5)}^{\frac{1}{3x}} < (\frac{3}{5})^{\frac{1}{2y}} < (\frac{3}{5})^{\frac{1}{z}} \)

Since \(\frac{3}{5} < 1 => {\frac{1}{3x}} > {\frac{1}{2y}} > {\frac{1}{z}}\)
Given x,y,z are single-digit and non-prime integers, x,y,z can be {1,4,6,8,9}
=> x = 1, y = 4, z = 9
=> x+y+z = 14

Answer: B.

Bunuel
If x, y, and z are distinct single-digit non-prime positive integers, and \(\sqrt[9x]{\frac{27}{125}} < \sqrt[4y]{\frac{9}{25}} < \sqrt[z]{\frac{3}{5}}\), what is the value of x + y + z?

A. 13
B. 14
C. 15
D. 16
E. 17

Gift
12 Days of Christmas Competition
This question is part of our holiday event
Win $40,000 in prizes: courses, tests, and more
Show more
   1   2   3   4 
Moderator:
Bunuel
Math Expert
113028 posts