Last visit was: 23 Apr 2026, 01:44 It is currently 23 Apr 2026, 01:44
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.
Close
Request Expert Reply
Confirm Cancel
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [7]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [7]
4
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [10]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [10]
3
Kudos
Add Kudos
6
Bookmarks
Bookmark this Post
General Discussion
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [3]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [3]
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
avatar
archiemuty
Joined: 27 Jun 2014
Last visit: 16 Apr 2020
Posts: 2
Own Kudos:
1
 [1]
Posts: 2
Kudos: 1
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi. Please clarify how you got St 2: (k-2)(k_4)>0
hence, k >4 or k <2.

According to my workings K>4 or K>2. How did you get K<2.Much appreciated

Posted from my mobile device
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [1]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [1]
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Use wavy curve method for solving the inequality range problems:



Steps of wavy curve method :

1. All zeros of the function f(x) contained on the left-hand side of the inequality should be marked on the number line with inked (black) circles.

2. All points of discontinuities of the function f(x) contained on the left-hand side of the inequality should be marked on the number line with un-inked (white) circles.

3. From right to left, beginning above the number line (in case of the value of f(x) is positive in step (iii), otherwise, from below the number line), a wavy curve should be drawn to pass through all the marked points so that when it passes through a simple point the curve intersects the number line, and, when passing through a double point the curve remains located on one side of the number line.

4. The appropriate intervals are chosen in accordance with the sign of inequality (the function f(x) is positive whenever the curve is situated above the number line, it is negative if the curve is found below the number line). Their union represents the solution of the inequality.


Remark :

i) A point of discontinuity will never be included in the answer.

ii) If you are asked to the interval where f(x) is non-negative or non-positive then make the interval closed corresponding to the roots of the numerator and let it remain open corresponding to the roots of the denominator.

The point where f(x) vanishes are called zeroes of the function .

The point x=bj are called point of discontinuity of f(x).

If the exponent of the factor is odd then the point is called simple point .

If the exponent of the factor is even the point is called a simple point.

Illustration :

Let f(x) = {(x-3)(x+2)(x+5) }/{(x+1)(x+7)} .

As we can find the critical points as x=3,-2,-5,-1,7.

Now plot the points on the number line as per the rule of the number line . (Left to Right)

Now check for each interval whether the value of f(x) is greater than zero or less than zero.

Note: Just take a value between the interval and calculate f(x). If f(x)>0 then curve lies above the number line or it will lie below the number line.

Follow the procedure for this given problem you will get the curve as shown above.

f(x) >0 for x ∈ [-5,-2] U (-1,3] U (7,∞)

f(x)<0 for x ∈ (-∞,-5] U [-2,-1) U [3,7)


archiemuty
Hi. Please clarify how you got St 2: (k-2)(k_4)>0
hence, k >4 or k <2.

According to my workings K>4 or K>2. How did you get K<2.Much appreciated

Posted from my mobile device

Attachment:
Capture.JPG
Capture.JPG [ 14.76 KiB | Viewed 12182 times ]
Attachment:
Capture.JPG
Capture.JPG [ 18.02 KiB | Viewed 12149 times ]
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [1]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
For the specific doubt on this problem, see the sketch.
Please acknowledge the solution.


archiemuty
Hi. Please clarify how you got St 2: (k-2)(k_4)>0
hence, k >4 or k <2.

According to my workings K>4 or K>2. How did you get K<2.Much appreciated

Posted from my mobile device
Attachment:
Capture.JPG
Capture.JPG [ 92.83 KiB | Viewed 12110 times ]
User avatar
Archit3110
User avatar
Major Poster
Joined: 18 Aug 2017
Last visit: 22 Apr 2026
Posts: 8,627
Own Kudos:
5,190
 [1]
Given Kudos: 243
Status:You learn more from failure than from success.
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1: 545 Q79 V79 DI73
GMAT Focus 2: 645 Q83 V82 DI81
GPA: 4
WE:Marketing (Energy)
Products:
GMAT Focus 2: 645 Q83 V82 DI81
Posts: 8,627
Kudos: 5,190
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GMATBusters
Is \(m*k^4\) > 0?
1) \(m^2-8m+15<0\)
2) \(k^2-6k+8>0\)


#1
\(m^2-8m+15<0\)

can be written as ; ( m-5)*(m-3)<0
which is possible only when m=4
value of k not know so insufficient
#2
k^2-6k+8>0

can be written as

(k-4)*(k-2)>0

value of k will fall ; 1<k>4
i.e k can be 0 or -ve or >4
value of m not know ; insufficient
from 1*2
\(m*k^4\) > 0
since k can be 0 or integer so answer to above is both yes and no
hence insufficient
IMO E
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
You are right... :thumbup:


Archit3110
GMATBusters
Is \(m*k^4\) > 0?
1) \(m^2-8m+15<0\)
2) \(k^2-6k+8>0\)


#1
\(m^2-8m+15<0\)

can be written as ; ( m-5)*(m-3)<0
which is possible only when m=4
value of k not know so insufficient
#2
k^2-6k+8>0

can be written as

(k-4)*(k-2)>0

value of k will fall ; 1<k>4
i.e k can be 0 or -ve or >4
value of m not know ; insufficient
from 1*2
\(m*k^4\) > 0
since k can be 0 or integer so answer to above is both yes and no
hence insufficient
IMO E
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Guys!!!

Let us schedule the doubt chat session at 10 PM IST at this group - Group link

Be ready with ur doubts

User avatar
gurmukh
Joined: 18 Dec 2017
Last visit: 30 Dec 2025
Posts: 258
Own Kudos:
Given Kudos: 20
Posts: 258
Kudos: 269
Kudos
Add Kudos
Bookmarks
Bookmark this Post
In rectangle ABCD, two diagonals cut at point O, if some ratio of length and breadth is given,can we find out angle AOB without using any trigonometry?

Posted from my mobile device
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The GMAT would ask angle only if the angle is evident owing to an isosceles triangle, or 30-60-90 or 45-45-90 triangle.



gurmukh
In rectangle ABCD, two diagonals cut at point O, if some ratio of length and breadth is given,can we find out angle AOB without using any trigonometry?

Posted from my mobile device

Attachment:
30-60-90-example-diagram.png
30-60-90-example-diagram.png [ 5.54 KiB | Viewed 11701 times ]
Attachment:
454590 triangle.jpg
454590 triangle.jpg [ 8.89 KiB | Viewed 11746 times ]
User avatar
gurmukh
Joined: 18 Dec 2017
Last visit: 30 Dec 2025
Posts: 258
Own Kudos:
Given Kudos: 20
Posts: 258
Kudos: 269
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Ok. That means there is no ratio.
I was also thinking same

Posted from my mobile device
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Yes, the trigonometric ratio is not in GMAT syllabus

gurmukh
Ok. That means there is no ratio.
I was also thinking same

Posted from my mobile device
User avatar
gurmukh
Joined: 18 Dec 2017
Last visit: 30 Dec 2025
Posts: 258
Own Kudos:
Given Kudos: 20
Posts: 258
Kudos: 269
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GMATBusters
Yes, the trigonometric ratio is not in GMAT syllabus

gurmukh
Ok. That means there is no ratio.
I was also thinking same

Posted from my mobile device
I think in rectangle ABCD in which two diagonals cut at point O. Angle AOB can be calculated when ratio of length and breadth are given Sq of 3: 1.
One angle at diagonal will come out 60 and one angle will come out 120.
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The angle in any triangle depends on the ratio of sides and vice versa using Sine rule. But that is not tested.

gurmukh
GMATBusters
Yes, the trigonometric ratio is not in GMAT syllabus

gurmukh
Ok. That means there is no ratio.
I was also thinking same

Posted from my mobile device
I think in rectangle ABCD in which two diagonals cut at point O. Angle AOB can be calculated when ratio of length and breadth are given Sq of 3: 1.
One angle at diagonal will come out 60 and one angle will come out 120.
User avatar
gurmukh
Joined: 18 Dec 2017
Last visit: 30 Dec 2025
Posts: 258
Own Kudos:
Given Kudos: 20
Posts: 258
Kudos: 269
Kudos
Add Kudos
Bookmarks
Bookmark this Post
When the ratio of sides is given sqt of 3 : 1 then we take out angle using 90:30:60 triangle, this identity of trigonometry we use in GMAT and through this we can take out angles. Otherwise trigonometry is not required.

Posted from my mobile device
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
yes, you are right... only specific cases are tested.

gurmukh
When the ratio of sides is given sqt of 3 : 1 then we take out angle using 90:30:60 triangle, this identity of trigonometry we use in GMAT and through this we can take out angles. Otherwise trigonometry is not required.

Posted from my mobile device
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
Kudos
Add Kudos
Bookmarks
Bookmark this Post
yes you are right.

for (k-2)(k-4)>0
hence, k >4 or k <2
now, k can be anything less than 2 even 0 or -ve

(
Izzy77
(k-2)(k_4)>0
hence, k >4 or k <2
now, k can also be equal to zero.
User avatar
shameekv1989
Joined: 14 Dec 2019
Last visit: 17 Jun 2021
Posts: 816
Own Kudos:
Given Kudos: 354
Location: Poland
Concentration: Entrepreneurship, Strategy
GMAT 1: 640 Q49 V27
GMAT 2: 660 Q49 V31
GMAT 3: 720 Q50 V38
GPA: 4
WE:Engineering (Consumer Electronics)
Products:
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello, I have a question regarding "3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?"

Solution to this is concentration after 3 rounds :- (1 - 3/30)^3 = 721/1000 (Because in each round the concentration for the previous round changes and eventually turns out to be this in the end)

However, I wanted to understand how can I extend this concept to much harder or rather complicated questions for eg:-

3L of solution is drawn from a container containing 30L of solution that is 20% concentrated. It is replaced by another solution that is 10% concentrated and the process is repeated 2 times. What is the concentration of solution after 3 rounds?

Thanks in advance,
Shameek
User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 21 Apr 2026
Posts: 1,921
Own Kudos:
6,855
 [3]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,921
Kudos: 6,855
 [3]
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Hi,
It can be done by finding the concentration at each step, but as the new solution added is not 0% milk, it will affect the new concentration differently each step.

Step 1)

3L of the solution is drawn from a container containing 30L of solution that is 20% concentrated. It is replaced by another solution that is 10% concentrated
it is same as mixing 2 solutions:
i) 27L of 20% and 3L of 10%
final concentration will be : (27*.2+3*.1)/30 = 0.19 or 19%

Step 2)

3L of the new solution is drawn from a container containing 30L of solution that is 19% concentrated. It is replaced by another solution that is 10% concentrated
it is same as mixing 2 solutions:
i) 27L of 19% and 3L of 10%
final concentration will be : (27*.19+3*.1)/30 = 0.181 or 18.1%

Step 3)

3L of the new solution is drawn from a container containing 30L of solution that is 18.1% concentrated. It is replaced by another solution that is 10% concentrated
it is same as mixing 2 solutions:
i) 27L of 18.1% and 3L of 10%
final concentration will be : (27*.181+3*.1)/30 = 0.1729 or 17.29%

Since, in this question, the pattern will not be there, and we need to calculate each step individually, it is highly unlikely that the question with more than 2 steps will be asked in GMAT.

I Hope, the concept is clear.

Feel free to tag me in case of any doubt.

Happy Learning.



shameekv1989
Hello, I have a question regarding "3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?"

Solution to this is concentration after 3 rounds :- (1 - 3/30)^3 = 721/1000 (Because in each round the concentration for the previous round changes and eventually turns out to be this in the end)

However, I wanted to understand how can I extend this concept to much harder or rather complicated questions for eg:-

3L of solution is drawn from a container containing 30L of solution that is 20% concentrated. It is replaced by another solution that is 10% concentrated and the process is repeated 2 times. What is the concentration of solution after 3 rounds?

Thanks in advance,
Shameek
 1   2   
Moderators:
Math Expert
109763 posts
498 posts
212 posts