Events & Promotions
|
|
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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Jun 0808:00 PM EDT
-10:00 PM EDT
Master the GMAT with expert live instruction, a personalized study plan, and real-time support. Includes 40 hours of online classes plus 6 months of access to the TTP GMAT OnDemand video course. Mon/Wed June 8, 2026 →August 12, 2026 8:00pm-10:00pm EST
Jun 1006:00 AM PDT
-06:15 PM PDT
Register for the GMAT Club Virtual MBA Spotlight Fair – the world’s premier event for serious MBA candidates. This is your chance to hear directly from Admissions Directors at nearly every Top 30 MBA program..- Jun 10
10:00 AM PDT
-11:00 AM PDT
Scoring 715 on the GMAT Focus Edition requires more than just learning formulas, memorizing concepts, or solving hundreds of questions. In this episode, Nishant shares how he improved his GMAT preparation by focusing on application of concepts, and more. 
Jun 1111:00 AM EDT
-01:00 PM EDT
TTP GMAT OnDemand gives serious students 400+ hours of expert video instruction, the full TTP course, AI support, weekly office hours, and a 715+ score guarantee—all built for elite GMAT score improvement.
06 Jun 2026, 10:13
Kudos
Bookmarks
N
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
(N/A)
Question Stats:
0% (00:00) correct
0%
(00:00)
wrong
based on 0
sessions
History
Date
Time
Result
Not Attempted Yet
5 Rules Worth Memorizing
More theory
[*]Difference of Squares
Formula:
N = a2 − b2 = (a+b)(a−b)
Key Rule:
A number can be written as a difference of squares if:
• It is odd
OR
• It is divisible by 4
Why?
(a+b) and (a−b) are either both odd or both even.
Odd × Odd = Odd
Even × Even = Multiple of 4
Example:
15 = 42 − 12 = (4+1)(4−1) = 5×3
20 = 62 − 42 = (6+4)(6−4) = 10×2
Counterexample:
6 cannot be written as a2 − b2
Reason:
6 = 4k+2
Numbers of form 4k+2 never work.
- Difference of squares ⇔ odd or multiple of 4.
- Odd sum of two primes ⇒ one prime must be 2.
- In factorials, cancel first; never expand.
- AM ≥ GM with equality at x = y.
- Equal numbers often create extrema:
• Maximum product for fixed sum
• Minimum sum for fixed product
• Minimum sum of squares for fixed sum.
More theory
- AM-GM Thinking
Formula:
(x+y)/2 ≥ √(xy)
Equality only when:
x = y
Most Important Learning:
For a fixed sum:
Product is maximum when numbers are equal.
Example:
Sum = 20
10×10 = 100
9×11 = 99
8×12 = 96
Maximum occurs at:
10 and 10
Second Learning:
For a fixed product:
Sum is minimum when numbers are equal.
Example:
Product = 36
6+6 = 12
4+9 = 13
3+12 = 15
Minimum occurs at:
6 and 6
GMAT Trigger:
See:
• Fixed sum + maximize product
OR
• Fixed product + minimize sum
Immediately test:
x = y
- Minimum/Maximum Strategy
Key Rule:
For a fixed sum:
x2 + y2
is minimized when:
x = y
Example:
x+y = 10
Case 1:
52 + 52
= 25 + 25
= 50
Case 2:
42 + 62
= 16 + 36
= 52
Case 3:
22 + 82
= 4 + 64
= 68
Notice:
The more unequal the numbers become, the larger the sum of squares becomes.
GMAT Example:
Given:
(x+y)2 > 200
Find minimum possible value of:
x2 + y2
Minimum occurs when:
x = y
So:
(2x)2 > 200
4x2 > 200
x2 > 50
Therefore:
x2 + y2 > 50 + 50 = 100
GMAT Trigger:
When asked to minimize:
x2+y2
a2+b2
sum of squares
Try making variables equal.
[*]Difference of Squares
Formula:
N = a2 − b2 = (a+b)(a−b)
Key Rule:
A number can be written as a difference of squares if:
• It is odd
OR
• It is divisible by 4
Why?
(a+b) and (a−b) are either both odd or both even.
Odd × Odd = Odd
Even × Even = Multiple of 4
Example:
15 = 42 − 12 = (4+1)(4−1) = 5×3
20 = 62 − 42 = (6+4)(6−4) = 10×2
Counterexample:
6 cannot be written as a2 − b2
Reason:
6 = 4k+2
Numbers of form 4k+2 never work.
- Prime Number Sums
If the sum of two primes is odd:
One of the primes must be 2.
Example:
19 = 2 + 17
Works because 17 is prime.
GMAT Trigger:
When asked:
"Can odd number N be written as sum of two primes?"
Check:
N − 2
If N−2 is prime, answer is Yes.
If not, answer is No.
Example:
45 − 2 = 43 (prime)
So:
45 = 2 + 43
Works.
06 Jun 2026, 10:26
Kudos
Bookmarks
The key idea:
When you add the same positive number b to both the numerator and denominator,
x/y → (x+b)/(y+b)
the new fraction moves closer to 1.
Why?
Because the numerator and denominator become more similar.
----------------------------------------------------------
Case 1: Fraction less than 1
2/5 = 0.4
Add 3 to both:
(2+3)/(5+3) = 5/8 = 0.625
The fraction got bigger and moved closer to 1.
Therefore:
If x/y < 1, then (x+b)/(y+b) > x/y
----------------------------------
Case 2: Fraction greater than 1
5/2 = 2.5
Add 3 to both:
(5+3)/(2+3) = 8/5 = 1.6
The fraction got smaller and moved closer to 1.
Therefore:
If x/y > 1, then (x+b)/(y+b) < x/y
----------------------------------
GMAT Shortcut:
If a fraction is less than 1, adding the same number to the numerator and denominator makes it larger.
If a fraction is greater than 1, adding the same number to the numerator and denominator makes it smaller.
In both cases, the fraction moves toward 1.
Key Rule
Start with x/y.
Add a to the numerator and b to the denominator:
(x + a)/(y + b)
The new fraction moves closer to a/b.
Decision Rule
If x/y < a/b, the fraction increases.
If x/y > a/b, the fraction decreases.
If x/y = a/b, the fraction stays the same.
Example 1
Start: 2/7
Add 3 to numerator and 5 to denominator.
Target fraction = 3/5.
Since 2/7 < 3/5, the new fraction is larger:
2/7 < 5/12
Example 2
Start: 11/12
Add 2 to numerator and 5 to denominator.
Target fraction = 2/5.
Since 11/12 > 2/5, the new fraction is smaller:
11/12 > 13/17
GMAT Shortcut
Compare the original fraction x/y with a/b.
If x/y < a/b → fraction goes up.
If x/y > a/b → fraction goes down.
If x/y = a/b → no change.
Think of (x + a)/(y + b) as a weighted average of x/y and a/b, so it always moves toward a/b.
Key rule:
(x+2)/(y+3) moves toward 2/3.
Need to know whether x/y is above or below 2/3.
(1) y > 20
No information about x. x/y could be above or below 2/3.
Insufficient.
(2) x < 5
No information about y. x/y could be above or below 2/3.
Insufficient.
Together:
x < 5 and y > 20
Largest possible fraction is 4/21.
Since 4/21 < 2/3, x/y < 2/3 always.
Moving toward 2/3 means the fraction increases.
Definite answer: Yes.
Answer = C.
Key rule:
(x+5)/(y+5) moves toward 1.
Need to know whether x/y is above or below 1.
(1) y = 5
No information about x.
x/y could be less than or greater than 1.
Insufficient.
(2) x > y
Since x/y > 1, the fraction is above 1.
Moving toward 1 makes the fraction smaller.
Definite answer: No.
Sufficient.
Answer = B.
When you add the same positive number b to both the numerator and denominator,
x/y → (x+b)/(y+b)
the new fraction moves closer to 1.
Why?
Because the numerator and denominator become more similar.
----------------------------------------------------------
Case 1: Fraction less than 1
2/5 = 0.4
Add 3 to both:
(2+3)/(5+3) = 5/8 = 0.625
The fraction got bigger and moved closer to 1.
Therefore:
If x/y < 1, then (x+b)/(y+b) > x/y
----------------------------------
Case 2: Fraction greater than 1
5/2 = 2.5
Add 3 to both:
(5+3)/(2+3) = 8/5 = 1.6
The fraction got smaller and moved closer to 1.
Therefore:
If x/y > 1, then (x+b)/(y+b) < x/y
----------------------------------
GMAT Shortcut:
If a fraction is less than 1, adding the same number to the numerator and denominator makes it larger.
If a fraction is greater than 1, adding the same number to the numerator and denominator makes it smaller.
In both cases, the fraction moves toward 1.
Key Rule
Start with x/y.
Add a to the numerator and b to the denominator:
(x + a)/(y + b)
The new fraction moves closer to a/b.
Decision Rule
If x/y < a/b, the fraction increases.
If x/y > a/b, the fraction decreases.
If x/y = a/b, the fraction stays the same.
Example 1
Start: 2/7
Add 3 to numerator and 5 to denominator.
Target fraction = 3/5.
Since 2/7 < 3/5, the new fraction is larger:
2/7 < 5/12
Example 2
Start: 11/12
Add 2 to numerator and 5 to denominator.
Target fraction = 2/5.
Since 11/12 > 2/5, the new fraction is smaller:
11/12 > 13/17
GMAT Shortcut
Compare the original fraction x/y with a/b.
If x/y < a/b → fraction goes up.
If x/y > a/b → fraction goes down.
If x/y = a/b → no change.
Think of (x + a)/(y + b) as a weighted average of x/y and a/b, so it always moves toward a/b.
Key rule:
(x+2)/(y+3) moves toward 2/3.
Need to know whether x/y is above or below 2/3.
(1) y > 20
No information about x. x/y could be above or below 2/3.
Insufficient.
(2) x < 5
No information about y. x/y could be above or below 2/3.
Insufficient.
Together:
x < 5 and y > 20
Largest possible fraction is 4/21.
Since 4/21 < 2/3, x/y < 2/3 always.
Moving toward 2/3 means the fraction increases.
Definite answer: Yes.
Answer = C.
Key rule:
(x+5)/(y+5) moves toward 1.
Need to know whether x/y is above or below 1.
(1) y = 5
No information about x.
x/y could be less than or greater than 1.
Insufficient.
(2) x > y
Since x/y > 1, the fraction is above 1.
Moving toward 1 makes the fraction smaller.
Definite answer: No.
Sufficient.
Answer = B.
Moderator:
