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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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If the median of 5 positive integers is 10, is their average (arithmetic mean) greater than 10?

1) The largest number is 40
2) The smallest number is 1

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Suppose the numbers satisfy a ≤ b ≤ 10 ≤ c ≤ d. The question asks if ( a + b + 10 + c + d ) / 5 > 10 or a + b + c + d + 10 > 50.
This is equivalent to the inequality, a + b + c + d > 40.
If a question includes the words “greater than”, then it asks us to look for a minimum.
Since a, b c, and d are positive, and d = 40 by condition 1), we must have a + b + c + d > 40.
Condition 1) is sufficient.

Condition 2)
If a = 1, b = 2, c = 11, and d = 40, then a + b + c + d > 40, and the answer is ‘yes’.
If a = 1, b = 2, c = 11, and d = 12, then a + b + c + d < 40, and the answer is ‘no’.
Thus, condition 2) is not sufficient since it does not yield a unique solution.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
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f(x)=x^2n+x^n+1, where n is an integer. Is f(x)=1?

1) x=-1
2) n is a multiple of 5.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
f(x)=1
⇔ x^2n+x^n+1=1
⇔ x^2n+x^n=0
⇔ x^n(x^n +1)=0
⇔ xn= 0 or xn =-1
⇔ ( x = 0 ) or ( x = -1 and n is odd )

Conditions 1) and 2)
If x = -1 and n = 5, then f(x) = (-1)10 + (-1)5 + 1 = 1 + (-1) + 1 = 1 and the answer is ‘yes’.
If x = -1 and n = 10, then f(x) = (-1)20 + (-1)10 + 1 = 1 + 1 + 1 = 1 and the answer is ‘no’.

Thus, both conditions together are not sufficient, since they do not yield a unique solution.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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The squares of two consecutive positive integers differ by 55. What is the smaller of the two integers?

A. 27
B. 29
C. 30
D. 32
E. 35

Let the two consecutive positive integers be n and n+1.
Then (n+1)^2 – n^2 = 55, so 2n+1 = 55.
It follows that 2n = 54 and n = 27.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

What is the greatest positive three-digit number that is divisible by 2, 3, 4, 5, 6 and 7?

A. 120
B. 140
C. 210
D. 420
E. 840

=>

The question asks for the value of the greatest positive three-digit multiple of lcm(2,3,4,5,6,7), where lcm(2,3,4,5,6,7) is the least common multiple of 2, 3, 4, 5, 6 and 7.
lcm(2,3,4,5,6,7) = 22*3*5*7 = 420.
There are two positive three-digit integers that are multiples of 420: 420 and 840.
The greatest positive three-digit number that is a multiple of 420 is 840.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If n is an integer, is (n+1)(n+2)(n+3) divisible by 12?

1) n is an even number.
2) n is a multiple of 4.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since n+1, n+2 and n+3 are three consecutive integers, (n+1)(n+2)(n+3) is a multiple of 3.

Condition 2) tells us that n+1 and n+3 are odd integers, and n+2 is an even number which is not a multiple of 4. Thus, (n+1)(n+2)(n+3) is not a multiple of 4.
CMT(Common Mistake Type 1) states “no” is also an answer and a condition giving rise to the unique answer “no” is sufficient. Thus, condition 2) is sufficient.

Condition 1)
If n = 2, then (n+1)(n+2)(n+3) = 3*4*5 = 60 is a multiple of 12 and the answer is “yes”.
If n = 4, then (n+1)(n+2)(n+3) = 5*6*7 = 210 is not a multiple of 12 and the answer is “no”.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

A is the set of 6-digit positive integers whose first three digits are same as their last three digits, written in the same order. Which of the following numbers must be a factor of every number in the set A?

A. 7
B. 11
C. 17
D. 19
E. 23

=>

Each number n in the set A is an integer of the form “xyz,xyz”. So,
n = 10^5x + 10^4y + 10^3z + 10^2x + 10y + z
= 10^3(10^2x + 10y + z) + (10^2x + 10y + z )
= 1000(10^2x + 10y + z) + (10^2x + 10y + z )
= 1001(10^2x + 10y + z )
= 11*91(10^2x + 10y + z )

Thus, n is a multiple of 11.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

Is a triangle, with one side of length 12, inscribed in a circle a right triangle?

1) The area of the circle is 36π.
2) The circumference of the circle is 12π.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Attachment: 3.8.png [ 7.42 KiB | Viewed 420 times ]

If a side of a triangle is the diameter of its circumscribed circle, then the triangle is a right triangle.

Condition 1)
A circle with area 36π has radius 6 and diameter 12. So, the answer is ‘yes’ and condition 1) is sufficient.

Condition 2)
A circle with circumference 12π has diameter 12. So, the answer is ‘yes’ and condition 2) is sufficient.

Note that if conditions 1) and 2) yield the same information, Tip 1 of the VA method tells us that D is most likely to be the answer.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If x and y are positive integers and (x-y)^2+y^2=25, which of the following could be the value of x?

A. 6
B. 7
C. 8
D. 9
E. 10

=>

The possible values of the pair (x-y,y), with x and y positive, are (0,5), (3,4), (-3,4), (4,3) and (-4,3).
So, (x,y) = (5,5), (7,4), (1,4), (7,3) or (-1,3).
Thus, x could be 5, 7 or 1.

_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If x and y are positive integers, x/y=?

1) 2^{x+y}3^{xy}=72
2) 2^x3^y=12

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Condition 2) tells us that 2^x3^y=12 = 2^23^1 and x = 2, y = 1.
Thus x/y = 2/1 = 2, and condition 2) is sufficient.

Condition 1)
2^{x+y}3^{xy}=72 = 2^33^2 yields the equations x+y=3 and xy=2.
So, x = 1 and y = 2, or x =2 and y = 1.
Thus, x/y = 1/2 or x/y = 2/1 = 2.
Condition 1) is not sufficient since it does not yield a unique solution.

_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

The point (p,q) lies in which quadrant of the x-y plane?

1) (p+1, q) lies in the 2nd quadrant
2) (q-1, p) lies in the 4th quadrant

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Condition 1) tells us that p + 1 < 0 and q > 0, which is equivalent to p < -1 < 0 and q > 0. Thus, (p, q) is in the 2nd quadrant.
Condition 1) is sufficient.

Condition 2) tells us that q - 1 > 0 and p < 0, which is equivalent to p < 0 and q > 1 > 0. Thus, (p, q) is in the 2nd quadrant.
Condition 2) is sufficient.

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.
_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If n, n/3 and n/4 are positive integers, and n is less than or equal to 100, how many values of n are possible?

A. 6
B. 7
C. 8
D. 9
E. 10

=>

The condition n/3 is a positive integer tells us that n is a positive multiple of 3.
The condition n/4 is a positive integer tells us that n is a positive multiple of 4.
Thus, n is a positive multiple of 12.

The number of positive multiples of 12 less than or equal to 100 is 8 since 100 = 12*8 + 4.

_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

If m and n are prime numbers, what is the value of m+n?

1) 15≤m<n≤20
2) mn = 323

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Condition 1)
15≤m<n≤20 tells us that m = 17 and n = 19. So, m + n = 17 + 19 = 36.
Condition 1) is sufficient since it yields a unique solution.

Condition 2)
mn=323 tells us that m = 17 and n = 19, or m = 19 and n = 17.
In both cases, m + n is 36.
Condition 2) is sufficient since it yields a unique solution.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

Jeonghee has 5 different red cards and 5 different blue cards. She shuffles the10 cards, and then places 5 of the cards in a row. What is the probability that all red cards are adjacent to each other and all blue cards are adjacent to each other in her row?

A. 2/5
B. 28/125
C. 31/126
D. 33/140
E. 25/216

=>

The total number of ways in which 5 cards can be chosen out of 10 cards is 10P5 = 10*9*8*7*6.

There are 5*4*3*2*1 arrangements of each of BBBBB and RRRRR.
There are 5*4*3*2*5 arrangements of each of BBBBR, RBBBB, RRRRB and BRRRR.
There are 5*4*3*5*4 arrangements of each of BBBRR, RRBBB, RRRBB and BBRRR.

Thus, the total number of arrangements with all red cards adjacent to each other and all blue cards adjacent to each other is (5*4*3*2*1)*2 + (5*4*3*2*5)*4 + (5*4*3*5*4)*4.
The required probability is ( 5*4*3*2*1*2 + 5*4*3*2*5*4 + 5*4*3*5*4*4 ) / 10*9*8*7*6 = { 5*4*3(4+40+80) } / { 10*9*8*7*6 } = 124 / 2*9*2*7*2 = 31/126.

_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

Express 2^20-2^19-2^18-2^17 as a power of 2.

A. 2^15
B. 2^16
C. 2^17
D. 2^18
E. 2^19

=>

2^20-2^19-2^18-2^17
=2^32^17- 2^22^17-2^12^17-2^17
= 2^17 (8-4-2-1)
= 2^17(1)
= 2^17

_________________
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Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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[GMAT math practice question]

p, q, and r are different prime numbers. What is the value of q?

1) (pq)^2=36
2) (qr)^2= 225

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 3 variables (p, q and r) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

Since p^2q^2=2^23^2 and q^2r^2 = 3^25^2, we have p = 2, q = 3 and r = 5.
Conditions 1) & 2) are sufficient, when applied together.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since p^2q^2=2^23^2, we must have p = 2, q = 3 or p = 3, q = 2.
Condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
Since q^2r^2=3^25^2, we have q = 3, r = 5 or q = 5, r = 3.
Condition 2) is not sufficient since it does not yield a unique solution.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
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Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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[GMAT math practice question]

The diagram below contains four right triangles with legs a and b. What is the area of the larger square?

Attachment: 3.29.png [ 8.7 KiB | Viewed 466 times ]

1) a = 12
2) b = 9

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

If we assume c is the length of the hypotenuse of the right triangle, we have c^2 = a^2 + b^2 and c^2 is the area the larger square.
We need the values of both a and b. Thus, conditions 1) & 2) are sufficient, when applied together, but neither condition is sufficient on its own.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

√r + 2/√r = 4. What is the value of r + 4/r?

A. 12
B. 14
C. 16
D. 32
E. 64

=>

(√r + 2/√r)^2 = r + 2(√r)(2/√r) + 4/r = r + 4/r + 4 = 4^2 = 16
So, r + 4/r = 16 -4 = 12.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If |2x|>|3y|, is x >y?

1) x>0
2) y>0

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

From condition 1), we have 3x > 2x = |2x| > |3y| ≥ 3y since |x| = x. So, x > y and the answer is ‘yes’.
Thus, condition 1) is sufficient.

Condition 2)
If x = 10, and y = 1, then x > y and the answer is ‘yes’.
If x = -10, and y = 1, then x < y and the answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

Is x < 0?

1) x^3 + 1 < 0
2) x^3 + x + 1 < 0

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
x^3 + 1 < 0
=> (x+1)(x^2-x+1) < 0
=> x + 1 < 0 since x^2-x+1 > 0
=> x < -1 < 0
Thus, condition 1) is sufficient, and the answer is ‘yes’.

Condition 2)
x^3 + x + 1 < 0
=> x^3 + x < -1
=> x(x^2 + 1) < -1
=> x < -1/(x^2 + 1) since x^2 + 1 > 0
=> x < -1/(x^2 + 1) < 0 since x^2 + 1 > 0
Thus, condition 2) is sufficient, and the answer is ‘yes’.

_________________
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Joined: 16 Aug 2015
Posts: 9138
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

x^3-y^3 = 90 and x-y = 3. What is the value of xy?

A. 3
B. 4
C. 5
D. 6
E. 7

=>

x^3-y^3 = (x-y)(x^2+xy+y^2) = 90
Since x – y = 3, x^2+xy+y^2 = 30
Now, 9 = (x-y)^2 = x^2 – 2xy + y^2 = x^2+xy+y^2 – 3xy = 30 – 3xy.
So, 3xy = 21 and xy = 7.

_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 11 Apr 2019, 18:24

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# Overview of GMAT Math Question Types and Patterns on the GMAT  