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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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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[Math Revolution GMAT math practice question]

If y=|x-1|+|x+1|, then y=?

1) x>-1
2) x<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.

There are three ranges of values of x to consider.
If x > 1, then y = | x – 1 | + | x + 1 | = x – 1 + x + 1 = 2x and we don’t have a unique value of y.
If -1 ≤ x ≤ 1, then y = | x – 1 | + | x + 1 | = - ( x – 1 ) + x + 1 = 2 and we have a unique value of y.
If x < 1, then y = | x – 1 | + | x + 1 | = -( x – 1 ) – ( x + 1 ) = -2x and we don’t have a unique value of y.

Asking for the value of y is equivalent asking if -1 ≤ x ≤ 1.
Both conditions yield the inequality -1 < x < 1, when applied together. Therefore, both conditions are sufficient, when taken together.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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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[Math Revolution GMAT math practice question]

m+n=?

1) (4^m)(2^n)=16
2) (2^{2m})(4^n)=64

=>

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) is equivalent to m + n = 3 as shown below:
(2^{2m})(4^n)=64
=> (2^{2m})(2^{2n})=2^6
=> 2^{2m+2n}=2^6
=> 2m+2n = 6
=> m + n = 3
Condition 2) is sufficient.

Condition 1)
(4^m)(2^n)=16
=> (2^{2m})(2^n)=2^4
=> 2^{2m+n}=2^4
=> 2m+n = 4
If m = 1 and n =2, then m + n = 3.
If m = 0 and n = 4, then m + n = 4.
Since it does not yield a unique solution, condition 1) is not sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°

=>

The sum of all interior angles of n-gon is (n-2)*180°.
A decagon is a 10-gon.
The sum of all interior angles is (10-2)* 180° = 8*180°.
And each interior angle of a regular decagon has measure 8*180° / 10 = 8*18° = 144°.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

If operation # represents one of addition, subtraction, multiplication, and division, what is the value of 0#1?

1) 2#1 = 2
2) 4#2 = 2

=>

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 operation is considered as a variable. Since we have 1 variable and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since 2#1 = 2, # is one of the operations, multiplication and division.
If # is the multiplication operation, then 0#1 = 0.
If # is the division operation, then 0#1 = 0.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since 4#2 = 2, # is one of the operations, subtraction and division.
If # is the subtraction operation, then 0#1 = -1.
If # is the division operation, then 0#1 = 0.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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]

What is the value of (2^8 + 2^9 + 2^{10} + 2^{11}) / 32?

A. 80
B. 96
C. 100
D. 120
E. 160

=>

(2^8 + 2^9 + 2^{10} + 2^{11}) / 32
= (2^8 + 2^82^1 + 2^82^2 + 2^82^3) / 2^5
= 2^8(1 + 2^1 + 2^2 + 2^3) / 2^5
= 2^3(1+2+4+8)
= 8*15
= 120

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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 a>b>c>d>0, is d<4?

1) 1/c + 1/d > 1/2
2) (1/a)+(1/b)+(1/c)+(1/d)=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.
The original condition a>b>c>d>0 is equivalent to 0 < 1/a < 1/b < 1/c < 1/d. The question asks if d < 4. This is equivalent to asking if 1/d > 1/4.

By condition 1, 1/d > 1/4 since 1/c < 1/d. So, d < 4. Condition 1) is sufficient.

Condition 2)
Since 0 < 1/a < 1/b < 1/c < 1/d and (1/a)+(1/b)+(1/c)+(1/d)=1, we have 1/a<1/d, 1/b<1/d, 1/c<1/d and 1/a +1/b + 1/c + 1/d < 1/d +1/d +1/d + 1/d = 4/d. Therefore, 1 < 4/d and d < 4.
Condition 2) is sufficient.

Note: This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is hard. For CMT4(B) questions, D is most likely to be the answer.
_________________
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Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

A. one
B. two
C. three
D. four
E. five

=>

y = x^4-3x^3+2x^2
=> y = x^2(x^2-3x+2)
=> y = x^2(x-1)(z-2)
The x-intercepts occur when y = 0. This occurs when x = 0, x = 1 and x = 2.
There are three x-intercepts.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

We define the harmonic mean of a set of numbers as the reciprocal of the average (arithmetic mean) of the reciprocals of the numbers. What is the harmonic mean of 20 and 30?

A. 22
B. 24
C. 25
D. 26
E. 28

=>

1 / { ( 1/20 + 1/30 ) / 2 } = 1 / { ( 3/60 + 2/60 ) / 2 } = 1 / { (5/60) / 2 } = 1 / { 5 / 120 } = 120 / 5 = 24.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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]

Is 3(a-b)>0?

1) a^3>b^3
2) a>b

=>

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.

Asking if 3(a-b)>0 is equivalent to asking if a – b > 0, or, equivalently, if a > b.
Thus, condition 2) is sufficient.

Condition 1)
a^3>b^3
=> a^3-b^3 > 0
=> (a-b)(a^2+ab+b^2) > 0
=> a - b > 0 since a^2+ab+b^2 > 0
=> a > b
Thus, condition 1) is sufficient too.

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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
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]

Is |a|<1?

1) a^2<1
2) 1/(1-a^2)>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.

|a|<1
=> |a|^2 < 1
=> a^2 < 1
=> a^2-1 < 0
=> (a+1)(a-1) < 0
=> -1 < a < 1

Condition 1) is sufficient, since it is equivalent to the question.

Condition 2)
1/(1-a^2) > 0
=> (1-a^2) > 0
=> a^2-1 < 0
=> (a+1)(a-1) < 0
=> -1 < a < 1
Thus, condition 2) is sufficient, since it is also equivalent to the question.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

Which of the following expressions is equal to 2^{32}-2^{31}-2^{30}?

A. 3*2^{29}
B. 5*2^{29}
C. 2^{30}
D. 3*2^{30}
E. 2^{31}

=>

2^{32}-2^{31}-2^{30}?
= (2^2)(2^{30})- (2^1)(2^{30})-2^{30}
= 4(2^{30})- 2(2^{30})-2^{30}
= (4-2-1)(2^{30})
=1(2^{30})=2^{30}

_________________
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Joined: 16 Aug 2015
Posts: 8425
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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.

_________________
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Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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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: 8425
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.

_________________
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Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
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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.

_________________
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Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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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: 8425
GMAT 1: 760 Q51 V42
GPA: 3.82
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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: 8425
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 364 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: 8425
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: 8425
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, 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.

_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 17 Mar 2019, 18:23

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