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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9032
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]

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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[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: 9032
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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: 9032
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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9032
GMAT 1: 760 Q51 V42
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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
Posts: 9032
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
Posts: 9032
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]

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 458 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: 9032
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.

_________________
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Joined: 16 Aug 2015
Posts: 9032
GMAT 1: 760 Q51 V42
GPA: 3.82
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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.

_________________
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Joined: 16 Aug 2015
Posts: 9032
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: 9032
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.

_________________
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Posts: 9032
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

x, y and z are different integers. Is their average equal to their median?

1) Their range is 11.
2) Their median is 11.

=>

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 x, y and z are different integers with x < y < z.
For their average ( x + y + z ) / 3 to be equal to their median y,
we must have z – y = y – x, and so their range is z – x = z – y + y – x = 2(y-x).
This implies that z – x is an even integer.

Condition 1)
Since condition 1) gives an odd value for the range, the answer is ‘no’. Thus, condition 1) is sufficient by CMT (Common Mistake Type) 1.

Condition 2)
If x = 10, y = 11 and z = 12, then the average and the median are the same, and the answer is ‘yes’
If x = 10, y = 11 and z = 15, then the average 12 is different from the median 11, 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: 9032
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(number property) If the sum of n consecutive positive integers is 42, which of these could be the value of n?

A. 7
B. 8
C. 9
D. 10
E. 11

=>

Recall that the sum of terms of an arithmetic sequence is {(a+l)/2}*n where a is the first term, l is the last term and n is the number of terms.

We are told that {(a+l)/2}*n = 42 or n(a+l) = 84.
Thus, n is a factor of 84.
7 is the unique factor of 84 among the choices.

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Joined: 16 Aug 2015
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[GMAT math practice question]

(geometry) What is the area of triangle ABC?
1) Triangle ABC has two sides of lengths 3 and 4
2) Triangle ABC is a right triangle

=>

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 a triangle has 3 variables in geometry, 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)

Attachment: 4.19.png [ 4.29 KiB | Viewed 379 times ]

There are two right triangles with sides of lengths 3 and 4 as shown above.
Thus, there are two possible areas: (1/2)*4*3 = 6 and (3/2) √7.

Since the conditions don’t yield a unique answer when applied together, they are not sufficient.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9032
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]

(number property) If a, b, and c are integers, is a+b+c an even integer?

1) a^2+b^2 is an even integer
2) b^2+c^2 is an even integer

=>

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 (x, y and z) 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)
If a = 1, b = 1 and c = 1, then a + b + c = 3 is not an even integer and the answer is ‘no’.
If a = 2, b = 2 and c = 2, then a + b + c = 6 is an even integer and the answer is ‘yes’.
Since the conditions don’t yield a unique answer when applied together, they are not sufficient.

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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Example 9:
Stations M and N are connected by two separate, straight and parallel rail lines that are 500 miles long. Freight train A and freight train B simultaneously left Station M and Station N, respectively, and each freight train traveled to the other’s point of departure. The two freight trains passed each other after traveling for 4 hours. When the two freight trains passed, which train was nearer to its destination?

(1) At the time when the two freight trains passed, freight train A had averaged a speed of 60 miles per hour.
(2) Freight train B averaged a speed of 130 miles per hour for the entire trip.

Statement 1 Analysis:
From question stem: Given that both the train traveled for 4hours when they meet. So, A traveled for 4hours.
I states that A had an average speed of 60mph when it passed B.

So, A had traveled 4*60 = 240miles, when both the trains passed each other.
This implies B traveled 260miles.

On a total distance of 500miles, B is near to its destination.

I this statement I is sufficient to answer the question.

But the OA is E.

Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(number properties) If n is a positive integer, what is the value of n?

1) n(n-1) is a prime number
2) n(n+1) has 4 factors

=>

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

Condition 1)
Only n =2 makes n(n-1) a prime number.
Thus, condition 1) is sufficient.

Condition 2)
Integers with four factors have the form p*q or p^3, where p and q are prime integers.
It is impossible to have n(n+1)=p^3, where n is an integer and p is a prime number.
The only time n(n+1) = pq is when n(n+1) = 2*3 and n =2.
Condition 2) is sufficient since it yields a unique solution.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9032
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(number properties) What is the smallest positive multiple of 15 that has only 0 and 1 as digits?

A.15
B. 30
C. 110
D. 1110
E. 111000

=>

A multiple of 15 is a number divisible by both 3 and 5.
Since it is divisible by 5, its units digit must be 0 or 5.
Since it is divisible by 3, the sum of its digits must be a multiple of 3.
1110 is the smallest number with only 0 and 1 as its digits that has a units digit of 0 or 5 and the sum of its digits a multiple of 3.

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[GMAT math practice question]

(inequality) a, b, c, d and e are real numbers with a<b<c<d<e. Is abcde negative?

1) abc < 0
2) cde < 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 5 variables and 4 equations, D is most likely to be the answer. So, we should consider each condition on its own first. If a question is an inequality, then inequalities in the original condition can be counted as equations.

Condition 1)
If a = -1, b = 1, c = 2, d = 3, e = 4, then abcde < 0, and the answer is ‘yes’
If a = -4, b = -3, c = -2, d = -1, e = 1, then abcde > 0, and the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
There are two cases to consider:

i) 0 lies between c and d
ii) 0 is greater than e.

If 0 lies between c and d, then abcde < 0 and the answer is ‘yes’.
If 0 is greater than e, then abcde < 0 and the answer is ‘yes’.
Condition 2) is sufficient since it gives a unique answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
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Joined: 16 Aug 2015
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[GMAT math practice question]

x^2 + 4x + 1 = 0. What is the value of x^2 + 1/x^2?

A. 10
B. 12
C. 14
D. 16
E.18

=>

x^2 + 4x + 1 = 0
=> x + 4 + 1/x = 0
=> x + 1/x = -4
So, (x + 1/x)^2 = (-4)^2 and x^2 +2x(1/x) + 1/x^2 = 16.
Therefore,
x^2 +2 + 1/x^2 = 16
and x^2 + 1/x^2 = 14.

_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 08 May 2019, 17:28

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