The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(inequality) Five consecutive integers satisfies a<b<c<d<e. what is the maximum of a+e?

1) the summation of five integers is negative
2) e is positive

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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.

Consecutive integers have two variables for the first number and the number of integers. Since the number of integers is 5, we need one more equation and D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
a + b + c + d + e = a + a + 1 + a + 2 + a + 3 + a + 4 = 5a + 10 < 0 or a < -2. Then the maximum of a is -3 and e = a + 4 = 1.
Thus the maximum value of a + e = (-3) + 1 = -2.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
If e = 1, then a = -3 we have a + e = -2.
If e = 2, then a = -2 we have a + e = 0.
If e = 3, then a = -1 we have a + e = 2.
…
As e increases, a + e increases and approaches infinity.
Thus we don’t have a maximum value of a + e.
Condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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.

The squares of 1, 2, 3, 4, … are 1, 4, 9, 16, …, respectively and they have remainders of 1, 0, 1, 2,
if squares are divided by 4 series comes 1,0,1,0, Can you please explain how 2 came ?

[GMAT math practice question]

(velocity) A river flows at a constant speed of 2 miles per hour. It takes 3 hours for a ship to go a miles upstream. How many hours does it take for the ship to go b miles downstream?

1) a = 6 miles
2) b is longer than a by 3 miles

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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 2 variables (a and b) and 0 equations, C 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)
The original speed of the ship is a/3 + 2 = 6/3 + 2 = 4 mph. When the ship goes downstream, its speed is 4 + 2 = 6 mph.
The time that the ship takes to travel b miles down stream is b / 6 = (a + 3)/6 = (6 + 3 ) / 6 = 9/6 = 1.5 hours.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(number properties) Let f(n) be the number of positive factors of n. What is minimum x satisfying f(420)*f(x)=96?

A. 4
B. 6
C. 8
D. 10
E. 12

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Since 420 = 22*3*5*7, we have f(420) = (2+1)(1+1)(1+1)(1+1) = 24 and f(x) = 96 / f(420) = 96 / 24 = 4.
The smallest integer with 4 factors is 6, which has 4 factors, 1, 2, 3 and 6.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(inequality) a, b, c, d satisfy a-1=b+2=c-3=d+4. Which of a, b, c, d is the largest?

A. a
B. b
C. c
D. d
E. all same

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Since a – 1 = b + 2, we have a – b = 3 > 0 or a > b.
Since b + 2 = c – 3, we have b = c – 5 or a – b = a – ( c – 5 ) = a – c + 5 = 3 or a – c = -2 < 0. Thus we have c > a > b.
Since c – 3 = d + 4, we have c – d = 7 and c > d.
Thus c is the largest value of a, b, c and d.

Therefore, C is the answer.
Answer: C


[GMAT math practice question]

(geometry) What is d?


1) b=√3
2) c= 2

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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. We should simplify conditions if necessary.

Remind that a 30°-60°-90° right triangle has the ration among sides 1:2: √3.

Condition 1)
We know all angles of the triangles, ABC and CDE. Since b = √3, we can find CE =1, c = 2 and BC = √3. It follows that d = BC + CE = √3 + 1.
Thus, condition 1) is sufficient.

Condition 2)
We know all angles of the triangles, ABC and CDE. Since c = 2, we can find CE = 1 and BC = √3. It follows that d = BC + CE = √3 + 1.
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(algebra) m/n is a fraction. What are the values of m and n?

1) the irreducible form of m/n is 3/4
2) if 11 is subtracted from numerator of m/n and 4 is added to denominator of m/n, then the result is 2/5

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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 2 variables (x and y) and 0 equations, C 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) and 2)
Using condition 1), m/n = ¾, we must have 4m = 3n.
Condition 2) tells us that ( m – 11 ) / ( n + 4 ) = 2/5. Thus, 5(m-11) = 2(n+4) and 5m – 55 = 2n + 8. Rearranging yields 5m = 2n + 63 and 15m = 6n + 189.
Since 6n = 8m, 15m = 8m + 189, and 7m = 189. Thus, m = 27 and n = 36.
Both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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.

[GMAT math practice question]

(number properties) a and b are positive integers. What is the value of b-a?

1) a/b = 2/7
2) a+b is a two-digit integer greater than 20

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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 2 variables (a and b) and 0 equations, C 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 a/b= 2/7, we have 7a = 2b.
If a = 4 and b = 14, then a/b = 4/14 = 2/7, a+ b = 21 > 20, and b – a = 10.
If a = 6 and b = 21, then a/b = 6/21 = 2/7, a + b = 27 > 20, and b – a = 15.
Since both conditions together don’t yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.


Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(Statistics) 21, 22, 23, … , and (20+n) are n consecutive integers. One integer is excluded, and the average (arithmetic mean) of the remaining integers is calculated. The minimum possible value of this average is 60. What is the maximum possible value of this average?

A. 60
B. 61
C. 62
D. 63
E. 64

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The minimum value of the average after excluding one of the numbers occurs when the largest integer, (20+n) is excluded. The average of the remaining numbers is then n/2 (21 + (20+n-1))/n = (n + 40) /2 = 60. So, n = 80.

The maximum value of the average after excluding one of the numbers occurs when the smallest integer, 21 is excluded. The average of the remaining numbers is then n/2 (22 + (20+n))/n = 122 / 2 = 61.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(ratio) Tom travels the same distance in 4 steps as Alice travels in 5 steps. If the length of one of Alice’s steps is 36 cm, what distance, in cm, does Tom cover in 18 steps?

A. 5184
B. 3688
C. 1800
D. 1243
E. 810

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Let t and a be stride-lengths of Tom and Alice, respectively.
Then 4t = 5a and a = 36.
So, t = (5/4)a = (5/4)*36 = 45 cm.
It follows that 18t = 18*45 = 810 cm.

Therefore, E is the answer.
Answer: E

[GMAT math practice question]

(geometry) ABCD and AFGE are rectangles and the area of rectangle ABCD is 120. What is the area of rectangle AFGE?

1) the area of triangle GBC is 24
2) the area of triangle EGD is 9


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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. We should simplify conditions if necessary.

Since finding the area of a rectangle requires 2 variables, C 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)


Let H be the point of intersection of the extension of FG with side CD. Since the area of rectangle BCHF is twice the area of triangle GBC, condition 1) tells us that the area of rectangle BCHF is 48. Since the area of rectangle EDGH is twice the area of triangle EGD, condition 2) tells us that the area of rectangle EDGF is 18.
Now, the area of rectangle AFGE is the area of rectangle ABCD minus the sum of the areas of rectangles BCHF and EDGH. It is 120 – ( 48 + 18 ) = 120 – 66 = 54.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(geometry) l is parallel to m, what is the value of x+y?

1) A=100
2) B=50


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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. We should simplify conditions if necessary.

<x = <A + <B and <y = 180 - <A.
Therefore, x + y = <A + <B + 180 - <A = <B + 180.

Thus, condition 2) is sufficient on its own.
Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) p, q and r are prime numbers. What is the value of p?

1) p+q=r
2) 1<p<q

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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 + q = r and p, q and r are prime numbers, one of p and q must be 2.
In addition, p = 2 because 1 < p < q.
Both conditions together are sufficient.

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)
If p = 2, q = 3 and r = 5, then p = 2.
If p = 3, q = 2 and r = 5, then p = 3.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
If p = 2 and q = 3, then p = 2.
If p = 3 and q = 5, then p = 3.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(algebra) The time is between 8 and 9 o’clock. The angle between the hour hand and the minute hand is 180o. What time is it?

A. 8 hours 9 and (9/10) minutes
B. 8 hours 10 and (10/11) minutes
C. 8 hours 11 and (11/12) minutes
D. 8 hours 12 and (12/13) minutes
E. 8 hours 13 and (13/14) minutes

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The minute hand moves 6° and the hour hand moves 0.5° for every minute.
The angle between the hour hand and the minute hand at 8:00 is 120°.
After x minutes, the angle between the hands is 120° + x * 6° - x * 0.5° = 120° + x * 5.5° = 180. So, x * 5.5° = 60° and x = 600 / 55 = 120 / 11 = 10 + (10/11).

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

(geometry) In the figure, ABCD is a square, EFG is a right triangle and H is the midpoint of AD. Moreover, point A lies on the line EF. What is the value of x?

A. 200
B. 216
C. 230
D. 236
E. 250


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Since triangles EFG and EAH are similar,
(34 + x):20 = 1775: (x/2). So,
x^2 + 34x – 1775*20*2 = 0.
Factoring yields (x+284)(x-250) = 0. Since x must be positive, x = 250.

Therefore, E is the answer.
Answer: E

[GMAT math practice question]

(geometry) In the figure, AB is parallel to EF. What is the value of z?


1) x =78°

2) y = 70°

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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)
We draw lines through C and D, parallel to AB, as shown in the diagram.
Since <a and <ABC=105° are supplementary, a=75°.
<d = 180° - <X - <a = 180° - 78° - 75° = 27°, and <c = <d = 27° as they are alternate interior angles.
Since y = 70°, b = y – c = 70°-27° = 43°
Since <z and <b are alternate interior angles, <z = <b =43°.
Both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(number properties) n is a positive integer. What is the value of n?

1) n is less than 200
2) the number of positive factors of n is 15

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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 (n) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.
Condition 1) is obviously not sufficient.

Condition 2)
If n has 15 factors, then n = p^4*q^2 or n=p^{14}, where p and q are prime numbers.
Condition 2) is not sufficient, since there are a lot of possibilities.

Conditions 1) & 2)
If p = 2 and q = 3 and n = p^4*q^2, then n = (2^4)(3^2) = 144. Note that 2^14 = 16384 > 200 and 3^4 2^2 = 324 > 200, so this is the only possible value of n satisfying conditions 1) and 2). For example, if p = 2 and q = 5, then n = 400.

Therefore, C is the answer.
Answer: C

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.