The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(number properties) The sum of the squares of two prime numbers is 173. What is the sum of the two prime numbers?

A. 5
B. 8
C. 10
D. 15
E. 16

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Let the prime numbers be p and q.
Then p^2 + q^2 =173, and one of p and q must be an even number and the other must be an odd number. Since the unique even prime is 2, we can set p = 2.
Then p^2 + q^2 = 4 + q^2 = 173 and q^2 = 169.
Therefore, q = 13 and p + q = 2 + 13 = 15.

Therefore, D is the answer.
Answer: D

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[GMAT math practice question]

(algebra) What is the value of (2x)/(x+y) + (3y)/(x-y) +(x^2)/(x^2 – y^2)?

1) x/2 = y/3
2) x=1

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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 the conditions if necessary.

(2x)/(x+y) + (3y)/(x-y) +(x^2)/(x^2 – y^2)
= (2x)(x-y)/(x+y)(x-y) + (3y)(x+y)/(x-y)(x+y) +(x^2)/(x^2 – y^2)
= (2x)(x-y)/ (x^2 – y^2) + (3y)(x+y)/ (x^2 – y^2) +(x^2)/(x^2 – y^2)
= {(2x)(x-y) + (3y)(x+y) +(x^2)}/(x^2 – y^2)
= ( 3x^2 +xy + 3y^2 )/(x^2 – y^2)

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that A is most likely to be the answer to this question.

Condition 1)
Rearranging x/2 = y/3 yields x = (2/3)y.
Therefore,
( 3x^2 +xy + 3y^2 )/(x^2 – y^2) = (3*(2/3)^2y^2 + (2/3)y^2 + 3y^2) / ((2/3)^2y^2 – y^2)
= ((4/3) + (2/3) + 3)y2 / ((4/9) – 1)y^2
= 5y^2 / (-5/9)y^2
= 9
Thus, condition 1) alone is sufficient.

Condition 2) is obviously not sufficient since it provides no information about y.

Therefore, A is the answer.
Answer: A

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that A is most likely to be the answer to this question.

[GMAT math practice question]8.7/q51 8.13

(number property) What are the values of x and y?

1) 1/x + 1/y = 1/5
2) x and y are positive integers

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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) & 2)
1/x + 1/y = 1/5
=> 5y + 5x = xy after multiplying by 5xy
=> xy - 5y - 5x = 0
=> xy - 5y - 5x + 25 = 25
=> (x-5)(y-5) = 25

Since x and y are positive integers, there are three possible pairs of values for (x-5,y-5). These are are (1,25), (25,1) and (5,5).
So, the possible pairs (x,y) are (6,30), (30,6) and (10,10).

Since we don’t have a unique solution, both conditions together are not sufficient.


Therefore, E is the answer.
Answer: E

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]

(geometry) What is the value of x?



1) ∠ABO = 15
2) ∠BOC = 30

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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 the angle at the circumference is half of the central angle, standing on the same arc, condition 2) is sufficient.



Condition 1)
<BAO is equal to <ABO, but we don’t know the measures of <OAC and <ACO. So, we can’t work out the measure of <OAC or <x. Therefore, condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(geometry) As the figure shows, a ball and cone are inscribed by a cylinder. What is the ratio of the volumes of the sphere to the cone to the cylinder?

A. 1:1:2
B. 1.5:1:3
C. 1.5:1.5:2
D. 2:1:3
E. 3:2:4



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From the figure, the radii of the sphere and the base circle are (9/2).
The volume of the sphere is (4/3) π(9/2)^3 = π(243/2).
The volume of the cone is (1/3) π(9/2)^2(9) = π(243/4).
The volume of the cylinder is π(9/2)^2(9) = π(729/4)

Thus, the volumes are in the ratio π(243/2) : π(243/4) : π(729/4) = 2 : 1 : 3.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(geometry)



What is the sum of ∠x and ∠y?
1) Triangle ABC is equilateral
2) BD = AE


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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) & 2)
Since △ABE and △BCD are congruent, <ABE = <BCD.
So, <EFC = <FBC + <BCD (exterior angle of triangle BFC) = <FBC + <ABE (corresponding angles of congruent triangles ABE and BCD) = <B = 60°
Thus, <x + <y = 180° - 60° (angle sum of triangle EFC) = 120°.

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]

(Function) What is f(g(2))?

1) f(x) =3x-2
2) g(x)=x^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.

We require infinitely many values to determine f(x) and g(x). Since the original condition includes infinitely many variables 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)
f(g(2)) = f(2^2) = f(4) = 3*4 – 2 = 12 – 2 = 10

Therefore, C is the answer.
Answer: C

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.

[GMAT math practice question]

(geometry)



In the figure, ∠AOB= 30°. What the length of arc BC?

1) AO is parallel to BC
2) the length of arc AB is 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.

In order to determine BC, we need to know the radius of the circle and the measure of angle <BOC. Thus, we have 2 variables and 0 equations, and 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 AO and BC are parallel, <OBC = 30° (alternate interior angles).
Since OB and OC are congruent (equal radii), <OCB 30°.
Thus <BOC = 180°-30°-30°=120°
The ratio between the arc lengths of AB and BC is 30:120 = 1:4.
Thus, the arc length of BC is 4 times the arc length of AB, so it is 20.
Both conditions together are sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the radius of the circle.

Therefore, the answer is C.
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]

(Work/ Rate Problems) A swimming pool is full of water. Pump A takes 6 hours to empty the pool completely, and pump B takes 7 hours to completely empty the pool. Phil starts to use pump A to drain the pool at 1 PM. Some time later, he completes the job using both pumps A and B together. If he wants to empty the pool completely by 6 PM, at what time should Phil start pump B?

A. 3:30 PM
B. 4 PM
C. 4:20 PM
D. 4:50 PM
E. 5 PM

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Suppose Phil starts pump B at time x.
Pump A works for 5 hours, and pump B works for 6 – x hours.
Pump A empties 1/6 of the pool in 1 hour and pump B empties 1/7 of the pool in 1 hour.
So,
(1/6)*5 + (1/7)(6-x) = 1, and, after multiplying both sides by 42 we obtain 35 + 36 – 6x = 42.
Thus, x = 4 + 5/6 and pump B begins work at 4:50 pm.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(Percent and Interest Problems) A store purchased 100 balls from a wholesaler. Initially, they sold them for the unit cost price plus 30%. After selling 70 balls at this price, they reduced the selling price by $15 and sold the remaining 30 balls at this marked-down price. The total profit made after selling all 100 balls was $2,400. What was the original unit cost of each ball?

A. $85
B. $90
C. $95
D. $100
E. $105

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Let x be the unit cost of each ball.
Then 70 * x * 1.3 + 30*( x*1.3 – 15 ) = 100x + 2,400.
Solving for x yields
130*x – 450 = 100*x + 2400,
And
30x = 2850.
Dividing both sides by 30 gives x = 95.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(geometry) In the figure, ABC is a right triangle. What is the area of ABC?



1) Circle O circumscribes triangle ABC and has diameter 13
2) Circle O’ is inscribed in triangle ABC and has diameter 6


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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.

2 variables are required to specify the two circles, and two variables are required to specify the right triangle. So, we have 4 variables. Since one side of the triangle is equal to the diameter of one of the two circles, we have 1 equation.
Since we have 4 variables and 1 equation, 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 the radius of O’ is 3, BE = BD = O’E = 3.
Suppose the length of AD is a.
Then AF = AD = a, and CF = CE = 13-a.
So, BC = BE + EC = 3 + 13 – a = 16 – a,
And AB = AD + DB = a + 3.
Thus, the area of triangle ABC is (1/2)(AB+BC+CA)(3) = (1/2)(a+3+16-a+13)(3) = 48

Therefore, C is the answer.
Answer: C

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.

[GMAT math practice question]

(set) A={x|(7/15)x+1/3 = 4/3} and B={y| 2m-(1/15)y = 3}, where m is a real number. What is the value of m?
1) A∩B≠Ø
2) B≠Ø

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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 (7/15)x+1/3 = 4/3 and 7x + 5 = 20 by definition of set A, we must have x = 15/7 and A = { 15/7 }.
Since 2m-(1/15)x = 3 and 30m – x = 45 by definition of set B, x = 30m – 45 and B = { 30m – 45 }.

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

Condition 1)
Since A∩B≠Ø, we must have 30m – 45 = 15/7 and condition 1) yields a unique solution. It is sufficient.

Condition 2)
Since m can be any value and condition 2) doesn’t yield a unique solution, it 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.

[GMAT math practice question]

(number properties) Given two different positive integers, what is the ratio of the larger number to the smaller one?

1) the sum of the two numbers is 1000 less than the product of the two numbers
2) one of the two numbers is a perfect square

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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 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)
Suppose x and y are the integers and x is a perfect square.

Then xy = x + y + 1000, and xy – x – y + 1 = 1001.
Thus, (x-1)(y-1) = 1001 = 7*11*13.
Since x is a perfect square, only 11*13 + 1 = 144 is a perfect square out of all possible values 7+1, 11+1, 13+1, 7*11+1, 7*13+1, 11*13+1, and 7*11*13+1.
Thus, x = 144 and y = 8.
Therefore, x : y = 144:8.
Since both conditions together yield a unique solution, they 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)
We have xy = x + y + 1000 or xy – x – y + 1 = 1001.
Thus (x-1)(y-1) = 1001 = 7*11*13.
We can find pairs of solutions x=2 and y=1002, and x=1002 and y=2.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
Since it provides no information about the second number, condition 2) is not sufficient.

Therefore, the answer is C.
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) What is the remainder when 9^n -1 is divided by 10?

1) n is a multiple of 2
2) n is a multiple of 3

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

The remainder when 9^n -1 is divided by 10 is the same as the units digit of 9^n -1. This is easily determined from the units digit of 9n.

9^1 = 9, 9^2 = 81 ~ 1, 9^3 ~ 9, 9^4 ~ 81 ~ 1, …
So, the units digits of 9n have period 2:
They form the cycle 9 -> 1.
Thus, 9^n has a units digit of 9, if n is an odd number and a units digit if 1, if n is an even number.

Thus, condition 1) is sufficient.

Condition 2) is not sufficient since a multiple of 3 can be either an even or an odd number.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

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

1) N is divisible by 75
2) N has 75 positive factors, including 1 and N.

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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)
The possible values of N are 75, 150, …. Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
2^{74} and 2^4*3^4*5^2 have 75 factors. Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
N = 3^{24}*5^2 and N=2^4*3^4*5^2 have 75 factors. Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E


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]

(Absolute Values) What is the value of x+y?

1) |x-2|=4
2) |x-y+3|=4

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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) & 2)
Since |x-2| = 4, we have x – 2 = ±4. So, x = 2 ± 4.
Thus, x = 6 or x = -2
Since |x – y + 3| = 4, we have x – y + 3 = ±4 and y = x + 3 ± 4.
Thus, y = x – 1 or y = x + 7.
The possible pairs (x,y) are (6,5), (6,13), (-2,-3) and (-2,5), and their sums are 11, 19, -5 and 3.
Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, the answer is E.
Answer: E

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.