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# The Ultimate Q51 Guide [Expert Level]

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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31 Jul 2019, 18:06
[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

=>

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).

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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01 Aug 2019, 18:13
[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

Attachment:

7.24ps.png [ 8.57 KiB | Viewed 260 times ]

=>

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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04 Aug 2019, 19:00
[GMAT math practice question]

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

Attachment:

7.29ds.png [ 8.92 KiB | Viewed 222 times ]

1) x =78°

2) y = 70°

=>

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.

Attachment:

7.31.png [ 22.29 KiB | Viewed 222 times ]

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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05 Aug 2019, 18:24
[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

=>

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.

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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The Ultimate Q51 Guide [Expert Level]  [#permalink]

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Updated on: 10 Aug 2019, 07:41
[GMAT math practice question]

(number properties) A and B are one-digit numbers. What is the value of A+B?

1) The 6 six-digit integer B6354A is a multiple of 99
2) A is less than 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.

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)
Condition 1) tells us that B6354A is a multiple of 9. So, A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, and A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
Since condition 2) tells us that A < B, (9,9) is not possible. For all remaining pairs, the value of of A + B is 9.
Therefore, conditions 1) and 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 B6354A is a multiple of 9 and A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
We test these pairs individually to check whether they give rise to multiples of 99:
963540 = 11*87594 + 6 is not a multiple of 11.
863541 = 11*78503 + 8 is not a multiple of 11.
763532 = 11*69412 + 10 is not a multiple of 11.
663543 = 11*60322 + 1 is not a multiple of 11.
563544 = 11*51231 + 3 is not a multiple of 11.
463545 = 11*42140 + 5 is not a multiple of 11.
363546 = 11*33049 + 7 is not a multiple of 11.
263547 = 11*23958 + 9 is not a multiple of 11.
163548 = 11*14868 is a multiple of 11.
963549 = 11*87595 + 4 is not a multiple of 11.
(8,1) is a unique pair of (A,B).
Than we have A + B = 9.
Condition 1) is sufficient, since it yields a unique solution.

Condition 2) is obviously not sufficient.

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Originally posted by MathRevolution on 06 Aug 2019, 18:22.
Last edited by MathRevolution on 10 Aug 2019, 07:41, edited 2 times in total.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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07 Aug 2019, 18:33
[GMAT math practice question]

(geometry) ABCD is a square and ADE is an equilateral triangle. What is the value of x?

Attachment:

7.29(ps).png [ 6.57 KiB | Viewed 598 times ]

A. 120°
B. 130°
C. 140°
D. 150°
E. 160°

=>

<AED = <EAD = <EDA = 60°
Then <EDC = < EAB = 30° and <CED = <BEA = 75°
Therefore, x = 360° – ( <AED + <BEA + <CED ) = 360° - (60° + 75° + 75°) = 360° - 210° = 150°

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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07 Aug 2019, 18:57
MathRevolution wrote:
[GMAT math practice question]

(number properties) A and B are one-digit numbers. What is the value of A+B?

1) The 6 six-digit integer B6354A is a multiple of 99
2) A is less than 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.

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)
Condition 1) tells us that B6354A is a multiple of 9. So, A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, and A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
Since condition 2) tells us that A < B, (9,9) is not possible. For all remaining pairs, the value of of A + B is 9.
Therefore, conditions 1) and 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 B6354A is a multiple of 9 and A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
We test these pairs individually to check whether they give rise to multiples of 99:
963540 = 11*87594 + 6 is not a multiple of 11.
863541 = 11*78503 + 8 is not a multiple of 11.
763542 = 11*69412 + 0 is a multiple of 11.
663543 = 11*60322 + 1 is not a multiple of 11.
563544 = 11*51231 + 3 is not a multiple of 11.
463545 = 11*42140 + 5 is not a multiple of 11.
363546 = 11*33049 + 7 is not a multiple of 11.
263547 = 11*23958 + 9 is not a multiple of 11.
163548 = 11*14868 is a multiple of 11.
963549 = 11*87595 + 4 is not a multiple of 11.
Two pairs (A,B), which are (2,7) and (8,1), satisfy the required conditions.
In both cases, A + B = 9.
Condition 1) is sufficient, since it yields a unique solution.

Condition 2) is obviously not sufficient.

Th highlighted is incorrect. 11*69412 = 763532 NOT 763542.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8429
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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08 Aug 2019, 17:52
[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

=>

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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09 Aug 2019, 01:06
Mo2men wrote:
MathRevolution wrote:
[GMAT math practice question]

(number properties) A and B are one-digit numbers. What is the value of A+B?

1) The 6 six-digit integer B6354A is a multiple of 99
2) A is less than 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.

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)
Condition 1) tells us that B6354A is a multiple of 9. So, A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, and A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
Since condition 2) tells us that A < B, (9,9) is not possible. For all remaining pairs, the value of of A + B is 9.
Therefore, conditions 1) and 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 B6354A is a multiple of 9 and A + B + 6 + 3 + 5 + 4 = A + B + 18 is a multiple of 9, A + B is a multiple of 9.
The possible pairs (A,B) are (0,9), (1,8), (2,7), … , (8,1) and (9,9).
We test these pairs individually to check whether they give rise to multiples of 99:
963540 = 11*87594 + 6 is not a multiple of 11.
863541 = 11*78503 + 8 is not a multiple of 11.
763542 = 11*69412 + 0 is a multiple of 11.
663543 = 11*60322 + 1 is not a multiple of 11.
563544 = 11*51231 + 3 is not a multiple of 11.
463545 = 11*42140 + 5 is not a multiple of 11.
363546 = 11*33049 + 7 is not a multiple of 11.
263547 = 11*23958 + 9 is not a multiple of 11.
163548 = 11*14868 is a multiple of 11.
963549 = 11*87595 + 4 is not a multiple of 11.
Two pairs (A,B), which are (2,7) and (8,1), satisfy the required conditions.
In both cases, A + B = 9.
Condition 1) is sufficient, since it yields a unique solution.

Condition 2) is obviously not sufficient.

Th highlighted is incorrect. 11*69412 = 763532 NOT 763542.

Hello,

Thank you for the correction.
We changed.
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Math Revolution GMAT Instructor
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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12 Aug 2019, 01:10
[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

=>

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.

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.
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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12 Aug 2019, 20:21
[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

=>

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.

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.
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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15 Aug 2019, 18:51
[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

Attachment:

8.5 PS.png [ 57.4 KiB | Viewed 413 times ]

=>

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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18 Aug 2019, 18:38
[GMAT math practice question]

(geometry)

Attachment:

812ds.png [ 7.88 KiB | Viewed 379 times ]

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

=>

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°.

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.
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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19 Aug 2019, 18:39
[GMAT math practice question]

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

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

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

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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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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20 Aug 2019, 18:23
[GMAT math practice question]

(geometry)

Attachment:

8.16 DS.png [ 12.5 KiB | Viewed 342 times ]

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

=>

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.

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.
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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21 Aug 2019, 18:38
[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

=>

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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22 Aug 2019, 18:05
[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

=>

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.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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25 Aug 2019, 18:55
[GMAT math practice question]

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

Attachment:

8.19 DS.png [ 11.32 KiB | Viewed 275 times ]

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

=>

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.

Attachment:

8.21ads.png [ 7.09 KiB | Viewed 274 times ]

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

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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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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03 Sep 2019, 00:22
[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.

=>

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.

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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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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03 Sep 2019, 18:55
[GMAT math practice question]

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

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

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.
_________________
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Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 03 Sep 2019, 18:55

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# The Ultimate Q51 Guide [Expert Level]

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