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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) a, b and c are positive integers. Is 2(a^4+b^4+c^4) a perfect square?

1) a + b + c = 0
2) abc = 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.
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.

Condition 2)
Condition 2) tells us that abc = 6. For three numbers to multiply to 6, the numbers must be 1, 2, and 3 and the order does not matter. We can substitute these numbers into the given equation to get:
2(a^4 + b^4 + c^4) = 2(1^4 + 2^4 + 3^4) = 2(1 + 16 + 81) = 2*98 = 196 = 14^2, which is a perfect square.
Thus condition 2) is sufficient.

Condition 1)
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
Then, we have a^2 + b^2 + c^2 = -2(ab + bc + ca) since a + b + c = 0.
We have (a^2 + b^2 + c^2)^2 = (-2(ab + bc + ca))2 when we square both sides.
Its left-hand side is
(a^2 + b^2 + c^2)^2
= (a^2 + b^2 + c^2) (a^2 + b^2 + c^2)
= a^4 + a^2b^2 +a^2c^2 + a^2b^2 + b^4 + b^2c^2 + a^2c^2 + b^2c^2 + c^4
= a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2c^2a^2.

Its right-hand side is
(-2(ab + bc + -ca))^2
= 4(ab + bc + ca)(ab + bc + ca)
= 4(a^2b^2+ ab^2c + a^2bc + ab^2c + b^2c^2 + abc^2 + a^2bc + abc^2 + c^2a^2
= 4(a^2b^2 + b^2c^2 + c^2a^2 + 2a^2bc + 2ab^2c + 2abc^2)
= 4(a^2b^2 + b^2c^2 + c^2a^2) + 2abc(a + b + c))
= 4(a^2b^2 + b^2c^2 + c^2a^2), since a + b + c = 0
We have a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2).
Then 2(a^4 + b^4 + c^4) = 4(a^2b^2 + b^2c^2 + c^2a^2) = (-2(ab + bc + ca))^2
Thus 2(a^4 + b^4 + c^4) is a square of -2(ab + bc + ca).
We realize we have only used condition 1), but haven’t used condition 2).
It means condition 1) alone is sufficient.
Thus, condition 1) is sufficient.

This question is a CMT 4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.

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.

Isn't it obvious that all a, b and c are 0 in condition 1 since it's mentioned that they are positive integers. Not sure why you followed a longer approach.

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[GMAT math practice question]

(Number Properties) x is an integer. What is the value of x?

1) x^2+4x+9 is a perfect square.
2) x is a non-zero 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Condition 1)
We have x^2 + 4x + 9 = k^2 for some integer k.
Then we have
k^2 – (x^2 + 4x + 4) = 5
k^2 – (x + 2)^2 = 5
or (k + x + 2)(k – x - 2) = 5.

We have four cases
Case 1) k + x + 2 = 1, k – x – 2 = 5
Adding both equations together gives us:
k + x + 2 + k – x – 2 = 1 + 5
2k = 6
k = 3

Then k + x + 2 = 1 becomes
3 + x + 2 = 1
x = -4
Then we have k = 3, x = -4.

Case 2) k + x + 2 = 5, k – x – 2 = 1
Adding both equations together gives us:
k + x + 2 + k – x – 2 = 5 + 1
2k = 6
k = 3
Then k + x + 2 = 5 becomes
3 + x + 2 = 5
x = 0
Then we have k = 3, x = 0.

Case 3) k + x + 2 = -1, k – x – 2 = -5
Adding both equations together gives us:
k + x + 2 + k – x – 2 = -1 + -5
2k = -6
k = -3
Then k + x + 2 = -1 becomes
-3 + x + 2 = -1
x = 0
Then we have k = -3, x = 0.

Case 4) k + x + 2 = -5, k – x – 2 = -1
Adding both equations together gives us:
k + x + 2 + k - x - 2 = -5 + -1
2k = -6
k = -3
Then k + x + 2 = -5 becomes
-3 + x + 2 = -5
x = -4
Then we have k = -3, x = -4.

Thus, we have two solutions for x, which are 0 and -4.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)

Since condition 2) does not provide enough information to yield a unique solution, it is not sufficient.

Conditions 1) & 2)
We have a unique solution -4.
Since both conditions together yield a unique solution, they are 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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[GMAT math practice question]

(Absolute Value) (x, y) satisfies |3x - 2y + 4| + |-x + 2y - 2| = 0. What is 2x - y?

A. 1/3
B. -5/2
C. 1
D. -4/3
E. 5/3

=>

Both |3x - 2y + 4| and |-x + 2y - 2| are greater than or equal to 0 and the sum of them is 0, so both of them are 0.
Then we have 3x - 2y + 4 = 0 and -x + 2y - 2 = 0.
Adding the 2 equations together gives us:
3x – 2y + 4 – x + 2y – 2 = 0 + 0
2x + 2 = 0
2x = -2
x = -1
Then 3x – 2y + 4 = 0 becomes:
3(-1) – 2y + 4 = 0
-2y + 1 = 0
-2y = -1
= 1/2
We have x = -1, y = ½.
Thus, we have 2x - y = 2(-1) – ½ = -2 – ½ = -5/2.

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[GMAT math practice question]

(Work rate) If A works a job alone, it takes x days. If B does it alone, it takes y days. If A and B work together, they can do 3/20 of the total amount of the job per day. A and B worked together for 5 days, and A worked the rest for 3 days. What is the value of x + y?

A. 20
B. 21
C. 25
D. 27
E. 29

=>

Assume 1 is the work amount of the job.
The work amount that machines A and B can do in a day is 1/x and 1/y, respectively.
Then we have 1/x + 1/y = 3/20.
And we have (1/x + 1/y)*5 + (1/x)*3 = 1.

(1/x + 1/y)*5 + (1/x)*3 = 1
⇔ (3/20)*5 + (1/x)*3 = 1, since 1/x + 1/y = 3/20
⇔ (3/4) + (1/x)*3 = 1
⇔ (1/x)*3 = 1/4
⇔ (1/x) = 1/12
⇔ x = 12

1/x + 1/y = 3/20
⇔ 1/12 + 1/y = 3/20, since x = 12
⇔ 1/y = 3/20 – 1/12 = 9/60 – 5/60 = 4/60 = 1/15
⇔ y = 15

Thus, we have x + y = 12 + 15 = 27.

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[GMAT math practice question]

(Equation) What is the value of 4a + 2b?

1) (a - b)^2 - 3(a - b) – 18 = 0
2) a + b = 8

=>

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 (a and b) and 0 equations, C is most likely 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)
From condition 1) we get:
(a - b)^2 - 3(a - b) – 18 = 0
⇔ Let x = (a + b) giving us:
⇔ x^2 – 3x - 18 = 0
⇔ (x + 3)(x – 6) (trinomial factoring)
⇔ ((a-b) + 3)((a - b) - 6) = 0 (substituting (a + b) back in)
⇔ a - b = -3 or a - b = 6

When we consider a - b = -3 from condition 1) and a + b = 8 from condition 2) and add them together we get 2a = 5, and a = 5/2. Substituting that back into the second equation gives us 5/2 + b = 8, and b = 11/2. Then 4a + 2b = 4(5/2) + 2(11/20) = 10 + 11 = 21.

When we consider a - b = 6 from condition 1) and a + b = 8 from condition 2) and add them together we get 2a = 14, and a = 7. Substituting that back into the second equation gives us 7 + b = 8, and b = 1. Then 4a + 2b = 4(7) + 2(1) = 28 + 2 = 30.

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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Joined: 16 Aug 2015
Posts: 9164
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[GMAT math practice question]

(Geometry) The figure shows a semi-circle. AC and BD are perpendicular to AB and CD is a tangent line to the semi-circle. What is the radius of the semi-circle?

Attachment: 5.12DS.png [ 7.38 KiB | Viewed 639 times ]

1) AC = 10
2) BD = 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.
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.

Since we have 5 variables (AC, CT, BD, TD, and AB) and 2 equations (AC = CT, and BD = TD), E is most likely 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: 5.12ds(a).png [ 24.23 KiB | Viewed 640 times ]

Conditions 1) & 2)

When we have DH perpendicular to AC, we have a right triangle CDH.
AB^2 = DH^2 = CD^2 – CH^2 = 15^2 – 5^2 = 200.
Thus, the radius = AB/2 = √200/2 = 10√2/2 = 5√2.

Since both conditions together yield a unique solution, they are 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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Function) f(x) is a function. What is the value of f(2020)?

1) f(10)=11
2) f(x+3)=f(x) - 1/f(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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have many variables to determine a function, E is most likely 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(13)=f(10) - 1/f(10) + 1=11 - 1/11 + 1=5/6.
f(16)=f(13) - 1/f(13) + 1=5/6 - 1/5/6 + 1=-1/11.
f(19)=f(16)-1/f(16)+1=-1/11-1/-1/11+1=-6/5
f(22)=f(16) - 1/f(16) + 1=-6/5 - 1/-6/5 + 1=11.

Since we have 2008 = 4*5002 + 0 has a remainder 0 when it is divided by 4, f(2008) = f(16) = -1/11.
Since both conditions together yield a unique solution, they are 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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Geometry) In the figure, what is BD^2 - CD^2?

1) AB = 8, AC = 5

Attachment: 5.11PS.png [ 5.49 KiB | Viewed 551 times ]

=>

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 5 variables (AB, BD, CD, CA, and AD) and 0 equations, E is most likely 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: 5.11PS(A).png [ 9.14 KiB | Viewed 551 times ]

Conditions 1) & 2)
Since triangles ABD and ACD are right triangles, we have AB^2 = BD^2 + AD^2 and AC^2 = CD^2 + AD^2. Rearranging the equations gives us BD^2 = AB^2 – AD^2 and CD^2 = AC^2 – AD^2.
Subtracting the equations gives us BD^2 – CD^2 = (AB^2 - AD^2) – (AC^2 – AD^2) = AB^2 - AC^2 = 64 - 25 = 39.

Since both conditions together yield a unique solution, they are 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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Equation) a and b are different integers. What is the root of (x - a)^2 = (x - b)^2?

1) a – b = 3
2) a + b = 7

=>

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 question (x - a)^2 = (x - b)^2 is equivalent to x = (a + b)/2 for the following reason.
(x - a)^2 = (x - b)^2
⇔ (x - a)^2 - (x - b)^2 = 0
⇔ ((x - a) - (x - b)) ((x - a) + (x - b)) = 0
⇔ (-a + b)(2x - (a + b)) = 0
⇔ (2x - (a + b)) = 0 (by dividing both sides by -a + b since a ≠ b)
⇔ 2x = (a + b) (by adding ( a + b) to both sides)
⇔ x = (a + b)/2 (by dividing both sides by 2)
So, we have to find the value of a + b.
Thus, look at condition (2). It tells us that a + b = 7, which is exactly what we are looking for. The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition (1) tells us that a – b = 3, from which we cannot determine the unique value of a + b. The answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition (2) ALONE is sufficient.
_________________
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Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Function) n is a positive integer. What is the value of f(48)?

1) f(2n) = f(n)
2) f(2n+1) = (-1)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 need many variables to determine a function, E is most likely 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)
48 = 2^4 · 3.
f(3) = f(2·1 + 1) = (-1)^1 = -1
f(6) = f(2·3) = f(3) = -1
f(12) = f(2·6) = f(6) = -1
f(24) = f(2·12) = f(12) = -1
f(48) = f(2·24) = f(24) = -1.

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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Algebra) x and y are real numbers. What is the value of x + y?

1) x/y = -√3
2) x + √3y = 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (x and y) and 0 equations, C is most likely 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 x = -√3, y = 1, then we have x + y = -√3 + 1.
If x = √3, y = -1, then we have x + y = √3 - 1.

Since both conditions together do not yield a unique solution, they are not sufficient.

Note: Tip 1) of the VA method states that D is 95% likely to be the answer if condition 1) gives the same information as condition 2). However, we have the answer E in this problem.

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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Joined: 16 Aug 2015
Posts: 9164
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
[GMAT math practice question]

(Geometry) A circular swimming pool is surrounded by a wall with a height of 2m. The area of the wall is 21% of the area of the swimming pool. What is the area of the swimming pool?

A. 400πm^2
B. 300πm^2
C. 250πm^2
D. 200πm^2
E. 100πm^2

=>
Assume the radius of the swimming pool is r.
Then the area of the wall is 2πr·2 = 4πr.
Since we have 4πr = (21/100) πr^2, we have 21r^2 – 400r + 400 = 0 or (21r + 20)(r - 20) = 0.
Thus, we have r = 20.
Then, the area of the swimming pool is πr^2 = 400π.

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1
[GMAT math practice question]

(Function) What is the value of f(2^100)?

1) f(2)=1/2
2) f(mn) = f(m) + f(n) for positive integers m 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 need many variables to determine a function, E is most likely 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(2^2) = f(2*2) = f(2) + f(2) = 1/2 + 1/2 = 1.
f(2^3) = f(2^2*2) = f(2^2) + f(2) = 1 + 1/2 = 3/2.
f(2^4) = f(2^3*2) = f(2^3) + f(2) = 3/2 + 1/2 = 2.

Then we can figure out f(2^n) = n/2.

Thus, f(2^100) = 100/2 = 50. The answer is unique, and the condition is sufficient according to the Common Mistake Type 2, which states that the number of
Both conditions (1) and (2) together are sufficient.
Therefore, C is the correct answer.

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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[GMAT math practice question]

(Number Property) A, B, and n are positive integers. What is the value of n?

1) x^4+x^2-n can be factored to (x^2+A)(x^2-B)
2) 1 ≤ n ≤ 10

=>

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.

Since we have 3 variables (x, y, and z) and 0 equations, E is most likely 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)
(x^2+A)(x^2-B) = x^4+(A-B)x^2-AB = x^4+x^2-n
Then we have A – B = 1 and AB = n
If A = 2 and B = 1, we have n = AB = 2.
If A = 3 and B = 2, we have n = AB = 6.

The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together 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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[GMAT math practice question]

(Coordinate Geometry) What is the value of a + b + c?

1) One of roots of the quadratic equation ax^2 + bx + c = 0 is 2.
2) The intersection of two functions y = ax^2 and y = -bx - c is (-1, 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (x, y, and z) and 0 equations, E is most likely 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)

When we replace x in the equation ax^2 + bx + c = 0 with 2 from condition 1), we have 4a + 2b + c = 0.
When we substitute x and y in the equation y = ax^2 with -1 and 2, respectively, we have 2 = a(-1)^2, and a = 2.
When we substitute x and y in the equation y = -bx - c with -1 and 2, respectively, we have 2 = -b(-1) – c, and b – c = 2.
If we replace a in the equation 4a + 2b + c = 0 with 2, we have 4(2) +2b +c = 0, and 2b + c = -8.
When we add the last two equations, we get b – c + 2b + c = 2 – 8, 3b = -6 or b = -2.
If we replace b in the equation b – c = 2 with -2, we have -2 – c = 2, -c = 4, and c = -4.
Thus, we have a + b + c = 2 +(-2) + (-4) = -4.

The answer is unique, and both conditions combined are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, C is the correct answer.

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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[GMAT math practice question]

(Number Property) Every page of a book is numbered from 1 without any omission. One page of the book is torn out. The summation of the numbers of the remaining pages is 1256. Which page is torn out?

A. 9~10
B. 21~22
C. 31~32
D. 43~44
E. 51-52

=>

Assume the original book has n pages and the page numbers of the torn page are k and k+1.

Then we have 1 + 2 + 3 + … + n = n(n+1)/2 = 1256 + k + ( k + 1 ).

Since k ≥ 1, we have n(n+1)/2 = 1256 + k + k + 1 ≥ 1256 + 1 + 2 = 1259, n(n+1)/2 ≥ 1259, or n(n+1) ≥ 2518.

Then we have n ≥ 50.

Since n ≥ k, we have n(n+1)/2 ≤ 1256 + k + k + 1 ≤ 1256 + n + n + 1, n(n+1)/2 ≤ 1257 + 2n, n(n+1) ≤ 2514+ 4n, n^2 + n - 4n ≤ 2514, or n^2 – 3n ≤ 2514. Then n ≤ 51.

Case 1) n = 50
We have (50·51)/2 = 1256 + k + (k + 1) or 1275 = 1256 + 2k + 1.
Then 1275 = 1257 + 2k, 2k = 18, and k = 9.
Then, we have k = 9.

Case 2) n = 51
(51·52)/2 = 1256 + k + (k + 1) or 1326 = 1256 + 2k + 1. Then 1326 = 1257 + 2k, 2k = 69, and k = 69/2.
Then, we have k = 69/2, which is not an integer.

Thus, the page numbers of the torn page are 9 and 10.

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[GMAT math practice question]

(Statistics) The table shows 5 people (Adam, Betty, Carol, David, and Eddy) and the difference between each weight and the average weight of the 5 people. What is the weight of the lightest person among those five people?

1) If another person, Fred, who is 8 kg heavier than Adam, is included, the average is increased by 4%.
2) x is an even prime number.

Attachment: 6.1 DS.png [ 3.19 KiB | Viewed 214 times ]

=>

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.

Since the sum of all the differences between the weights and their average is zero, we have 4 + (-7) + 6 + (-5) + x = 0 or x = 2.
Thus, we don’t need condition 2), since the table already tells us that x = 2.

Let’s look at the condition 1).

Assume the new average, including Fred, is m’.
We have m’ = [5m + (m + 12)] / 6, m’ = (6m + 12) / 6, m’ = m + 2. We also m’ = 1.04m, since we are told the average increases by 4%.
Putting the 2 equations together gives us m + 2 = 1.04m.
Then we have 0.04m = 2 or m = 50.
Thus, the lightest weight is Betty’s weight m – 7 = 43.

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.
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[GMAT math practice question]

(Equation) If A = {x|x^2 - 2(k - 1)x + 4 = 0}, what is set A?

1) k is a positive integer.
2) The is 1 element in set A.

=>

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.

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

Condition 1) tells us that k is a positive integer, from which we cannot determine the unique set of A. If k = 3, then x^2 - 2(k - 1)x + 4 = x^2 – 2(3 – 1)x + 4 = x^2 - 4x + 4 = (x - 2)^2 = 0 and A = {2}. However, if k = 1, then x^2 - 2(k - 1)x + 4 = x^2 - 2(1 – 1) + 4 = x^2 + 4 = 0, which does not have a root and A is an empty set.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that the number of elements in A is 1, from which we get that the discriminant of the quadratic equation is 0.

Since the number of roots of the equation is 1, its discriminant
(b^2 – 4ac)
= [2(k – 1)]^2 - 4·1·4
= 4(k - 1)^2 - 16
= 4(k2 - 2k + 1) - 4·4
= 4(k^2 – 2k + 1 - ) 4 (taking out a common factor of 4)
= 4(k^2 - 2k - 3)
= 4(k + 1)(k - 3) is zero.
Then, we have 4(k + 1)(k - 3) = 0 and k = -1 or k = 3.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2)
Then only k = 3 is the unique answer.

The answer is unique, and both conditions together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together are 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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[GMAT math practice question]

(Algebra) What is the value of m?

1) The difference between the two roots of x^2 + (1 + m)x + 20 = 0 is 1.
2) m > 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Condition 1) tells us that the difference between the two roots of x^2 + (1 + m)x + 20 = 0 is 1.
We can assume p and p+1 are the roots of the equation x^2 + (1 + m)x + 20 = 0.
We have (x - p)(x - (p + 1)) = x^2 - (2p + 1)x + p(p + 1) = x^2 + (1 + m)x + 20.
Then, we have 1 + m = -2p – 1 or m = -2p – 2. We also have p(p + 1) = 20 or p^2 + p - 20 = (p - 4)(p + 5) = 0.
Thus p = 4 or p = -5, which we can substitute into the first equation giving us m = -2p – 2 = -2·4 – 2 = -10, or m = -2p – 2 = -2·(-5) – 2 = 8 .
Then we have m = -10 or m = 8.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that m > 0, from which we cannot determine the value of m. For example, m can be 2 or 3.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2) together tell us that the answer, m = 8 is unique, and both conditions are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together are 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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[GMAT math practice question]

(Statistics) The table shows the differences between the test results and the average by subject. The average is 6, and the variance is 8.8. If a is the test result of Physics and b is that of Chemistry, what is the value of ab?

Attachment: 6.2ps.png [ 3.41 KiB | Viewed 135 times ]

A. 2
B. 5
C. 7
D. 9
E. 11

=>

Since the sum of all differences is 0, we have 3 – 1 + x + y = 0 or x + y = -2.
Since the variance is 8.8, we have [3^2 + (-1)^2 + x^2 + 02 + y^2] / 5 = 8.8, 9 + 1 + x^2 + y^2 = 44, or x^2 + y^2 = 44 - 10 = 34.
Since we have x + y = -2 or y = -x - 2, we have x^2 + y^2 = x^2 + (-x - 2)^2 = 2x^2 + 4x + 4 = 34 or 2x^2 + 4x – 30 = 0.
Dividing by 2, we have x^2 + 2x - 15 = 0 or (x + 5)(x - 3) = 0.
Then we have x = -5, y = 3 or x = 3, y = -5.
Since the average is 6, we have a = 6 – 5 = 1, b = 6 + 3 = 9 or a = 6 + 3 = 9, b = 6 - 5 = 1.
Thus, we have ab = 9.

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# The Ultimate Q51 Guide [Expert Level]

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