The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

If x<y, is x(1+x)<y(1+y)?

1) x>1/2
2) x+y>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.

Now,

x(1+x)<y(1+y)
=> x+x^2 - y - y^2 < 0
=> (x-y) + (x^2 - y^2) < 0
=> (x-y) + (x-y)(x+y) < 0
=> (x-y)(1+x+y) < 0
=> 1+x+y > 0, since x < y.

Condition 1)
Since y > x > 1/2, we have x + y + 1 > 0.
Thus, condition 1) is sufficient.

Condition 2)
Since x + y > 1, we have x + y > 0.
Thus, condition 2) is sufficient too.

Therefore, D is the answer.

Answer: D

[GMAT math practice question]

What is the difference between the average (arithmetic mean) of the first 9 positive integers and the average (arithmetic mean) of the second 9 non-negative integers?

A. 0
B. 1
C. 2
D. 3
E. 4

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The first 9 positive integers are 1, 2, 3, …, 9. Their average is 5.
The second 9 non-negative integers are 1, 2, 3, …, 9, since the first 9 non-negative integers are 0, 1, 2, …, 8. Their average is 5 too.

Thus the difference between those averages is 0.

Therefore, A is the answer.


Hi ammuseeru







I am not an expert, but here are my two cents. Hope you have not overlooked highlighted text.
The reason we divided by 2 is that the pairs can also be interchanged within themselves. It is same
as 36C2.

[Math Revolution GMAT math practice question]

If the greatest common divisor of (n-1)!, n!, and (n+1)! is 5040, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 8

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Since n! and (n+1)! are multiples of (n-1)!, (n-1)! is their gcd.
It follows that n – 1 = 7 or n = 8, since 7! = 5040.

Therefore, E is the answer.
Answer: E


[Math Revolution GMAT math practice question]

Does x^2+px+q = 0 have a root?

1) p<0
2) q<0

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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.

The discriminant of the equation is p^2 – 4q. If the discriminant is greater than or equal to zero, then the quadratic equation has roots.
The question asks if p^2-4q ≥ 0 or not.

Since p^2 ≥ 0, if q < 0, then p^2-4q ≥ 0. Thus, condition 2) is sufficient.

Condition 1)
If p = -1 and q = 0, then the discriminant is positive and the equation has 2 roots, which are 0 and 1. So, the answer is ‘yes’.
If p = -1 and q = 1, then the discriminant is negative and the equation has no real roots. So, the answer is ‘no’.
Since we don’t have a unique solution, condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

What is the sum of the solutions of the equation (x-1)^2=|x-1|?

A. -1
B. 0
C. 1
D. 2
E. 3

=>

(x-1)^2=|x-1|
=> |x-1|^2=|x-1|
=> |x-1|^2-|x-1|=0
=> |x-1| (|x-1|-1)=0
=> |x-1| = 0 or |x-1|-1=0
=> |x-1| = 0 or |x-1|=1
=> x-1 = 0 or x-1=±1
=> x=1 = 0 or x=1±1
=> x=1, x=0 or x=2

The sum of the solutions is 0 + 1 + 2 = 3.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

If m and n are integers, is m+m2-n an even number?

1) m is an even number
2) n is an even number

=>
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.

m+m2-n = m(m+1) – n. Now, m(m+1) is an even number since m(m+1) is the product of two consecutive integers. Thus, the parity of m+m2-n depends on the parity of n only.
Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

If the average (arithmetic mean) of p, q, and r is 6, what is the value of r?

1) p=-r
2) p=-q

=>

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.

The question asks for the value of r, where r = 18 – ( p + q ). If we know the value of p + q, then we can determine the value or r. Condition 2) implies that p + q = 0. Therefore, r = 0 + r = ( p + q ) + r = 18, and condition 2) is sufficient.



Condition 1)
If p = -1, q =18, and r = 1, then p + q + r = 18 and r = 1.
If p = -2, q =18, and r = 2, then p + q + r = 18 and r = 2.
Since it doesn’t give a unique value of r, condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

The four-digit number 486X, where X is the units digit of 486X, is a multiple of 36. What is the value of X?

A. 0
B. 2
C. 4
D. 6
E. 8

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Since X is a one-digit integer, 0 ≤ X ≤ 9.
36 = 4*9, so 486X is a multiple of both 4 and 9.
Since 486X is a multiple of 4, 6*10 + X must be a multiple of 4.
Thus, X is one of values 0, 4 and 8.
In addition, since 486X is a multiple of 9, the sum of all of its digits, 4 + 8 + 6 + X = X + 18 is a multiple of 9.
Thus, X must be 0 or 9.

0 is the only one of these digits for which 486X is a multiple of 36.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

For positive integers m and n, is m^n a perfect square?

1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question asks if n is an even integer or m is a perfect square. Condition 2):
“23,4n5 is a multiple of 9” is equivalent to the statement that 2 + 3 + 4 + n + 5 = n + 14 is a multiple of 9. For this to occur, we must have n = 4 and m^n = m^4 = (m^2)^2 is a perfect square. Condition 2 is sufficient.

Condition 1)
“12,3m0 is a multiple of 4” is equivalent to the statement that m is an even integers, since this is what is required for 12,3m0 to be a multiple of 4.
Thus, condition 1) tells us that m = 0, 2, 4, 6 or 8. Since we don’t know the exponent n, condition 1) is not sufficient.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

A right triangle has hypotenuse 10. If its perimeter is 25, what is its area?

A. 125/4
B. 125/2
C. 125
D. 225/4
E. 225/2

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Let a and b be the legs of the right triangle.
Since the hypotenuse is 10, a^2+b^2=100.
Since the triangle’s perimeter is 25, we have a + b + 10 = 25 and a + b = 15.
Recall that (a+b)^2 = a^2 + 2ab + b^2.
2ab = (a+b)^2 – (a^2+b^2) = 225 – 100 = 125.
Thus, the area of the triangle is (1/2)ab = 125 / 4.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(geometry) What is the value of x?



1) ∠ABO = 15
2) ∠BOC = 30

=>

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]

(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

=>

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]

(Statistics) A and B are subsets of positive integers. Are the standard deviations of A and B equal?

1) A is the set of all odd numbers between 1 and 100, inclusively.
2) B is the set of all even numbers between 1 and 100, inclusively.

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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)

B = { 2, 4, 6, … , 100 } = A + 1 = { 1, 3, 5, … , 99 } + 1.
Since set B is the shift of set A by 1, sets A and B have the same standard deviation.

Since both conditions together yield a unique solution, they are sufficient.

Since this question is a statistics 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 condition 1) does not have any information regarding set B, it is not sufficient obviously.

Condition 2)

Since condition 2) does not have any information regarding set A, it is not sufficient obviously.

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]

(Fractions) 3 positive numbers a, b and c satisfy a/2b-c=2b/3a+c=a/b . What is a/b?

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

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Assume a/2b-c=2b/3a+c=a/b=k.

Then we have a = k(2b - c), 2b = k(3a + c).
When we add those equations, we have
a + 2b =k(2b-c) + k(3a+c)
a + 2b = 2bk – ck + 3ak + ck
a + 2b = 2bk + 3ak
a + 2b = k(2b + 3a)
or a/b + 2 = k(3a/b + 2) (dividing by b).
Since (a/b) = k , we have
k + 2 = k(3k + 2)
k + 2 = 3k^2 + 2k
3k^2 + 2k – k - 2 = 0
3k^2 + k - 2 = 0
or (k + 1)(3k - 2) = 0.
Then k = -1 or k = 2/3
Then k = a/b = 2/3 since a and b are positive numbers.

Therefore, B is the answe