The Ultimate Q51 Guide [Expert Level]

[Math Revolution GMAT math practice question]

Five letters A, P, P, L and E are listed in a row. How many arrangements have at least one letter between the two Ps?

A. 24
B. 30
C. 36
D. 42
E. 48

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When we encounter “at least” in counting questions or probability questions, we should consider complementary counting.
The total number of arrangements of the 5 letters is 5!/2! (5! Counts each arrangement of the two Ps 2! times).
The number of arrangements with no letter between the two Ps is 4!.
Thus, the number of arrangements in which at least one letter lies between the two Ps is 5!/2! – 4! = 60 – 24 = 36.

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

If 3 different numbers are selected from the first 8 prime numbers, what is the probability that the sum of the 3 numbers selected is even?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/8

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Suppose p, q and r are prime numbers.
In order for p + q + r to be even, one of them must equal 2, since 2 is the only even prime number.
Once 2 has been selected, there are 7 prime numbers remaining from which to select 2 numbers. Thus, the number of selections in which the sum of the 3 numbers is even is 7C2 = 21.
The total number of selections is 8C3 = 56.
Thus, the probability that the sum of the three numbers is even is 21/56 = 3/8.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

If x^2-3x=10, what is the value of x?

1) x^2-4 = 0
2) x<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.

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


The original condition x2-3x=10 is equivalent to x = -2 or x = 5 as shown below:

x^2-3x=10
=> x^2-3x-10= 0
=> (x+2)(x-5) = 0
=> x = -2 or x = 5

Condition 1)
x^2-4 = 0
=> (x-2)(x+2) = 0
=> x = -2 or x = 2
Only x = -2 also satisfies the original condition, so we have a unique solution.
Thus, condition 1) is sufficient.



Condition 2)
Since x < 6 from condition 2) and x = -2 or x = 5 from the original condition, x = -2 or x = 5.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

[Math Revolution GMAT math practice question]

If x=0.abcabc........(a repeating infinite decimal), what is the value of a+b+c?

1) 1-x=0.123123123…….
2) 0.8<x<0.9

=>
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 4 variables (x, a, b and c) and 1 equation (x = 0.abcabc….), E is most likely to be the answer. However, condition 1) includes 3 hidden equations (matching the values of the decimal places) and allows us to determine the values of all four variables as follows:
x = 1 – 0.123123123… = 0.876876876… = 0.abcabcabcabc…
Thus a = 8, b = 7, c = 6 and a + b + c = 8 + 7 + 6 = 21.
Condition 1) is sufficient.

Condition 2)
If x = 0.811811811…, then a = 8, b = 1, c = 1 and a + b + c = 10.
If x = 0.812812812…, then a = 8, b = 1, c = 2 and a + b + c = 11.
Since we don’t have a unique solution, condition 2) is not sufficient.


Therefore, A is the answer.
Answer: A

[Math Revolution GMAT math practice question]

When A and B are positive integers, is AB a multiple of 4?

1) The greatest common divisor of A and B is 6
2) The least common multiple of A and B is 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.

Asking if AB is a multiple of 4 is equivalent to asking if AB = 4k for some integer k.

Condition 1)
Since A = 6a = 2*3*a and B = 6b = 2*3*b for some integers a and b, AB = 2^2*3^2*ab = 4*3^2ab.
Thus, AB is a multiple of 4 and condition 1) is sufficient.

Condition 2)
If A = 6 and B = 5, then lcm(A,B) = 30, but AB = 30 is not a multiple of 4, and the answer is ‘no’.
If A = 6 and B = 10, then lcm(A,B) = 30, and AB = 60 is a multiple of 4. The answer is ‘yes’.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A


[Math Revolution GMAT math practice question]

What is the greatest prime factor of 1+2+3+….+36?

A. 2
B. 3
C. 9
D. 31
E. 37

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Since 1 + 2 + 3 + … + n = n(n+1)/2, we have 1 + 2 + 3 + … + 36 = (36*37)/2 = 18*37 = 2*3^2*37.
Thus, 37 is the greatest prime factor of 2*32*37.

Therefore, the answer is E.
Answer: E


[Math Revolution GMAT math practice question]

If 2 numbers are selected from the first 8 prime numbers, what is the probability that the sum of the 2 numbers selected is an even number?

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

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In order for the sum to be even, both primes selected must be odd. As 2 is the only even prime number, the number of selections with an even sum is equal to the number of ways to select 2 numbers from these 7 odd prime numbers, or 7C2.
The total number of selections of 2 prime numbers from the first 8 prime numbers is 8C2.
Therefore, the probability that the sum of the two numbers selected is even is
7C2 / 8C2 = {(7*6)/(1*2)}/{(8*7)/(1*2)} = 6/8 = 3/4.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

If |x+1|=2|x-1|, x=?

1) x<1
2) x>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.

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

The original condition |x+1|=2|x-1| is equivalent to x=1/3 or x=3 as shown below:

|x+1|=2|x-1|
=> |x+1|^2=(2|x-1|)^2
=> (x+1)^2=4(x-1)^2
=> x^2+2x+1=4(x^2-2x+1)
=> x^2+2x+1=4x^2-8x+4
=> 3x^2-10x+3 = 0
=> (3x-1)(x-3) = 0
=> 3x-1=0 or x-3 = 0
=> x=1/3 or x=3

Thus, condition 1) is sufficient since it gives a unique solution.

Condition 2) is not sufficient, since it allows both possible solutions.

Therefore, A is the answer.
Answer: A

[Math Revolution GMAT math practice question]

If a and b are positive integers such that when a is divided by b, the remainder is 10, what is the value of b?

1) b>10
2) b<12

=>

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.

By the quotient-remainder theorem, we can write a = b * q + 10, where the remainder 10 is less than b, that is, b > 10.

Thus, condition 2) “b<12” is sufficient since it gives the unique solution b = 11.

Note: Condition 1) does not give a unique solution. For example, we might have b = 11 or b = 12. Thus, it is not sufficient.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

If A={x| x^3>8}, B={x| 1<x^3<64}, C={x| x^3<27}, which inequality represents A∩B∩C?

A. x^3<27
B. 1< x^3<64
C. x^3<64
D. 1<x^3<27
E. 8<x^3<27


=>

A∩B∩C is the set of all numbers that are in all three of the sets A, B and C. So,
A∩B∩C = { x | x^3>8 and 1<x^3<64 and x^3<27} = { x | 8<x^3<27}

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

If -1<x<0, Which of the following is listed in ascending order?

A. x^{-1}, x, x^2, 1, x^{-2}
B. x^{-1}, x^2, x, 1, x^{-2}
C. x^{-2}, x, x^2, 1, x^{-1}
D. x, x^{-1}, x^2, 1, x^{-2}
E. x^{-1}, x, x^2, x^{-2}, 1


=>

We can find the correct order by plugging in a value for x.
Suppose x = -1/2. Then -1 < x < 0, and
x^2 = 1/4, x^{-1} = -2 and x^{-2} = 4.
Thus, x^{-1} < x < x^2 < 1 < x^{-2}

Therefore, the answer is A.
Answer: A

[Math Revolution GMAT math practice question]

Is x^3-y^3>x^2+xy+y^2?

1) x > y + 1
2) 0 < y < x

=>

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 original condition x^3-y^3>x^2+xy+y^2 is equivalent to x > y + 1 as shown below:
x^3-y^3 > x^2+xy+y^2
=> (x-y)(x^2+xy+y^2)>x^2+xy+y^2
=> x – y > 1 after dividing both sides by x^2+xy+y^2, since x^2+xy+y^2 > 0.

Since the final inequality is equivalent to x > y + 1, condition 1) is sufficient.

Condition 2)
If x = 3 and y = 1, then x – y = 2 > 1, and the answer is ‘yes’.
If x = 1 and y = 1/2, then x – y = 1/2 < 1, and the answer is ‘no’.
Since it doesn’t give a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

[Math Revolution GMAT math practice question]

If the average (arithmetic mean) of 5 numbers is 20, what is their standard deviation?

1) Their minimum is 20.
2) Their maximum is 20.

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

Note that if the average and the maximum of a data set are the same, then all of the data values are the same and the standard deviation is 0. Similarly, if the average and the minimum of a data set are the same, all of the data values are the same and the standard deviation is 0.

Thus, each of conditions is sufficient on its own since the minimum and the maximum are the same as the average.

Therefore, D is the answer.
Answer: D

[Math Revolution GMAT math practice question]

What is the number of roots of the equation (x^3-1)/(x^2+x+1)=x?

A. 0
B. 1
C. 2
D. 3
E. no solution

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The original condition (x^3-1)/(x^2+x+1)=x is equivalent to x – 1 = x as shown below:

(x^3-1)/(x^2+x+1)=x
=> (x-1) (x^2+x+1) / (x^2+x+1) = x by factoring
=> x – 1 = x

But x – 1 = x has no solution.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

Suppose f(x)=ax^4+bx^2+1, where a and b are constants. If f(3)=9, what is the value of f(-3) ?

A. -9
B. -3
C. 0
D. 3
E. 9

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f(3) =a(3)^4 + b(3)^2 + 1 = 81a + 9b + 1 = 9
f(-3) =a(-3)^4 + b(-3)^2 + 1 = 81a + 9b + 1 = f(3) = 9

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

x and y are non-negative integers. If xy+2x+3y=0, then y=?

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

=>

xy+2x+3y=0
=> xy+2x+3y+6=6
=> (x+3)(y+2)=6
Since x and y are integers, x+3 and y+2 are integers.
The possible pairs (x+3,y+2) are as follows:
( x+3, y+2 ) = (1,6), (2,3), (3,2), (6,1), (-1,-6), (-2,-3), (-3,-2) and (-6,-1).
The corresponding pairs (x,y) are:
(x,y) = (-2,4), (-1,2), (0,0), (3,-1), (-4,-8), (-5,-5), (-6,-4) and (-9,-3).
The unique pair with non-negative x and y is (x,y)=(0,0).
Thus, y = 0.

Therefore, the answer is A.
Answer: A

[Math Revolution GMAT math practice question]

(x^2-3x+2)(y^2-5y+6)=?

1) x=1
2) y=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.

Factoring the question gives (x^2-3x+2)(y^2-5y+6) = (x-1)(x-2)(y-2)(y-3).

Condition 1)
If x = 1, then (x-1)(x-2)(y-2)(y-3) = 0.
Condition 1) is sufficient.

Condition 2)
If y = 1, then (x-1)(x-2)(1-2)(1-3) = 2(x-1)(x-2).
Since we don’t know the value of x, condition 2) is not sufficient.

Note: Tip 4) of VA (Variable Approach) method states that if both conditions are trivial, they are not usually sufficient. Thus, conditions 1) & 2) are not sufficient by Tip 4).

Therefore, A is the answer.
Answer: A