The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(Algebra) If a = 1/√2-1-2, what is a^2 + 2a + 4?

A. 2
B. 3
C. 4
D. 5
E. 6

=>

1/√2-1=(√2+1)/(√2-1)(√2+1)=√2+1
Then we have
a = √2 + 1 – 2
a = √2 - 1
or a + 1 = √2.

(a + 1)^2 = (√2)^2 = 2
a^2 + 2a + 4 = a^2 + 2a + 1 + 3 = (a + 1)^2 + 3 = 2 + 3 = 5.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(algebra) If x^2 - 4x + 1 = 0, what is x^2 + 1/x^2?

A. 14
B. 16
C. 18
D. 20
E. 22

=>

x^2 - 4x + 1 = 0
x – 4 + 1/x = 0 (dividing both sides by x)
x + 1/x = 4 (adding 4 to both sides)
(x+1/x)^2 = 16 (squaring both sides)
x^2 +2x(1/x) + 1/x^2 = 16 (foiling the left side)
x^2 + 1/x^2 + 2 = 16 (simplifying)
x^2 + 1/x^2 = 14 (subtracting 2 from both sides)

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Statistics) A company has 2 departments, department A has 5 employees, and department B has 6 employees. All employees of the company did some “push-ups”. What is the standard deviation of these 11 employees?

1) The average and the standard deviation of A are 7 and 1, respectively.
2) The average and the standard deviation of B are 7 and 3, respectively.

=>

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 the standard deviations of two sets, we have 2 variables 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)
Assume A1, A2, … , A5 are the “push-up” numbers of the employees in department A and B1, B2, … , B6 are the “push-up” numbers of the employees in department B.
The combined average of departments A and B is
( 5*7 + 6*7 ) / 11 = 11*7/11 = 77/11 = 7.
The variances of A and B are squares of the standard deviations of A and B, respectively.
The variance of A is { (A1-7)^2 + … + (A5-7)^2 } / 5 = 1 and we have { (A1-7)^2 + … + (A5-7)^2 = 5.
The variance of B is { (B1-7)^2 + … + (B6-7)^2 } / 6 = 1 and we have { (B1-7)^2 + … + (B6-7)^2 = 6.

The combined variance of sets A and B is
{ (A1-7)^2 + … + (A5-7)^2 + (B1-7)^2 + … + (B6-7)^2 } / 11
= { 5 + 6 } / 11 = 11/11 = 1.
The standard deviation of the sets A and B is the square root of the combined variance equal to 1.

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 we don’t have any information about department B, it is not sufficient, obviously.

Condition 2)
Since we don’t have any information about department 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]

(Number Properties) If √980xy is a positive integer, what is the value of xy?

1) x and y are positive integers
2) x ≥ y

=>

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)
We have 980 = 2^25^17^2.

X = 5, y = 2 and x = 20, y = 2 are possible solutions.
Then we have xy = 10 and 40.
Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(Algebra)What is a^3b + a^2b^2 + ab^3?

1) a – b = 3
2) ab = 3

=>

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.

a^3b + a^2b^2 + ab^3?
= ab(a^2 + ab + b^2) (taking out a common factor of ab)
= ab((a^2 - 2ab + b^2 + 3ab)) (because -2ab + 3ab = ab from the equation in the previous line)
= ab((a-b)^2+3ab) (factoring)

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

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(Function) What is the minimal value of f(n) = 2n^2 - 7n + 8, if n is an integer?

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

=>

f(n) = 2n^2-7n+8
= 2(n^2 - (7/2)n) + 8 (taking out a common factor of 2 from the first 2 terms)
= 2(n^2 - 2(7/4)n + (49/16) - (49/16)) + 8 (completing the square)
= 2(n - 7/4)^2 - 49/8 + 8 (factoring and multiplying 2 times 49/8)
= 2(n - 7/4)^2 + 15/8 (adding like terms)
f(n) has a minimum value when n = 7/4
The closest integer to 7/4 is 2.
Thus the minimum value of f(n) is f(2) = 2*2^2 - 7*2 + 8 = 8 – 14 + 8 = 2.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(Algebra) p = a/(a+b)(a+c)+b/(b+c)(b+a)+c/(c+a)(c+b), q = b+c/a+c+a/b+a+b/c. What is p-q?

1) a, b, and c are integers.
2) 1/a+1/b+1/c = 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.

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.

p = a/(a+b)(a+c)+b/(b+c)(b+a)+c/(c+a)(c+b)
= a(b+c)/(a+b)(a+c)(b+c)+b(c+a)/(b+c)(b+a)(c+a)+c(a+b)/(c+a)(c+b)(a+b) (getting a common denominator)
= ab+ca+bc+ab+ca+bc/(a+b)(b+c)(c+a) (multiplying through the brackets and combining in one fraction)
= 2(ab+ca+bc)/(a+b)(b+c)(c+a) (adding like terms and taking out a common factor of 2)

q = b+c/a+c+a/b+a+b/c
= b+c+a-a/a+c+a+b-b/b+a+b+c-c/c (adding and subtracting variables in order to get a numerator of a + b + c)
= a+b+c/a-a/a+a+b+c/b-b/b+a+b+c/c-c/c (separating fractions)
= a+b+c/a-1+a+b+c/b-1+a+b+c/c-1 (simplifying fractions)
= (a+b+c)(1/a+1/b+1/c)-3 (adding the constants and taking out a common factor of a + b + c)


Condition 2)
1/a+1/b+1/c=0
=> bc/abc+ca/abc+ab/abc=0 (getting a common denominator)
=> ab+bc+ca/abc=0 (combining into one fraction)
=> ab + bc + ca = 0 (multiplying both sides by abc)
Then, we have p = 2(ab+ca+bc)/(a+b)(b+c)(c+a) = 0/(a+b)(b+c)(c+a) = 0 and q = (a+b+c)(1/a+1/b+1/c)-3 = (a+b+c)*0-3 = -3.
Thus, p – q = 0 – (-3) = 3.
Since condition 2) yields a unique solution, it is sufficient.

Condition 1)

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

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(Function) f(x) and g(x) are functions. What is the value f(3) + g(3)?

1) f(x + g(y)) = ax + y + 1 where a is a constant.
2) f(0) = -2, g(0) = 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.

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 a) 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 substitute 0 for y, we have f(x + g(0)) = f(x+1) = ax + 1.
Then, we have f(0) = f((-1) + 1) = a(-1) + 1 = 1 – a = -2 or a = 3.
f(3) = f(2+1) = 3*2+1 = 7.
We have f(x + g(y) – 1 + 1) = 3(x + g(y) - 1) + 1 = 3x + 3g(y) - 2 = 3x + y + 1 or 3g(y) = y – 3.
Then g(y) = y/3 + 1 and g(3) = 2.
Thus f(3) + g(3) = 7 + 2 = 9.

Since both conditions together yield a unique solution, they are sufficient.
Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

[GMAT math practice question]

(Geometry) The figure shows the right triangle ABC and inscribed circle, which is tangential at D. AD = p and CD = q. What is the area of △ABC?

1) p = 2, q = 3.
2) The radius of the inscribed circle is 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.

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.

We can put BC = BF = x.
Put BC = a = q + x, AC = b = p + q and AB = c = p + x.
Then the area of triangle ABC = (1/2)ac and x = (a-b+c)/2, p = (b-(a-c))/2 and q = (b+(a-c))/2
Then we have
pq = {(b-(a-c))/2}{(b+(a-c))/2}
pq = {b^2 + b(a-c) – b(a-c) - (a-c)^2}/4
pq = (b^2 – (a-c)^2)/4
pq = {b^2 – (a-c)(a-c)}/4
pq = {b^2 – (a^2 – ac – ac + c^2)}/4
pq = (b^2 - a^2 - c^2 + 2ac)/4
pq = ac/2.
Thus, the area of the triangle ABC is pq, and condition 1) is sufficient.

Condition 2)

Since there are a lot of possibilities for p and q, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Geometry) How many right triangles are there using 3 lines among 7 lines with lengths of 5, 6, 7, 8, 10, 12, and 13?

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

=>

Assume c is the length of the hypotenuse of the right triangle, while a and b are the lengths of the legs with a < b.
If c = 13, then a = 5, b =1 2 is the only possible solution.
If c = 12, then there is no possible solution of a and b.
If c = 10, then a = 6, b = 8 is the only possible solution.
If c = 8, then there is no possible solution of a and b.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(Algebra) x, y and z are real numbers with xyz = 1. What is the value of (x - 1)(y - 1)(z - 1)?

1) x + y + z = 3
2) xy + yz + zx = -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.

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.

(x - 1)(y - 1)(z - 1)
= (x – 1)(yz – y – z + 1)
= xyz – xy – xz + x – yz + y + z - 1
= xyz – (xy + yz + zx) + (x + y + z) – 1

Since we have x + y + z = 3 and xy + yz + zx = -4 from both conditions 1) and 2), we have (x - 1)(y - 1)(z - 1) = xyz – (xy + yz + zx) + (x + y + z) – 1 = 1 – (-4) + 3 – 1 = 7.
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(Number Property) N is an integer. Is N a perfect square?

1) N is 1 greater than the product of 4 consecutive integers.
2) N is a summation of squares of 4 consecutive odd 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Condition 1)

Assume N is 1 greater than a product of four consecutive integers, x, x+1, x+2, and x+3 where x is an integer.
We have
N = x(x + 1)(x + 2)(x + 3) + 1
N = x(x + 3)(x + 1)(x + 2) + 1
N = (x^2 + 3x)(x^2 + 3x + 2) + 1
N = (x^2 + 3x)^2 + 2(x^2 + 3x) + 1
N = (x^2 + 3x + 1)^2
Thus, N is a perfect square.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

If N = 1 + 3 + 5 + 7 = 16, then N is a perfect square and the answer is ‘yes’.
If N = 3 + 5 + 7 + 9 = 24, then N is not a perfect square and the answer is ‘no’.
Since condition 2) does not 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 Property) a, b, and c are integers. Is 2(a^4 + b^4 + c^4) a perfect square?

1) a = 1, b = 1, and c = -2
2) a + b + c = 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.

Condition 1)

Since we have a = 1, b = 1, and c = 2, we have
2(a^4 + b^4 + c^4) = 2(14 + 14 + 24)
= 2(1 + 1 + 16)
= 2*18
= 36.
2(a^4 + b^4 + c^4) = 36 is a perfect square and the answer is ‘yes’.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca),
we have a^2 + b^2 + c^2 = -2(ab + bc + ca).

Then, rearranging the second formula gives us:
(a^2 + b^2 + c^2)^2 = (-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 + a^2c^2)
= 4((a^2b^2 + b^2c^2 + a^2c^2 + 2ab^2c + 2a^2bc + 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

Following the pattern in the first equation gives us:
(a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2)

We now have two equations:
(a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2b^2 +b^2c^2+ c^2a^2)
(a^2 + b^2 + c^2)^2 = 4(a^2b^2 + b^2c^2 + c^2a^2)

Combining the two equations gives us:
a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2+ c^2a^2) = 4(a^2b^2 + b^2c^2 + c^2a^2)
a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2)
2(a^4 + b^4 + c^4) = 4(a^2b^2 + b^2c^2 + c^2a^2) = (a^2 + b^2 + c^2)^2.
Thus, 2(a^4 + b^4 + c^4) is a perfect square.

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

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

[GMAT math practice question]

(Geometry) The figure shows a right triangle ABC with ∠A = 90, BH = 8, and CH = 4. Moreover, AH and BC are perpendicular to each other. What is the length of AH?

A. √2
B. 3√2
C. 4√2
D. 3
E. 4



=>

Since triangles ABH and CAH are similar to each other, we have AH:BH = CH:AH.
Thus, we have x:8 = 4:x or x^2 = 32 and x = 4 √2.

Therefore, C is the answer.
Answer: C

Why cant we use 30-60-90 triangle method, A is clearly divided in the ratio of 2:1 ?

√
[GMAT math practice question]

(Number Properties) What is the solution of (x, y)’s satisfying √500 = √x +√y and x < y?

1) x and y are positive integers.
2) x = 5t^2 with 0 < t < 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.

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

Condition 1)
x = 5 = 5*1^2, y = 5*9^2 = 405 and x = 5*2^2, y = 5*8^2 = 320 are possible solutions.

Condition 2)
x = 5 = 5*1^2, y = 5*9^2 = 405 and x = 5*2^2, y = 5*8^2 = 320 are possible solutions.

Conditions 1) & 2)

x = 5 = 5*1^2, y = 5*9^2 = 405 and x = 5*2^2, y = 5*8^2 = 320 are possible solutions.

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

Therefore, E is the answer.
Answer: E

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

[GMAT math practice question]

(Statistics) The standard deviation of x1, x2, …, x5 is 2√2. What is the standard deviation of √2x1, √2x2, …, √2x5?

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

=>

Remind the property that S(aX+b) = |a|S(X) where S(X) is the standard deviation of set X and X is a data set.

S(√2x1, √2x2, …, √2x5) = √2S(x1,x2,x3,x4,x5) =√2*(2√2) = 4.

Therefore, E is the answer.
Answer: E