The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(Inequality) What are the values of x and y?

1) x and y are numbers such that 3x - 2y = 6(x - 1).
2) x and y are integers with -3 < x ≤ 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.

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)

3x - 2y = 6(x - 1)
=> 3x - 2y = 6x - 6
=> -2y = 3x - 6
=> -2y = 3(x - 2)
=> 2y = -3(x - 2)
=> 2y = 3(2 - x)
=> y = (3/2)(2 - x)

Since x and y are integers from condition 2) and x - 2 is an even number, x must be an even number. Also, y is a multiple of 3.

Since we have -3 < x ≤ 3 from condition 2, we have -2 ≤ x ≤ 3. The possible values of x are -2, 0 and 2. Substituting these values into y = (3/2)(2 - x) gives the possible pairs of (x, y), which are (-2, 6), (0, 3) and (2, 6).

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]

(number properties) N is a 3-digit positive integer s with the expression, abc. What is the value of N?

1) b > 2a + c.
2) a + c > 12.

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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.

Since we have 3 variables (a, b, and c) 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)
a, b, and c are integers with 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, and 0 ≤ c ≤ 9.

If a ≥ 5, then b > 10 + c > 10 never happens and we have a ≤ 4 from condition 1).

Since condition 2) tells us that the minimum value of a is 4, a ≥ 4, and the unique value of a is 4.

Since b > 8 + c and c ≥ 0, we have b = 9 and c = 0.
Therefore, N is 490, and it is a unique answer.

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)
If a = 4, b = 9, and c = 2, then N=492 is an answer.
If a = 3, b = 9, and c = 2, then N=392 is also an answer.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
If a = 4, b = 9, and c = 2, then N=492 is an answer.
If a = 5, b = 9, and c = 2, then N=592 is also an answer.
Since condition 2) does not yield a unique solution, it is not 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]

(algebra) For x, y satisfying (x-3)/3= (y+2)?2, x and y always satisfies ax + by = 3. What is (a, b)?

A. (1, 0)
B. (2, 3)
C. (1/2, -3/4)
D. (1/2, -3/2)
E. (4/3, -3/4)

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(x-3)/3 = (y+2)/2
=> 2(x - 3) = 3(y + 2) (cross multiplying)
=> 2x - 6 = 3y + 6 (multiplying through the brackets)
=> 2x - 3y = 12 (adding like terms)
=> (1/2)x - (3/4)y = 3 (dividing both sides by 4).
Then we have a = 1/2 and b = -3/4.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(statistics) 160 students from School A and School B are chosen to take the mathematics test and the average of the test is 66.5. How many students from School A are chosen?

1) The average of School A is 1.5 points higher than the total average.
2) The average of School B is 2.5 points lower than the total average.

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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.

We assume that a and b are the numbers of students in School A and School B, and x and y are the average scores of Schools A and B, respectively.



a + b = 160

Their average is (ax + by) / (a + b) = (ax + by) / 160 = 66.5 and it is equivalent to ax + by = 10640.

The question asks for the value of a.

Since we have 4 variables and 2 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 x = 66.5 + 1.5 = 68 from condition 1).
We have y = 66.5 - 2.5 = 64 from condition 2).
Then we have 68a + 64b = 10640.
Since we have a + b = 160, which we can multiply by 64 to get 64a + 64b = 64*160 = 10240.
Then we can say 4a = (68a + 64b) - (64a + 64b) = 10640 - 10240 = 400 or a = 100.

Therefore, both conditions together 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)
We have x = 66.5 + 1.5 = 68 from condition 1).
The original condition tells us that a + b = 160 and ax + by = 10640.
a = 100, b = 60, x = 68 and y = 64 and a = 120, b = 40, x = 68, y = 62 are solutions.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
We have y = 66.5 - 2.5 = 64 from condition 2).
The original condition tells us that a + b = 160 and ax + by = 10640.
a = 100, b = 60, x = 68 and y = 64 and a = 64, b = 96, x = 70.25, y = 64 are solutions.
Since condition 2) does not yield a unique solution, it is not sufficient.

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]

(inequality) a, b, and c are the lengths of the sides of an obtuse triangle. What is the maximum value of a?

1) a < b < c = 20.
2) a, b, and c are integers.

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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.

Since we have 3 variables (a, b, and c) 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)

Since the triangle is obtuse, we have c^2 > a^2 + b^2 or 400 > a^2 + b^2 from condition 1). Since a < b or a^2 < b^2 and a is an integer, we have a^2 < 200 or a ≤ 13 from condition 2).

The maximum value of a is 13.

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]

(work rate) If A works a job alone, it takes x days, if B does it alone, it takes y days, and if A and B work together, they can do 320 of the total job a day. A and B worked together for 5 days, and A worked alone for 3 days to finish the job. What is x + y?

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

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Assume the total amount of work is 1.
We have an equation 1/x + 1/y = 3/20.
Then we have (1/x + 1/y)*5 + (1/x)*3 = 1 and 8/x + 5/y = 1
When we multiply both sides of 1/x + 1/y = 3/20 by 5, we have 5/x + 5/y = 3/4.
We subtract it from 8/x + 5/y = 1 to get (8/x + 5/y) - (5/x + 5/y) = 1 - 3/4, and we have 3/x = 1/4 or x = 12.
Then we have 1/12 + 1/y = 3/20, 1/y = 3/20 - 1/12, 1/y = 9/60 - 5/60, 1/y = 4/60, 1/y = 1/15, and y = 15.
Therefore x + y = 12 + 15 = 27.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(algebra) x, y and z are the number of $2 stamps, $3 stamps and $10 stamps, respectively. What is the value of xyz?

1) The total number of stamps is 20.
2) The total value of all the stamps is $100.

=>

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)

We have x + y + z = 20 from condition 1) and 2x + 3y + 10z = 100 from condition 2).
When we subtract two times the first equation from the second equation, we have y + 8z = (2x + 3y + 10z) - 2(x + y + z) = 100 – 2*20, 2x + 3y + 10z - 2x - 2y - 2z = 100 - 40, y + 8z = 60, and y = 60 - 8z.
Then we notice that x = 9, y = 4, z = 7 and x = 2, y = 12, z = 6 are solutions of the equation system with x + y + z = 20 and 2x + 3y + 10z = 100.

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

Therefore, E is the answer.
Answer: E

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]

(algebra) There are 3 kinds of gifts A, B and C in a box. The number of gifts A, B and C are a, b and c, respectively. The prices of A, B and C are $3, $2 and $1. The total price of all gifts in the box is $48. What is the total price of gift A?

1) a < b < c.
2) a, b and c are all even numbers.

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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.

We have 3a + 2b + c = 48 from the original condition.

Since we have 3 variables (a, b and c) and 1 equation, 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 a = 2, then we have 2b + c = 42 and we have c = 34 when we have b = 4.
If a = 4, then we have 2b + c = 36 and we have c = 24 when we have b = 6.
Thus, a =2, b = 4, c = 34 and a = 4, b = 6, c = 24 are solutions.

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]

(inequalities) There is an inequality ax > b in terms of x. Does it have a solution?

1) a = 0.
2) b ≥ 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.
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)
Since a = 0 and b ≥ 0, 0x = 0 > b≥ 0 does not have a solution.
Thus, the number of solutions is zero and the answer is ‘No’.

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

Since this question is an inequality 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)
If a = 0 and b = 1, then we have 0x > 1 and the inequality does not have a solution and the answer is ‘No’.
If a = 0 and b = -1, then we have 0x > -1 and the inequality has an infinite number of solutions and the answer is ‘Yes’.

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

Condition 2)
If a = 0 and b = 1, then we have 0x > 1 and the inequality does not have a solution and the answer is ‘No’.

If a = 1 and b = 1, then we have x > 1 and the inequality has an infinite number of solutions and the answer is ‘Yes’.

Since condition 2) does not yield a unique solution, it is not sufficient.

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]

(inequalities) What is the solution set of the inequality, bx - (4a + b) < 0?

1) a is negative and a = -b.
2) The solution range of ax - (a - 2b) > 0 is x < 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.
The inequality bx - (4a + b) < 0 is equivalent to bx < (4a + b).

Condition 1)
Since a < 0 and a = - b, we have b = -a and b is negative.
bx < (4a + b) is equivalent to x > (4a + b)/b, (the direction of the inequality sign changes because we are dividing by a negative number). Then x > (-4b + b)/b (by replacing a with -b), x > (-3b)/b, or x > -3.
The solution is x > -3, so we get yes as the answer.
Since condition 1) yields a unique solution ‘yes’, it is sufficient.

Condition 2)
The inequality ax - (a - 2b) > 0 is equivalent to ax > (a - 2b).
Since its solution set is x < 3, we have x < (a - 2b)/a = 3. So, 3a = a - 2b or a = -b and a is negative. It is same as condition 1).

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

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).

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]

(Inequalities) Is b-a positive?

1) The solution set of (a - 3b)x + (b - 3a) < 0 is x > 5/3.
2) ab < 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.
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 1)
(a - 3b)x + (b - 3a) < 0
=> (a - 3b)x < (3a - b)
=> x > (3a - b) / (a - 3b) under the assumption a - 3b < 0.
Then, we have (3a - b)/(a - 3b) = 5/3 or 3(3a - b) = 5(a - 3b).
It is equivalent to 9a – 3b = 5a – 15b, 4a = -12b or a = -3b.
Since we have the assumption a – 3b < 0, we have (-3b) - 3b = -6b < 0 or b > 0. Since b is positive, a = -3b is negative.
Thus b – a is positive and condition 1) is sufficient, since it yields a unique solution.

Condition 2)
If a = 1 and b = -1, then b – a = (-1) - 1 = -2 is negative and the answer is ‘no’.
If a = -1 and b = 1, then b – a = 1 - (-1) = 2 is positive and the answer is ‘yes’.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Inequalities) Is xy + 1 greater than x + y?

1) 0 ≤ x < 1 and 0 ≤ y < 1.
2) x + y is negative.

=>

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 xy + 1 > x + y is equivalent to (x - 1)(y - 1)>0 for the following reason.
xy + 1 > x + y
=> xy + 1 – x – y > 0
=> xy - x - y + 1 > 0 (rearranging the equation)
=> x(y - 1) - 1(y - 1) > 0
=> (x - 1)(y - 1) > 0
=> x > 1, y > 1 or x < 1, y < 1

Condition 1)
Condition 1) tells us that 0 ≤ x < 1 and 0 ≤ y < 1. This tells us that x < 1, and y < 1, which fits our modified condition. Therefore, the answer is unique, 'yes,' and the condition is sufficient.

Condition 2)
If x = -1, and y = -1, then xy + 1 = (-1)(-1) + 1 = 1 + 1 = 2, x + y = (-1) + (-1) = -2. In this case, xy + 1 is greater than x + y and the answer is ‘yes’.
If x = 2, and y = -3, then xy + 1 = (2)(-3) + 1 = -6 + 1 = -5, x + y = 2 + (-3) = -1. In this case, xy + 1 is less than x + y 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

[GMAT math practice question]

(Algebra) a, b, and c are integers greater than 1 with a < b < c. What is the value of a + b + c?

1) (ab - 1)(bc - 1)(ca - 1) is divisible by abc.
2) a, b, and c are prime numbers less than 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)
Since a, b and c are prime numbers less than 6 with a < b < c, we have a = 2, b = 3 and c = 5.
Thus we have a + b + c = 10.
Since condition 2) provides a unique solution, it is sufficient.

Condition 1)
(ab - 1)(bc - 1)(ca - 1)
= (ab2c – ab – bc + 1)(ca - 1)
= a2b2c2 – ab2c – a2bc – abc2 + ab + bc + ca – 1
= a2b2c2 – abc(a + b + c) + (ab + bc + ca) – 1
= abc{abc – (a + b + c)} + (ab + bc + ca) – 1
Since (ab - 1)(bc - 1)(ca - 1) is divisible by abc, we notice that ab + bc + ca – 1 is divisible by abc.
Then we have ab + bc + ca – 1 = abc*n for some integer.
When we divide both sides of the equation by abc, we have n = 1/a + 1/b + 1/c – 1/abc < 1/2 + 1/2 + 1/2 = 3/2, since 2 ≤ a < b < c or 1/c > 1/b > 1/a ≥ 1/2.

Then the positive integer n equals 1.

When we divide both sides of the equation ab + bc + ca – 1 = abc by bc, we have a = 1 + a/c + a/b – 1/bc < 1 + 1 + 1 – 1/bc < 3.
Then we have a = 2.

When we substitute a with 2 in the equation ab + bc + ca - 1 = abc, we have 2b + bc + 2c - 1 = 2bc or bc – 2b – 2c + 1 = 0.
Then we have bc – 2b – 2c + 4 = 3 or (b - 2)(c - 2) = 3.
Then we have b – 2 = 1, c – 2 = 3 or b = 3, c = 5.

Thus, we have a + b + c = 10.

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

Therefore, D is the answer.
Answer: D

This question is a CMT4 (B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT4 (B) questions, D is most likely to be the answer.

[GMAT math practice question]

(Inequalities) An inequality |2x - a| < a - 4 has a solution. What is the range of a?

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

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Since we have an absolute value on the left side, the value of the right side must be positive in order for the inequality to have a solution.
Thus we have a – 4 > 0 or a > 4.

Therefore, E is the answer.
Answer: E

[GMAT math practice question]

(Inequalities) The solution set of an inequality (a + b)x + 2a - 3b < 0 is x < -1/3. What is the solution set of the inequality (a - 3b)x + b - 2a > 0, in terms of x?

A. x < -3
B. -3 < x < 3
C. -1 < x < 0
D. -1 < x < 3
E. 0 < x < 4

=>

(a + b)x + 2a - 3b < 0
=> (a + b)x < 3b - 2a
=> x < (3b - 2a) / (a + b) under the assumption a + b > 0
Since its solution set is x < -(1/3), we have (3b - 2a) / (a + b) = -(1/3) or (-3)(3b - 2a) = a + b. Then we have 6a – 9b = a + b or a = 2b.
Since a = 2b, the inequality (a - 3b)x + b - 2a > 0 is equivalent to (2b - 3b)x + b – 4b > 0 or -bx – 3b > 0.
Then we have bx < -3b or x < -3.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Functions) A function f(x) = -3x + 16 is a linear function and f(a+b) = c. What is f(|c|)?

1) f(a) = -a.
2) f(b) = 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.
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.

f(a+b) = c
=> -3(a+b) + 16 = -3c + 16
=> -3a - 3b + 16 = -3c + 16
=> -3a - 3b = -3c (by subtracting 16 on both sides)
=> a + b = c (by dividing by -3 on both sides)

Since we have 3 variables (a, b, and c) and 1 equation (a + b = c), 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 a = 8 from condition 1) for the following reason.
f(a) = -a
=> -3a + 16 = -a
=> 2a = 16
=> a = 8

We have b = 4 from condition 2) for the following reason.
f(b) = b
=> -3b + 16 = b
=> 4b = 16
=> b = 4.

Then we have c = a + b = 8 + 4 = 12.

f(|c|) = f(|12|) = f(12) = -3(12) + 16 = -36 + 16 = -20.

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

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]

(Geometry) What is the measure of the angle ∠EAC?

1) BD = DE = EA = AC
2) ∠ACE = ∠DBE + 40°



=>

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 9 variables from 3 triangles and 5 equations, ∠BDE + ∠EDA = 180, ∠BED + ∠DEA + ∠EAD = 180, ∠ABC + ∠BCA + ∠CAB = 180, ∠EBD + ∠EDB + ∠BED = 180 and ∠EAC + ∠ECA + ∠AEC = 180, 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)



Since we have BD = DE = EA = AC from condition 1), we have the measures of the interior angles, as shown in the above figure.
Since ∠ACE = ∠DBE + 40° from condition 2), we have 3x = x + 40°, 2x = 40° or x = 20°.
Thus, we have ∠EAC = 180° – 6x = 180° – 6(20), ∠EAC = 180° = 120 = 60°.

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.