The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(Algebra) There are 3 consecutive odd integers. Three times of the sum of the largest and the smallest of them is 6 less than 8 times the remaining odd integer. What is the smallest integer?

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

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Assume x – 2, x, and x + 2 are the three consecutive odd integers, where x is an odd number.
We have 3(x – 2 + x + 2) = 8x – 6, 3x – 6 + 3x + 6 = 8x - 6 or 6x = 8x – 6.
Then we have 2x – 6 = 0 or x = 3.
Thus, the smallest integer is x – 2 = 3 – 2 = 1.
Therefore, A is the correct answer.
Answer: A

[GMAT math practice question]

(Geometry) Are triangles △APC and △ABQ congruent to each other?

1) △PBC and △QAC are equilateral triangles.
2) △ABC is an equilateral triangle.



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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.

Let’s look at condition 1). It tells us that PA = AB, AC = AQ, and ∠PAC = ∠BAQ.
Since ∠PAC = 60° + ∠A and ∠BAQ = 60° + ∠A, we have ∠PAC = ∠BAQ.
Then triangles △APC and △ABQ are congruent according to the SAS congruency property, so we get yes as an answer.

The answer is unique, yes, so the condition is sufficient according to Common Mistake Type 1, which states that the answers must be in terms of a unique “yes” or “no.”

Let’s look at condition 2). It tells us that it is not sufficient.
If triangles △ABC, △APB, and △ACQ are congruent, then triangles △APC and △ABQ are congruent, so we get yes as an answer.
If triangles △ABC and △APB are congruent with sides 3 and AQ = CQ, then triangles △APC and △ABQ are not congruent, so we get no as an answer.

The answer is not unique, yes and no, so the condition is not sufficient according to Common Mistake Type 1, which states that if we get both yes and no as an answer, it is not sufficient.

Condition 1) ALONE is sufficient.

Therefore, A is the correct 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]

(Statistics) 100 students take a test. What is their test average?

1) There are 40 female students.
2) The female students’ average is 70, and the male students’ average is 60.

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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 can set this question up with a 2x2 matrix.



We have a + b = 100.

Since we have 4 variables (a, b, x, and y) and 1 equation, E is most likely the answer in general. However, since we have 1 equation in condition 1) and 2 equations in condition 2, C is most likely the answer in this question. 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) together tell us that a = 60, b = 40, x = 60 and y = 70.

The average is the value when the total score is divided by the total number of students, which is (ax + by) / (a + b).
We have a = 60 and b = 40 from condition 1) and we have x = 60 and y = 70.
Thus, the average is (60·60 + 40·70) / (60 + 40) = (3600 + 2800) / 100 = 6400/100 = 640.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that a = 60 and b = 40.
If x = 60 and y = 70, the average is (60·60 + 40·70) / (60 + 40) = (3600 + 2800) / 100 = 6400/100 = 64.
If x = 60 and y = 60, the average is (60·60 + 40·60) / (60 + 40) = (3600 + 2400) / 100 = 6000/100 = 60.

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.

Let’s look at condition 2). It tells us that x = 60 and y = 70.
If a = 60 and b = 40, the average is (60·60 + 40·70) / (60 + 40) = (3600 + 2800) / 100 = 6400/100 = 64.
If a = 50 and b = 50, the average is (50·60 + 50·70) / (50 + 50) = (3000 + 3500) / 100 = 6500/100 = 65.

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.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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]

(Number Properties) m and n are positive integers. What is the value of mn?

1) 2.03(n/m) = (0.3)^2
2) m and n are relatively prime 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.

Since we have 2 variables (m and n) 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) together tell us that m and n are relatively prime integers with 203n = 9m.

We have 203/100*n/m=(3/10)^2=9/100 or 203n = 9m from condition 1).
Since m and n are relatively prime, we have m = 203 and n = 9.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that 203n = 9m.
If m = 203 and n = 9, then we have mn = 1827.
If m = 406 and n = 18, then we have mn = 7308.

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.

Let’s look at condition 2). It tells us that m and n are relative primes.
If m = 203 and n = 9, then we have mn = 1827.
If m = 2 and n = 3, then we have mn = 6.

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.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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) a and b are integers. If a/504 is a terminating decimal, what is the value of a - b?

1) 3/b is the simplest fraction of a/504.
2) 150 ≤ a ≤ 200.

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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 a/504=a/2^3∙3^2∙7 is a terminating decimal, a is a multiple of 3^2 ·7 = 63.

Since we have 2 variables (a and b) and 1 equation, D 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) together tell us that we have ab = 3·504 = 2^3 · 3^3 · 7 and a = 189.
Thus, we have b = (3·504)/a = (3·504)/189 = 504/63 = 8.
Then we have a – b = 189 – 8 = 181

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that ab = 3·504 = 2^3 · 3^3 · 7.
If a = 189 and b = 8, then we have a – b = 189 – 8 = 181.
If a = 63 and b = 24, then we have a – b = 39.

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.

Let’s look at the condition 2). It tells us that a = 189.
If a = 189 and b = 8, then we have a – b = 189 – 8 = 181.
If a = 189, b = 1, then we have a – b = 189 – 1 = 188.

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

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.
Answer: C

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]

(Algebra) What is the value of (a-b)^2?

1) b/a < 0.
2) |a| = 4 and |b| = 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 (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) together give us that a = ±4, b = ±3 and ab < 0.
If a = 4 and b = -3, then (4-(-3))^2 = 7^2 = 49.
If a = -4 and b = 3, then (-4-3)^2 = 7^2 = 49.

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

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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) a and b are positive integers. What is the value of 2^a + 2^b?

1) a is the units digit of 7^1020 and b is the units digit of 3^224.
2) a and b are neither prime numbers nor composite 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.

Since we have 2 variables (a and b) and 0 equations and each condition has 2 equations, C is most likely to be the answer. Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately.

Let’s look at the condition 1). It tells us that a = 1 and b = 1.

Units of powers of 7 are 7^1~7, 7^2~9, 7^3~3, 7^4~1, 7^5~7, …
So, the units digits of 7^n have a period of 4:
They form the cycle 7 -> 9 -> 3 -> 1.
Thus, 7^n has a units digit of 1 if n has a remainder of 0 when it is divided by 4.
The remainder is 0 when 224 is divided by 4, so the units digit of 7^1020 is 1.

Units of powers of 3 are 3^1~3, 3^2~9, 3^3~7, 3^4~1, 3^5~3, …
So, the units digits of 3^n have a period of 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, 3^n has a units digit of 1 if n has a remainder of 0 when it is divided by 4.
The remainder is 0 when 224 is divided by 4, so the units digit of 3^224 is 1.

2^1 + 2^1 = 2 + 2 = 4.

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

Let’s look at the condition 2), it tells us that a = 1 and b = 1.

2^1 + 2^1 = 2 + 2 = 4.

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Also, the original condition needs 2 equations.
Condition 1) has 2 equations.
Condition 2) has 2 equations.

Each condition ALONE is sufficient.

Therefore, D is the correct 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).

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]

(Geometry) In the figure, what is the measure of ∠DHE?



1) □ABCD is a square.
2) □ECFG is a 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.
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 two quadrilaterals, we have 10 variables 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) together give us that:
Since we have BC = CD, EC = CF and ∠BCD = ∠DCF, triangles EBC and FDC are congruent according to the SAS property.

Since ∠EBC + ∠BED = 90° and ∠DEH = ∠BED, we have ∠DEH + ∠EDH = 90° and ∠DHE = 180° – 90° = 90°.

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

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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) △ABC and △CDE are equilateral triangles, as the figure shows. What is the measure of the ∠x?



A. 45°
B. 50°
C. 55°
D. 60°
E. 65°

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We have AC = BC, CD = CE and ∠BCE = ∠DCE = 60°.
Triangles ACD and BCE are congruent.
Since ∠EBC = ∠DAC, we have ∠x = ∠ACB = 60°.

Therefore, D is the correct answer.
Answer: D

[GMAT math practice question]

(Algebra) What is the value of a + b?

1) The equation 2(x + a) = bx - 4 has more than one solution.
2) a and b have the same absolute value and 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 have to find the value of a + b.

Follow the second and the third step: From the original condition, we have 2 variables (a and b). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

Recall 3 Principles and choose C as the most likely answer.

Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately.

Condition 1) tells us that a = -2 and b = 2. In order for the equation to have more than one solution, the corresponding coefficients on both sides must be equal, respectively. Then we have the left-hand side 2(x + a) = 2x + 2a and we have 2x + 2a = bx – 4. Since the equation 2x + 2a = bx – 4 has more than one solution, we have 2 = b and 2a = -4. Thus, condition 1) tells us that a = -2 and b = 2.
Then we have a + b = -2 + 2 = 0.

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

Condition 2) tells us that a = -b. Since we have |a| = |b| and ab < 0, a and b have different signs and a = -b. Thus, we have a + b = 0.
The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D

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) If (-4)^3÷(-2)^m=-2^{n-6} what is the value of mn?

1) m and n are positive even integers.
2) m < 6 and n > 6.

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Forget conventional ways of solving math questions. F or 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.
(-4)^3 ÷ (-2)^m=-2^{n-6}
=> -2^6 ÷ (-2)^m = -2^{n-6}
=> -2^{6-m} = -2^{n-6}, where m is an even integer.
=> 6 - m = n - 6
=> m + n = 12.

Follow the second and the third step: From the original condition, we have 2 variables (m and n) and 1 equation (m + n = 12). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer. Let’s look at each condition separately,

Condition (1) tells us that m and n are positive even integers, from which we get (m, n) = (2, 10), (4, 8), (6, 6), (8, 4), and (10, 4). If m = 2, and n = 10 we get mn = 20 and if m = 4, and n = 8, we get mn = 32.
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 < 6 and n > 6, from which we cannot get the unique values of m and n. For example, if m = 2, and n = 10, then we get mn = 20 and if m = 4, n = 8, we get mn = 32.
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.

Both conditions (1) and (2) together also do not give us unique values for m and n. For example, if m = 2 and n = 10, we get mn = 20 and if m = 4, and n = 8, we get mn = 32.

The answer is not unique, so both conditions (1) and (2) 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.

Therefore, E is the correct 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]

(Inequalities) If a rational number r satisfies 0 < r < 1, which one is the minimum among the following choices?

A. r
B. -r
C. -1/r
D. -r^2
E. -1/r^2

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We can substitute in a number satisfying the condition.
Assume r = 1/2.
Then -r = -1/2, -1/r = -2, -r^2 = -1/4 and -1/r^2 = -4
Thus -1/r^2 = -4 is the minimum number.

Therefore, E is the correct answer.
Answer: E

[GMAT math practice question]

(Ratio) At an entrance examination, the ratio of successful male applicants to successful female applicants is 5:2. What is the total number of applicants?

1) The ratio of male applicants to female applicants is 3:2.
2) The number of successful applicants is 140.

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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.

Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let x and y be the number of male and female applicants, respectively. Let 5k be the number of successful male applicants and 2k be the number of successful female applicants, giving us x = 5k. Then we have to find the total number of applicants, which is equal to x + y.

Follow the second and the third step: From the original condition, we have 3 variables (x, y, and k). To match the number of variables with the number of equations, we need 3 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

We know that the ratio of successful male applicants to successful female applicants is 5:2; there are 100 successful male applicants and 40 successful female applicants.

Then the number of male applicants is greater than or equal to 100, and the number of female applicants is greater than or equal to 40.

If the number of male applicants is 120 and that of female applicants is 80, then the total number of applicants is 200.
If the number of male applicants is 150 and that of female applicants is 100, then the total number of applicants is 250.

The answer is not unique, and both conditions 1) and 2) together 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.

Therefore, E is the correct 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]

(Function) A function satisfies f(xy) = f(x) + f(y) for any positive numbers x and y. We have f(2) = 1. What is the value of f(8)?

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

=>

f(4) = f(2·2) = f(2) + f(2) = 1 + 1 = 2.
f(8) = f(2·4) = f(2) + f(4) = 1 + 2 = 3.

Therefore, C is the correct answer.
Answer: C.

[GMAT math practice question]

(Number Properties) p, q, and r are positive integers. What is the value of p + q + r?

1) p, q, and r are prime numbers.
2) The product of p, q, and r is 5 times the sum of p, q, and r.

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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.

Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to determine the value of p + q + r.

Follow the second and the third step: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

Since we have 3 variables (p, q, and r) 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) together give us that:
Since we have pqr = 5(p + q + r), we assume p = 5.
Then we have 5qr = 5(5 + q + r) or qr = q + r + 5.

Since we have qr – q – r = 5, we have qr – q – r + 1 = 6 (by adding 1 to both sides so that it can easily be factored). Factoring gives us (q - 1)(r - 1) = 6.

Then we have the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).
If we have q – 1 = 1 and r – 1 = 6, we have q = 2 and r = 7.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3 and r = 4.
If we have q – 1 = 3 and r – 1 = 2, we have q = 4 and r = 3.
If we have q – 1 = 6 and r – 1 = 1, we have q = 7 and r = 2.

However, p, q, and r are prime numbers from condition 1), and their possible numbers are 2, 5, and 7.
Thus, we have p + q + r = 2 + 5 + 7 = 14.

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

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition 1) tells us that we don’t have a unique solution, obviously.
If p = 2, q = 3, and r = 5, we have p + q + r = 2 + 3 + 5 = 10.
If p = 2, q = 5, and r = 7, we have p + q + r = 2 + 5 + 7 = 14.

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.

Let’s look at the condition 2). It tells us that we don’t have a unique solution.
Assume p = 5 and since we have qr – q – r = 5, we have qr – q – r + 1 = 6 or (q - 1)(r - 1) = 6.

Then the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).
If we have q – 1 = 1 and r – 1 = 6, we have q = 2, r = 7 and p + q + r = 5 + 2 + 7 = 14.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3, r = 4 and p + q + r = 5 + 3 + 4 = 12.
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.
Thus, really, both conditions 1) & 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.

[GMAT math practice question]

(Geometry) The figure shows that BP is a line passing the center O and PT is a tangent line to the circle at point T. If ∠APT = 20°, what is ∠x?



A. 40°
B. 45°
C. 50°
D. 55°
E. 60°

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Since PT is tangent to the circle, we have ∠OTP = 90° and ∠AOT = 180° - 20° = 70°.
Since the triangle is an isosceles triangle with AO = OT, we have ∠OTA = ∠OAT = ∠x.
Thus ∠x = (180° - 70°)/2 = 55°.

Therefore, D is the correct answer.
Answer: D

Solution:

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.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of f(2) + f(3) + f(5).

Follow the second and the third step: From the original condition, we have many variables to determine a function f(x). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since f(1) = 1, we have f(2) = f(1+1) = f(1) + f(1) + 1·1 = 1 + 1 + 1 = 3 using condition 2).
Then we have f(3) = f(2+1) = f(2) + f(1) + 2·1 = 3 + 1 + 2 = 6.

f(5) = f(3+2) = f(3) + f(2) + 3·2 = 6 + 3 + 6 = 15.

Thus, we have f(2) + f(3) + f(5) = 3 + 6 + 15 = 24.

The answer is unique, so 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) & 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.