Math Revolution DS Expert - Ask Me Anything about GMAT DS

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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 \(\frac{203}{100}*\frac{n}{m}=(\frac{3}{10})^2=\frac{9}{100}\) 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.

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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 \(\frac{a}{504}=\frac{a}{2^3∙3^2∙7}\) 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 \(b = \frac{(3·504)}{a} = \frac{(3·504)}{189} = \frac{504}{63} = 8.\)

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) \(\frac{b}{a} < 0\).

2) \(|a| = 4\) and \(|b| = 3\).

[GMAT math practice question]

(Statistics) 100 students take a test. 20 students are in class A, 30 students in class B, and 50 students in class C. What is the average of the 100 students?

1) The average of class B is 10 points higher than that of class A.
2) The average of class C is 20 points higher than that of class B.

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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.

Assume \(a, b\) and \(c\) are the averages of classes \(A, B\), and \(C\), respectively.

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) together give us that \(b = a + 10\) and \(c = b + 20:\)

If \(a = 60, b = 70\) and \(c = 90\), then the average is
\(\frac{60·20 + 70·30 + 90·50}{100} = \frac{1200 + 2100 + 4500}{100 }= \frac{7800}{100} = 78\)

If \(a = 50, b = 60\) and \(c = 80\), then the average is
\(\frac{50·20 + 60·30 + 80·50}{100} = \frac{1000 + 1800 + 4000}{100} = \frac{6800}{100} = 68\)

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) & 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]

(Number Properties) \(x, y,\) and \(z\) are positive integers and \(z < y < x\). What is the value of \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}?\)

1) \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\) is an integer.

2) \(x = yz\) and \(y\) and \(z\) are consecutive integers and prime 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.

Since we have 3\(\) variables (\(x, y\), and \(z\)) and \(0\) equations, E 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 since we have \(z ≥ 1, y ≥ 2,\) and \(x ≥ 3\), we have \(\frac{1}{z} ≤ 1, \frac{1}{y} ≤ \frac{1}{2}\), and \(\frac{1}{x} ≤ \frac{1}{3}.\)

\(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} ≤ 1 + \frac{1}{2} + \frac{1}{3} = 1 + \frac{3}{6} + \frac{2}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6} < 2\) from condition 1).

Since the unique positive integer less than \(2\) is \(1\), we have \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1.\)

The actual values of \(x, y\), and \(z\) are \(6, 3,\) and \(2\), respectively.

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

Since \(2\) and \(3\) are unique consecutive integers and prime numbers, we have \(y = 3\) and \(z = 2.\)

If \(x = 6, y = 3\) and \(z = 2\), then \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{6} + \frac{1}{3} + \frac{1}{2} = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} = \frac{6}{6} = 1.\)

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

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.

=>

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]

(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 \(ab < 0\).

Is answer B ?

ab<0 => a and b are of opposite signs. They have absolute value so a+b=0.

[GMAT math practice question]

(Number Properties) \(m\) is a three-digit positive integer. What is the value of \(m\)?

1) The digits of \(m\) are \(5, 6,\) and \(7\) without repetition.
2) \(m\) is a product of two consecutive positive 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 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 \(n > 6\).