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MathRevolution
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Overview of GMAT Math Question Types and Patterns on the GMAT 13 Aug 2021, 03:16
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[Math Revolution GMAT math practice question]

(absolute value) Is \(ab<0\)?

\(1) |a+b| = - ( a + b )\)
\(2) |a+b| + 1 = |a| + |b|\)
Show more

=>

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

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

You should remember that the inequality \(|x+y| < |x| + |y|\) is equivalent to the inequality \(xy < 0.\)

Condition 2) tells us that \(|a+b| + 1 = |a| + |b|.\) Thus, \(|a + b| < |a| + |b|\) and \(ab < 0\).
Thus, condition 2) is sufficient.

Condition 1)
If \(a = -2\) and \(b = 1\), then the answer is ‘yes’.
If \(a = -1\) and \(b = -1\), then the answer is ‘no’.
Since it does not yield a unique solution, condition 1) is not sufficient.

Therefore, the answer is B.
Answer: B
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Overview of GMAT Math Question Types and Patterns on the GMAT 15 Aug 2021, 02:49
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MathRevolution
[GMAT math practice question]

(number properties) What is the remainder when \(1+n+n^2 +…+ n^8\) is divided by \(5\)?

1) The remainder when \(n\) is divided by \(5\) is \(3\)

2) \(n\) is less than \(5\)
Show more

=>

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

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

The easiest way to solve remainder questions is to plug in numbers.
The units digits of \(3^n\) for \(n = 1, 2, 3, 4, …\)are \(3, 9, 7, 1, 3, 9, 7, 1, …\)

So, the units digits of \(3^n\) have period \(4\):

They form the cycle \(3 -> 9 -> 7 -> 1\).

Thus, if \(n\) has remainder \(3\) when it is divided by \(5, 1+n+n^2 +…+ n^8\) has the same remainder as \(1 + 3 + 9 + 7 + 1 + 3 + 9 + 7 + 1 = 21\) when it is divided by \(5\). It has a remainder of \(1\) when it is divided by \(5.\)

Condition 1) is sufficient.

Condition 2)
If \(n = 1\), then \(1+n+n^2 +…+ n^8 = 9,\) which has remainder \(4\) when it is divided by \(5\).

If \(n = 3,\) then \(1+n+n^2 +…+ n^8\) has remainder \(1\) when it is divided by \(5.\)

Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A
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MathRevolution
[Math Revolution GMAT math practice question]

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

\(1) x=1\)
\(2) y=1\)
Show more

=>

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

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

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

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

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

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

Therefore, A is the answer.
Answer: A
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[Math Revolution GMAT math practice question]

(functions) In the x-y plane, line l passes through points \((-1,-1)\) and \((3,k)\). What is the value of \(k\)?

1) The y-intercept of line l is \(1\)
2) The slope of line l is \(2\)
Show more

=>

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

Since we have \(1\) variable (\(k\)) and \(0\) equations in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first.


We consider the equation of the line \(l, y = mx + b.\) Since it passes through the points \((-1,-1)\) and \((3,k),\) we can plug these points into its equation to yield \(-1 = -m + b\) and \(k = 3m + b.\)

Since we have \(3\) variables and \(2\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since the y-intercept of line \(l\) is \(1\), we have \(b = 1\) and \(m = b + 1 = 2.\)
Thus, \(k = 3m + b = 3*2 + 1 = 7.\)
Condition 1) is sufficient.

Condition 2)
Since the slope of line \(l\) is \(2\), we have \(m = 2\) and \(b = m – 1 = 1.\)
Thus, \(k = 3m + b = 3*2 + 1 = 7.\)
Condition 2) is sufficient.

Answer: D

Note: When we checked the two conditions, we showed that both were equivalent in terms of b. So, each condition is sufficient by Tip 1)
of the VA method, which states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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.
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Overview of GMAT Math Question Types and Patterns on the GMAT 21 Aug 2021, 02:34
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[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) \(xy + yz + zx = -4\)
Show more

=>

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
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[GMAT math practice question]

(number properties) We define ‘a mod n’ to be the remainder when a is divided by n. What is the value of ‘a mod 12’?

1) a mod 3=1
2) a mod 4=a mod 2
Show more

=>

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

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

Condition 1)
The numbers satisfying condition 1) are \(1, 4, 7, 10, 13, 16, 19, 22, …\) .

Their remainders, when they are divided by \(12\), are \(1, 4, 7\) and \(10.\)

Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
The numbers satisfying condition 2) are \(0, 1, 4, 5, 8, 9, 12, 13, 16, 17, …\) .

Their remainders, when they are divided by \(12\), are \(0, 1, 4, 5, 8\) and \(9\).

Condition 2) is not sufficient, since it doesn’t yield a unique answer.

Conditions 1) & 2)
The numbers satisfying conditions 1) & 2) are \(1, 4, 13, 16, … .\)

Their remainders, when they are divided by \(12\), are \(0\) and \(1.\)

Both conditions together are not sufficient, since they don’t yield a unique answer.

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.
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Overview of GMAT Math Question Types and Patterns on the GMAT 25 Aug 2021, 01:47
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[GMAT math practice question]

(statistics) \(f(x)\) is a function. What is the value of \(f(x)\)?

\(1) f(x)+f(1-x)=7\)

\(2) x + f(\frac{x}{3})= \frac{f(x)}{2}\)
Show more

=>

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 many variables to determine a function and 0 equations, E is most likely to be 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 \(0 + f(0) = (\frac{1}{2})f(0)\) or \(f(0) = 2\), when we substitute \(0\) for \(x\), since we have \(x+f(\frac{x}{3})=\frac{f(x)}{2}.\)

We have \(f(0) + f(1) = 7\), when we substitute \(0\) for \(x\), since we have \(f(x)+f(1-x)=7.\)

When we substitute \(1\) for \(x\) in condition 2), we have \(1+f(\frac{1}{3}) = (\frac{1}{2})f(1)\) or \(f(\frac{1}{3})=(\frac{1}{2})f(1)-1=\frac{7}{2}-1-\frac{5}{2}.\)

When we substitute \(\frac{1}{3}\) for \(x\) in condition 2), we have \(\frac{1}{3}+f(\frac{1}{9})=(\frac{1}{2})f(\frac{1}{3})\) or \(f(\frac{1}{9})=(\frac{1}{2})f(\frac{1}{3})-\frac{1}{3} = (\frac{1}{2})(\frac{5}{2})-\frac{1}{3}=\frac{5}{4}-\frac{1}{3}=\frac{11}{12}.\)

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 when the answer is A, B, C, or D.
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[Math Revolution GMAT math practice question]

(absolute value) Is \(\frac{|x|}{x}\) equal to \(-1\)?

\(1) x>0\)
\(2) x<1\)
Show more

=>

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

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

The question \(\frac{|x|}{x} = -1\) is equivalent to \(x ≤ 0\) as shown below:
\(\frac{|x|}{x} = -1\)
\(=> |x| = -x\)
\(=> x ≤ 0\)

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
If \(x > 0\), then “\(x ≤ 0\)” is always false, and the answer is ‘no’.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
Condition 2) is not sufficient, since the solution set of the question does not include the solution set of condition 2).

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.
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[GMAT math practice question]

(algebra) What is the value of \(xyz\)?

\(1) x+ \frac{1}{y} = 2\)

\(2) y – \frac{1}{z} = \frac{1}{2}\)
Show more

=>

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. 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 \(x + \frac{1}{y} = 2\), we have \(x = 2 – \frac{1}{y} = \frac{(2y-1)}{y}.\)

Since \(y – \frac{1}{z} = \frac{1}{2}\), we have \(\frac{1}{z} = y – \frac{1}{2} = \frac{(2y-1)}{2}\) or \(z = \frac{2}{(2y-1).}\)

Then we have \(xyz = [\frac{(2y-1)}{y}]*y*[\frac{2}{(2y-1)}] = 2.\)

Since both conditions 1) & 2) 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.
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MathRevolution
[GMAT math practice question]

(algebra) \(a≠b\) and \(b≠c\) are given. Is it true that \(a=c\)?

1) \((a-b)(b-c)(c-a) = 0\)

2) \(\frac{(a^2+3a)}{(a+1)} = \frac{(b^2+3b)}{(b+1)} = \frac{(c^2+3c)}{(c+1)}\)
Show more

=>

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) tells that \(a = b\) or \(b =c\) or \(c = a\). However, we have \(a≠b\) and \(b≠c\) from the original condition, so we have \(a = c\).

Thus, condition 1) is sufficient.

Condition 2)
Assume \(\frac{(a^2+3a)}{(a+1)} = \frac{(b^2+3b)}{(b+1)} = \frac{(c^2+3c)}{(c+1)} = k\)

We have \((a^2+3a) = k(a+1), (b^2+3b) = k(b+1)\) and \((c^2+3c) = k(c+1)\)

When we subtract the first two equations, we have
\((a^2+3a) - (b^2+3b) = k(a+1) - k(b+1)\)

=> \((a^2-b^2+3a-3b) = ka+k-kb-k\)

=>\( (a^2-b^2) + 3(a-b) = ka-kb\)

=> \((a^2-b^2) + 3(a-b) = k(a-b) \)

=> \((a+b)(a-b) + 3(a-b) = k(a-b)\)

=> \((a+b+3)(a-b) = k(a-b)\)

=> \(a+b+3 = k\) since \(a≠b\)

When we subtract the last two equations, we have
\((b^2+3b) - (c^2+3c) = k(b+1) - k(c+1)\)

=> \(b^2-c^2+3b-3c = kb+k-kc-k\)

=> \((b^2-c^2) + 3(b-c) = kb-kc\)

=> \((b^2-c^2) + 3(b-c) = k(b-c)\)

=> \((b+c)(b-c) + 3(b-c) = k(b-c)\)

=> \((b+c+3)(b-c) = k(b-c) \)

=> \(b+c+3 = k\) since \(b≠c\)

Since we have \(a+b+3 = k\) and \(b+c+3 = k\), we have \(a = c.\)
Thus condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that D is most likely to be the answer to this question, since each condition includes a ratio.

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

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.
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[GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(m^2-n^2\) divisible by \(4\)?

1) \(m^2+n^2\) has remainder \(2\) when it is divided by \(4\)

2) \(m*n\) is an odd integer
Show more

=>

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

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

The statement “\(m^2-n^2\) is divisible by \(4\)” means that \((m+n)(m-n)\) is divisible by \(4.\) This is equivalent to the requirement that \(m\) and \(n\) are either both even integers or both odd integers.

Since condition 2) tells us that both \(m\) and \(n\) are odd integers, condition 2) is sufficient.

Condition 1)
The square of an odd integer \((2a+1)^2 = 4a^2 + 4a + 1 = 4(a^2 + a) + 1\) has remainder \(1\) when it is divided by \(4\).
The square of an even integer \((2b)^2 = 4b^2\) has remainder \(0\) when it is divided by \(4\).
Thus, if “\(m^2+n^2\) has remainder \(2\) when it is divided by \(4\)”, both \(m\) and \(n\) must be odd integers.
Condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.
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[GMAT math practice question]

(statistics) If \(x > y,\) then what is the median of \(x, y, 9\) and \(9\)?

1) The average (arithmetic mean) of \(x\) and \(y\) is \(9\).

2) The average of \(x, y\) and \(18\) is \(12\).
Show more

=>

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

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions, if necessary.

Condition 1)
Since \(x > y\) and the average of \(x\) and \(y\) is \(9\), we have \(x > 9 > y.\)

Thus, the median of \(x, 9, 9\) and \(y\) is \(9\).

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

Condition 2)
Since \(\frac{( x + y + 18 )}{3} = 12\) or \(x + y + 18 = 36\), the average of \(x\) and \(y\) is \(9.\)

Condition 2) is sufficient by the same reasoning as condition 1).

Therefore, D is the answer.
Answer: D

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

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

(integer) If \(m\) and \(n\) are positive integers, what is the greatest common divisor of \(m\) and \(n\)?

\(1) m=n+1\)
\(2) m*n\) is divisible by \(2\)
Show more

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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.

Since two consecutive integers are always relatively prime, the greatest common divisor of m and n is 1. Thus, condition 1) is sufficient.

Condition 2)
If \(m = 2\) and \(n = 3\), then the greatest common divisor of \(m\) and \(n\) is \(1\).
If \(m = 2\) and \(n = 4,\) then the greatest common divisor of \(m\) and \(n\) is \(2\).
Thus, condition 2) is not sufficient since it does not yield a unique solution.

Therefore, the correct answer is A.
Answer: A
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[GMAT math practice question]

(algebra) \(\frac{m}{n}\) is a fraction. What are the values of \(m\) and \(n\)?

1) the irreducible form of \(\frac{m}{n}\) is \(\frac{3}{4}\)

2) if \(11\) is subtracted from numerator of \(\frac{m}{n}\) and \(4\) is added to denominator of \(\frac{m}{n}\), then the result is \(\frac{2}{5}\)
Show more

=>

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


Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be 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) and 2)
Using condition 1), \(\frac{m}{n} = \frac{3}{4}\), we must have \(4m = 3n.\)

Condition 2) tells us that \(\frac{( m – 11 )}{( n + 4 )} = \frac{2}{5}\). Thus, \(5(m-11) = 2(n+4)\) and \(5m – 55 = 2n + 8.\) Rearranging yields \(5m = 2n + 63\) and \(15m = 6n + 189\).

Since \(6n = 8m, 15m = 8m + 189\), and \(7m = 189\). Thus, \(m = 27\) and \(n = 36.\)

Both conditions together are sufficient.

Therefore, C is the 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.
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[GMAT math practice question]

(geometry) a, b and c are positive numbers. Is a>b-c?

1) a, b, and c are the lengths of three different sides of a triangle
2) a^2+b^2=c^2
Show more

=>

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

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

Condition 1)
If \(a, b\) and \(c\) are the lengths of the three sides of a triangle, we must have

\(a > b – c\) since the length of each side of a triangle is always greater than the difference between the lengths of the other two sides. Thus, condition 1) is sufficient.

Condition 2)
We can assume \(a, b\) and \(c\) are the sides of a right triangle. The above reasoning tells us that \(a > b – c\). Thus, condition 2) is also sufficient.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
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[GMAT math practice question]

(inequality) Five consecutive integers satisfies \(a<b<c<d<e.\) what is the maximum of \(a+e\)?

1) the summation of five integers is negative

2) \(e\) is positive
Show more

=>

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.

Consecutive integers have two variables for the first number and the number of integers. Since the number of integers is \(5\), we need one more equation and D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\(a + b + c + d + e = a + a + 1 + a + 2 + a + 3 + a + 4 = 5a + 10 < 0\) or \(a < -2.\) Then the maximum of a is \(-3\) and \(e = a + 4 = 1.\)

Thus the maximum value of \(a + e = (-3) + 1 = -2.\)

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

Condition 2)
If \(e = 1\), then \(a = -3\) we have \(a + e = -2.\)

If \(e = 2,\) then \(a = -2\) we have \(a + e = 0.\)

If \(e = 3,\) then \(a = -1\) we have \(a + e = 2.\)
…
As \(e\) increases, \(a + e\) increases and approaches infinity.

Thus we don’t have a maximum value of \(a + e.\)

Condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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

(number properties) If \(x\) and \(y\) are positive integers and \(y=\sqrt{5-x}\), then \(y\)=?

\(1) x>1\)
\(2) y<2\)
Show more

=>

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

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

Modifying the original condition:
Note that for \(\sqrt{5-x}\) to make sense, we must have \(x ≤ 5.\)
Also,
\(y=\sqrt{5-x}\),
\(=> y^2 = 5 – x.\)
Since \(y^2 = 5 – x\) is the square of an integer and \(0< x ≤ 5\), the only possible solutions are \(x = 1, y = 2\) and \(x = 4, y = 1.\)
Thus, if a condition allows us to figure out the value of either \(x\) or \(y\), it is sufficient.

Condition 1)
Since \(x > 1\), we must have \(x = 4\) and \(y = \sqrt{5-x} =\sqrt{5-4} = 1.\)
Thus, condition 1) is sufficient since it gives a unique solution.

Condition 2)
Since \(y < 2\), we have \(y = 1\) from the original condition.
Thus, condition 2) is sufficient since it gives a unique solution.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
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[GMAT math practice question]

(statistics) If the median and average (arithmetic mean) of a set of 4 different numbers are both 10, what is the smallest number?

1) The range of the 4 numbers is 10
2) The sum of the smallest and the largest numbers is 20
Show more

=>

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

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

Let \(a, b, c\) and \(d\) be the \(4\) numbers, and suppose \(a < b < c < d.\)

Then \(\frac{( a + b + c + d )}{4} = 10\) and \(\frac{( b + c )}{2} = 10.\)

Since \(b + c = 20\) and \(a + b + c + d = 40\), we must have \(a + d = 20.\)

Condition 1)

Since \(d – a = 10\) by condition 1), we can figure out the values of \(a\) and \(d\). Thus, condition 1) is sufficient.

Condition 2)

\(a + d = 20\) can be deduced from the original condition as shown above.

So, condition 2) provides no additional information.

If \(a = 1, b = 9, c = 11\) and \(d = 19\), then the smallest number is \(1\).

If \(a = 2, b = 9, c =11\) and \(d = 18\), then the smallest number is \(2\).

Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A
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[GMAT math practice question]

(inequality) Is \(3(a-b)>0\)?

\(1) a^3>b^3\)
\(2) a>b\)
Show more

=>

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

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

Asking if \(3(a-b)>0\) is equivalent to asking if \(a – b > 0\), or, equivalently, if \(a > b.\)
Thus, condition 2) is sufficient.

Condition 1)
\(a^3>b^3\)
\(=> a^3-b^3 > 0\)
\(=> (a-b)(a^2+ab+b^2) > 0\)
\(=> a - b > 0\) since \(a^2+ab+b^2 > 0\)
\(=> a > b\)
Thus, condition 1) is sufficient too.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.
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[GMAT math practice question]

(number properties) \(m\) and \(n\) are positive integers greater than \(1\). Is \(m^n\) a perfect square?

1) \(m\) is an odd integer
2) \(n\) is an odd integer
Show more

=>

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

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be 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 \(m = 9\) and \(n = 3\), then \(mn = 9^3 = (3^2)^3 = 3^6 = (3^3)^2 = 27^2,\) which is a perfect square, and the answer is ‘yes’.
If \(m = 3\) and \(n = 3\), then \(mn = 3^3 = (3)^3 = 27\), which is not a perfect square, and the answer is ‘no’.
Both conditions together are not sufficient, since they don’t yield a unique answer.

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.
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