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MathRevolution
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The Ultimate Q51 Guide [Expert Level] 02 Jan 2022, 02:42
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MathRevolution
[Math Revolution GMAT math practice question]

(number properties) \(x\) and \(y\) are positive integers. Is \(y\) an even integer?

\(1) x^2+x=y+2\)
\(2) x = 2\)
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.

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):
Since \(y = x^2 + x – 2\) and \(x = 2\), we have \(y = 4\).
Since this answer is unique, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since \(y = x^2 + x – 2 = (x-1)(x+2)\) and \(x\) is an integer, one of \(x – 1\) and \(x + 2\) is an even integer.
Thus, \(y\) is always an even integer and condition 1) is sufficient.

Condition 2)
Since it provides no information about y, condition 2) is not sufficient.


Therefore, A is the answer.
Answer: A

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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[GMAT math practice question]

(Number properties) \(m\) and \(n\) are positive integers such that \(m(n+10) = 75.\) What is the value of \(m\)?

1) \(n\) is not less than \(m\)

2) \(m\) is not a prime number
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 (\(m\) and \(n\)) and \(1\) equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We can find two pairs of solutions: \(m = 1\) and \(n = 65\), and \(m = 3\) and \(n = 15\).

Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
We can find only one pair of solutions: \(m = 1\) and \(n = 65.\)

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

Therefore, B is the answer.
Answer: B

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 \(a^3 + a^2b + ab^2 + b^3\)?

1) \(a + b = 2\sqrt{2}, ab = 1\)

2) \(a = \sqrt{2}+1 \)
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.

We have
\(a^3 + a^2b + ab^2 + b^3\)

\(= a^3 + 3a^2b + 3ab^2 + b^3 - (2a^2b + 2ab^2)\)

\(= (a + b)^3 – (2a^2b + 2ab^2)\)

\(= (a + b)^3 – 2ab(a + b)\)

Then we have \(a^3 + a^2b + ab^2 + b^3 = (a+b)^3 – 2ab(a + b) = (2\sqrt{2})^3 – 2*1*( 2\sqrt{2}) = 16\sqrt{2} - 4\sqrt{2} = 12\sqrt{2}\) from condition 1).

Thus, condition 1) is sufficient. since condition 1) has \(2\) equations

Condition 2)
Since we don’t have any information about b, condition 2) is not sufficient.

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

(Algebra) What is the value of \(a^2 + b^2 + c^2 – ab – bc - ca\)?

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

2) \(b-c=\sqrt{2}\)
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.

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

Since we have \(3\) variables (\(x, y,\) and \(z\)) and \(0\) equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

The question \(a^2 + b^2 + c^2 – ab – bc - ca\) is equivalent to \(\frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2}\) for the following reason
\(a^2 + b^2 + c^2 – ab – bc - ca\)
\(= \frac{1}{2}{2a^2 + 2b^2 + 2c^2 – 2ab – 2bc - 2ca}\)
\(= \frac{1}{2}{(a^2-2ab + b^2)+(b^2-2bc + c^2)+(c^2 - 2ca+a^2)}\)
\(=\frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2}\)

Conditions 1) & 2)
Since we have \(a-b=-1\) and \(b-c=\sqrt{2}\), we have \(c - a = -( c – b + b – a ) = -( √2 + (-1)) = 1 - √2.\)

\(a^2 + b^2 + c^2 – ab – bc - ca\)

\(= \frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2} \)

\(= \frac{1}{2}{(-1)^2+\sqrt{2}^2+(1-\sqrt{2})^2}\)

\(= \frac{1}{2}(1+2+3-2\sqrt{2})\)

\(= 3 - \sqrt{2}\)

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

Therefore, C is the answer.
Answer: C

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

(number properties) \(k\) is a positive integer. When \(2^{\frac{-k}{2}} ≤ \frac{1}{1},024, k=?\)

1) \(k\) is an integer less than \(21\)
2) \(k\) is an integer greater than \(12\)
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.

\(2^{\frac{-k}{2}}≤\frac{1}{1,024}\)
\(=> 2^{\frac{k}{2}}≥1,024\)
\(=> 2^{\frac{k}{2}}≥1,024=2^{10}\)
\(=> \frac{k}{2} ≥ 10\)
\(=> k ≥ 20\)
We have the simplified original condition \(k ≥ 20\).

Condition 1)
“\(k < 21\)” yields the unique solution \(k = 20\), since \(k\) is an integer and \(k ≥ 20\). Condition 1) is sufficient.

Condition 2)
We have \(k > 12\) from condition 2) and \(k ≥ 20\) from the original condition.
The integer \(k\) has many possible values.
Since it does not yield a unique solution, condition 2) is not sufficient.

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

(inequality) Is \(x < 0\)?

\(1) x^3 + 1 < 0\)
\(2) x^3 + x + 1 < 0\)
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 (\(x\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\(x^3 + 1 < 0\)
\(=> (x+1)(x^2-x+1) < 0\)
\(=> x + 1 < 0\) since \(x^2-x+1 > 0\)
\(=> x < -1 < 0\)
Thus, condition 1) is sufficient, and the answer is ‘yes’.

Condition 2)
\(x^3 + x + 1 < 0\)
\(=> x^3 + x < -1\)
\(=> x(x^2 + 1) < -1\)
\(=> x < \frac{-1}{(x^2 + 1)}\) since \(x^2 + 1 > 0\)
\(=> x <\frac{-1}{(x^2 + 1)} < 0\) since \(x^2 + 1 > 0\)
Thus, condition 2) is sufficient, and the answer is ‘yes’.

Therefore, D is the answer.
Answer: D
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The Ultimate Q51 Guide [Expert Level] 22 Jan 2022, 02:32
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[GMAT math practice question]

(functions) Is \(ax + 2y - 3 = 4x + by + 5\) an equation of a line on the xy-plane?

1) \(a ≠ 4.\)

2) \(b ≠ 2.\)
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.

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.
\(ax + 2y - 3 = 4x + by + 5\)

\(⇔ (a - 4)x + (2 - b)y – 8 = 0.\)

If we have \(a = 4\) and \(b = 2\), then the equation is equivalent to \(-8 = 0\), which is not an equation of a line.

So, each condition alone is sufficient.

Therefore, D is the answer.
Answer: D
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The Ultimate Q51 Guide [Expert Level] 26 Jan 2022, 03:46
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MathRevolution
[Math Revolution GMAT math practice question]

(absolute values) If \(y=|x-1|+|x+1|,\) then \(y=\)?

\(1) x>-1\)
\(2) x<1\)
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.

There are three ranges of values of x to consider.
If \(x > 1\), then \(y = | x – 1 | + | x + 1 | = x – 1 + x + 1 = 2x\) and we don’t have a unique value of \(y\).
If \(-1 ≤ x ≤ 1\), then \(y = | x – 1 | + | x + 1 | = - ( x – 1 ) + x + 1 = 2\) and we have a unique value of \(y\).
If \(x < 1\), then \(y = | x – 1 | + | x + 1 | = -( x – 1 ) – ( x + 1 ) = -2x\) and we don’t have a unique value of \(y\).

Asking for the value of \(y\) is equivalent asking if \(-1 ≤ x ≤ 1\).
Both conditions yield the inequality \(-1 < x < 1\), when applied together. Therefore, both conditions are sufficient, when taken together.

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.
Therefore, C is the answer.
Answer: C
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The Ultimate Q51 Guide [Expert Level] 29 Jan 2022, 10:04
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[Math Revolution GMAT math practice question]

(absolute value, geometry) Is \(a > b - c\)?

\(1) |c-b| < a\)
\(2) a, b\), and \(c\) are the lengths of the \(3\) sides of a triangle.
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.

Condition 1) is sufficient since it yields \(b – c ≤ | b - c | = | c – b | < a.\)

Condition 2)
Since the sum of the lengths of two sides of a triangle is greater than the length of the third side, we must have \(a + c > b\). Thus, condition 2) 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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The Ultimate Q51 Guide [Expert Level] 01 Feb 2022, 07:30
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[GMAT math practice question]

(number properties) \(a, b, c\) and \(d\) are integers. Is \(abcd + abc + ab + a\) an even number?

1) \(abc\) is an odd integer

2) \(bcd\) 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. We should simplify the conditions if necessary.

Modifying the question:
For \(abcd + abc + ab + a = a(bcd+bc+b+1)\) to be even, either a must be even or \(bcd + bc + b + 1\) must be even.

Condition 2):
If \(bcd\) is an odd integer, then \(b, c\) and \(d\) are odd integers. This implies that \(bc\) is odd, and \(bcd + bc + b + 1\) is an even integer. Condition 2) is sufficient.

Condition 1)
If \(a = b = c = d = 1\), then \(abcd + abc + ab + a = 1 + 1 + 1 + 1 = 4\), which is an even integer, and the answer is ‘yes’.
If \(a = b = c = 1\) and \(d = 2\), then \(abcd + abc + ab + a = 2 + 1 + 1 + 1 = 5\), which is an odd integer, and the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique solution.

Therefore, B is the answer.
Answer: B
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The Ultimate Q51 Guide [Expert Level] 04 Feb 2022, 01:22
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[Math Revolution GMAT math practice question]

(exponents) \(m+n=?\)

\(1) (4^m)(2^n)=16\)
\(2) (2^{2m})(4^n)=64\)
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.

Condition 2) is equivalent to \(m + n = 3\) as shown below:
\((2^{2m})(4^n)=64\)
\(=> (2^{2m})(2^{2n})=2^6\)
\(=> 2^{2m+2n}=2^6\)
\(=> 2m+2n = 6\)
\(=> m + n = 3\)
Condition 2) is sufficient.

Condition 1)
\((4^m)(2^n)=16\)
\(=> (2^{2m})(2^n)=2^4\)
\(=> 2^{2m+n}=2^4\)
\(=> 2m+n = 4\)
If \(m = 1\) and \(n =2\), then \(m + n = 3\).
If \(m = 0\) and \(n = 4,\) then \(m + n = 4.\)
Since it does not yield a unique solution, condition 1) is not sufficient.

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

(number properties) If \(m\) and \(n\) are positive integers, is \(\frac{m}{n}\) a terminating decimal?

1) \(m\) is divisible by \(9\)

2) \(n\) is divisible by \(30\)
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 \(n = 30\), then \(\frac{m}{n} = \frac{9}{30} = \frac{3}{10} = 0.3\) is a terminating decimal and the answer is ‘yes’.
If \(m = 9\) and \(n = 210,\) then \(\frac{m}{n} = \frac{9}{210} = \frac{3}{70}\) is not a terminating decimal since the denominator, \(70\), has a prime factor other than \(2\) and \(5\). The answer is ‘no’.
Both conditions together are not sufficient, since they don’t yield a unique solution.

Therefore, E is the answer.
Answer: E

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

(number property) If \(p\) and \(q\) are prime numbers, what is the number of factors of \(6pq\)?

1) \(p\) and \(q\) are different

2) \(p<3<q\)
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.

Recall that if \(n = p^aq^br^c\), where \(p, q\) and \(r\) are different prime numbers, and \(a, b\) and \(c\) are non-negative integers, then \(n\) has \((a+1)(b+1)(c+1)\) factors.

Condition 2)
We must have \(p = 2\) since \(p\) is prime and \(p < 3.\)

The prime factorization of \(6pq\) is \(2*3*p*q = 2^2*3*q\) since \(q\) is prime and \(q > 3\).

The number of factors of \(2^2*3*q is (2+1)(1+1)(1+1) = 12\). Condition 2) is sufficient since it yields a unique answer.

Condition 1)
If \(p = 2\) and \(q = 3\), then \(6pq = 2^2*3^2\) and the number of factors is \((2+1)(2+1)= 9.\)

If \(p = 5\) and \(q = 7\), then \(6pq = 2*3*5*7\) and the number of factors is \((1+1)(1+1)(1+1)(1+1) = 8.\)

Condition 1) is not sufficient since it does not yield a unique solution.

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

(number property) If \(a, b\), and \(c\) are integers, is \(a+b+c\) an even integer?

1) \(a^2+b^2\) is an even integer
2) \(b^2+c^2\) is an even 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 \(3\) 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)
If \(a = 1, b = 1\) and \(c = 1\), then \(a + b + c = 3\) is not an even integer and the answer is ‘no’.

If \(a = 2, b = 2\) and \(c = 2\), then \(a + b + c = 6\) is an even integer and the answer is ‘yes’.

Since the conditions don’t yield a unique answer when applied together, they are not sufficient.

Therefore, E is the answer.
Answer: E

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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The Ultimate Q51 Guide [Expert Level] 05 Mar 2022, 04:14
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[Math Revolution GMAT math practice question]

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

\(1) x>0\)
\(2) x<1\)
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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.

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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The Ultimate Q51 Guide [Expert Level] 10 Mar 2022, 10:35
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[Math Revolution GMAT math practice question]

(number properties) What is the value of the integer \(n\)?

1) \(n\) is a prime factor of \(21\)
2) \(n\) is a factor of \(49\)
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.

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)
\(n\) is a prime factor of \(21 = 3*7\) and \(n\) is \(3\) or \(7\).
Since it does not give a unique answer, condition 1) is not sufficient.

Condition 2)
If \(n\) is a factor of \(49 = 7^2\), then \(n\) is \(1, 7\) or \(49\).
Since it does not give a unique answer, condition 2) is not sufficient.

Conditions 1) & 2)
The unique integer satisfying both conditions is \(n = 7.\)
Both conditions are sufficient when taken together.

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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The Ultimate Q51 Guide [Expert Level] 13 Mar 2022, 04:11
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[GMAT math practice question]

(algebra) What is the value of the integer \(a\)?

1) \(x - (\frac{2}{3})(x-4a) = 7\) has a positive integer solution

2) \(a\) 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.

Since we have \(2\) variables (\(x\) and \(a\)) 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)
\(x - (\frac{2}{3})(x - 4a) = 7\) is equivalent to \(3x – 2(x-4a) = 21\) or \(x = 21 – 8a.\)

The possible pairs \((x,a)\) are \((13,1)\) and \((5,2).\)

Since both conditions together don’t yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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The Ultimate Q51 Guide [Expert Level] 21 Mar 2022, 02:54
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[Math Revolution GMAT math practice question]

(number property) \([x]\) is the greatest integer less than or equal to \(x\). \(<x>\) is the least integer greater than or equal to \(x\). What is the value of \(x\)?

\(1) [x] = 2\)
\(2) <x> = 2\)
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.

\([x]\) is analyzed as follows.
If \(n ≤ x < n + 1\) for some integer \(n\), then \([x] = n.\)
\(<x>\) is analyzed as follows.
If \(n – 1 < x ≤ n\) for some integer \(n\), then \(<x> = n.\)

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

Condition 1)
\([x] = 2\)
\(=> 2 ≤ x < 3\)
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
\(<x> = 2\)
\(=> 1 < x ≤ 2\)
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only \(x = 2\) satisfies both conditions.
Since the answer is unique, 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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The Ultimate Q51 Guide [Expert Level] 26 Mar 2022, 10:14
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[GMAT math practice question]

(statistics) If the median of \(5\) positive integers is \(10\), is their average (arithmetic mean) greater than \(10\)?

1) The largest number is \(40\)
2) The smallest number is \(1\)
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.

Suppose the numbers satisfy \(a ≤ b ≤ 10 ≤ c ≤ d\). The question asks if \(\frac{( a + b + 10 + c + d )}{5} > 10\) or \(a + b + c + d + 10 > 50.\)
This is equivalent to the inequality, \(a + b + c + d > 40.\)
If a question includes the words “greater than”, then it asks us to look for a minimum.
Since \(a, b c,\) and \(d\) are positive, and \(d = 40\) by condition 1), we must have \(a + b + c + d > 40.\)
Condition 1) is sufficient.

Condition 2)
If \(a = 1, b = 2, c = 11\), and \(d = 40\), then \(a + b + c + d > 40\), and the answer is ‘yes’.
If \(a = 1, b = 2, c = 11\), and \(d = 12\), then \(a + b + c + d < 40\), and the answer is ‘no’.
Thus, condition 2) is not sufficient since it does not yield a unique solution.

Therefore, A is the answer.
Answer: A
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The Ultimate Q51 Guide [Expert Level] 30 Mar 2022, 03:00
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Que: A color “code” is defined as a sequence of three dots arranged in a row. Each dot is colored either “red” or “black.” How many distinct codes can be formed?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 1
Show more


Solution: Number of ways arranging ‘n’ objects out of which ‘r’ objects are identical is n!/ r!

Case I: Let us assume we use two BLACK and 1 RED dot. Thus, the total way of arranging them is 3!/2! = 3

Case II: Similarly, let us assume we use two RED and 1 BLACK dot. Thus, the total way of arranging them is 3!/2! = 3

Case III: All three BLACK: 3!/3! = 1

Case IV: All three RED: 3!/3! = 1

Total number of codes: 3 + 3 + 1 + 1 = 8

The first dot can use RED or BLACK, the second dot also can use RED or BLACK, and the third dot also can do, so 2*2*2=8

Therefore, D is the correct answer.

Answer D
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