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Math Revolution GMAT Instructor
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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MathRevolution wrote:
[GMAT math practice question]

(algebra) What is the value of \(\frac{(3x+y)}{(x-3y)}\)?

\(1) 2x-y = 2\)

\(2) 3x-y = 0\)


=>

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.

When the question asks for a ratio, a fraction, a percent, a proportion or a rate, if one of conditions provides a ratio and the other condition provides a number, the condition with a ratio could be sufficient.

This question asks for a ratio.
Condition 1) provides a number and condition 2) provides the ratio, \(\frac{x}{y} = 2.\)
Thus, condition 2) is likely to be sufficient.

Condition 1) :
If \(x = 1\) and \(y = 0,\) then \(\frac{(3x+y)}{(x-3y)} = \frac{(3+0)}{(1-0)} = 3.\)

If \(x = 2\) and \(y = 2\), then \(\frac{(3x+y)}{(x-3y)} = \frac{(6+2)}{(2-6)} = \frac{8}{(-4)} = -2.\)

Since we do not obtain a unique answer, condition 1) is not sufficient.

Condition 2) :
Since \(3x = y, \frac{(3x+y)}{(x-3y)} = \frac{(3x+3x)}{(x-9x)} = \frac{6x}{(-8x)} = -(\frac{3}{4})\)
Condition 2) is sufficient.

Therefore, B is the answer.
Answer: B
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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MathRevolution wrote:
[GMAT math practice question]

(algebra) What is the value of \((x+y)(y+z)(z+x)+5\)?

\(1) xyz=-3\)

\(2) x+y+z=0\)


=>

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 \(x + y + z = 0\) from condition 2), \(x + y = -z, y + z = -x,\) and \(z + x = -y.\) Since condition 1) tells us that \(xyz = -3, (x+y)(y+z)(z+x)+5 = (-x)(-y)(-z) + 5 = -xyz + 5 = -(-3) + 5 = 8.\)

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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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(number properties) What is the value of \(\frac{2a}{(a+1)}+\frac{2b}{(b+1)}\)?

1) \(a, b\) are integers

2) \(ab=1\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(absolute value) On the number line, the mid-point of the two integers \(x\) and \(y\) is \(4\). What is the value of \(y\)?

1) the absolute value of \(x\) is \(6\)
2) \(x < y\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(number properties) What is the value of \(\frac{2a}{(a+1)}+\frac{2b}{(b+1)}\)?

1) \(a, b\) are integers

2) \(ab=1\)


=>

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 \(\frac{2a}{(a+1)} + 2b(b+1) = (2a*(b+1) + \frac{2b(a+1))}{(a+1)(b+1)} = \frac{(2ab + 2a + 2ab + 2b)}{(ab+a+b+1)} = \frac{(4ab + 2a + 2b)}{(ab+a+b+1)},\) the question asks the value of \(\frac{(4ab + 2a + 2b)}{(ab+a+b+1).}\)

Condition 1) is obviously not sufficient as it tells us nothing about the values of \(a\) and \(b.\)

Condition 2):
If \(ab = 1\), then \(\frac{(4ab + 2a + 2b)}{(ab+a+b+1)} = \frac{(2a+2b+4)}{(a+b+2)} = \frac{2(a+b+2)}{(a+b+2)} = 2.\) Condition 2) is sufficient.

Therefore, the answer is B.
Answer: B

Originally posted by MathRevolution on 11 Sep 2019, 00:57.
Last edited by MathRevolution on 15 Jul 2021, 03:42, edited 1 time in total.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(number properties) Let \(aob\) denote the greatest common divisor of \(a\) and \(b\), and let \(a□b\) denote the least common multiple of \(a\) and \(b.\) What is \((xoy)□(x□y)\)?

\(1) (63o99)x=540\)

\(2) 3y-(18□45)=0\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(absolute value) On the number line, the mid-point of the two integers \(x\) and \(y\) is \(4\). What is the value of \(y\)?

1) the absolute value of \(x\) is \(6\)
2) \(x < y\)


=>

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 the mid-point of x and y is 4, (x+y)/2 = 4, and x + y = 8.

Since we have 2 variables (x and y) and 1 equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since |x|=6, we must have x=6 or x=-6.
Since 4 lies midway between x and y, we must have either x=6 and y=2, or x=-6 and y=14.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
There are many possible pairs of integers, x and y, satisfying condition 2) and the original condition.
Examples are x=1 and y=7, and x=2 and y=6.
Condition 2) is not sufficient since it does not yield a unique solution.

Conditions 1) & 2)
Condition 1) tells us that x=6 and y=2, or x=-6 and y=14.
Since condition 2) tells us that x<y, we must have x=-6 and y=14.
Both conditions together are sufficient, since they yield a unique solution.

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.

Originally posted by MathRevolution on 13 Sep 2019, 08:11.
Last edited by MathRevolution on 15 Jul 2021, 03:42, edited 1 time in total.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(absolute values) What is the value of \(2x-y\)?

\(1) |3x-2y+4| + |-x+2y-2|=0\)

\(2) x\) and \(y\) are rational numbers
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(number properties) Let \(aob\) denote the greatest common divisor of \(a\) and \(b\), and let \(a□b\) denote the least common multiple of \(a\) and \(b.\) What is \((xoy)□(x□y)\)?

\(1) (63o99)x=540\)

\(2) 3y-(18□45)=0\)


=>

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 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)

The following reasoning shows that condition 1) implies that x = 60:
(63o99)x=540
=> (9*7o9*11)x=540
=> 9x=540
=> x=60

The following reasoning shows that condition 2) implies that y = 60:
3y-(18□45)=0
=> 3y-(9*2□9*5)=0
=> 3y-9*2*5=0
=> 3y-90=0
=> y-30=0
=> y=30

Thus, we may calculate
(xoy)□(x□y) = (60o30)□(60□30) = 30□60=60

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.

However, each of the conditions only gives us information about one of the variables, so neither is sufficient on its own.

Therefore, C is the answer.
Answer: C

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

(number properties) \(a, b\) and \(c\) are integers. What is the value of \(a-b-c\)?

\(1) |a|<|b|<|c|\)

\(2) a*b*c=12\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(absolute values) What is the value of \(2x-y\)?

\(1) |3x-2y+4| + |-x+2y-2|=0\)

\(2) x\) and \(y\) are rational numbers


=>

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) is equivalent to two equations: \(3x-2y+4=0\) and \(–x+2y-2 = 0\). When we solve this system of occasions, we obtain \(x = -1\) and \(y = \frac{1}{2}\). Thus, condition 1) is sufficient on its own.
Note the VA tells us this will be the case. The question has two variables, but condition 1) provides \(2\) equations.

Condition 2) is clearly not sufficient on its own.

Therefore, the answer is A.
Answer: A
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(number properties) \(a, b\) and \(c\) are integers. What is the value of \(a-b-c\)?

\(1) |a|<|b|<|c|\)

\(2) a*b*c=12\)


=>

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 (\(a, b\) and \(c\)) 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 \(a=1, b=2\) and \(c=6\), then \(|a| < |b| < |c|, abc = 12\) and \(a-b-c=-7.\)
If \(a=-1, b=-2\) and \(c=6\), then \(|a| < |b| < |c|, abc = 12\) and \(a-b-c=-5.\)

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

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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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(number properties) \((k, p)\) denotes the remainder when \(k\) is divided by \(p\). What is the value of \((mn, 5)\)?

\(1) (m, 5)=4\)

\(2) (n, 5)=2\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(algebra) \(<x,y>\) denotes \(x + \frac{y}{2}.\) What is the value of \(x\)?

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

\(2) <2x,2y>+1=<y,x> - 2\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(number properties) \((k, p)\) denotes the remainder when \(k\) is divided by \(p\). What is the value of \((mn, 5)\)?

\(1) (m, 5)=4\)

\(2) (n, 5)=2\)


=>

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 \(2\) variables (\(m\) and \(n\)) 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)
We have \(m = 5a + 4\) and \(n = 5b + 2\) for some integers \(a\) and \(b\).

Then we have \(mn = (5a+4)(5b+2) = 25ab + 10a + 20b + 8 = 5(5ab+2a+4b+1)+3\) and \(mn\) has a remainder \(3\) when it is divided by \(5.\)

Thus, 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)
We have \(m = 5a + 4\) from condition 1)
If we have \(m = 4\) and \(n = 1,\) then \(mn = 4\) has a remainder \(4\) when it is divided by \(5.\)

If we have \(m = 4\) and \(n = 2\), then \(mn = 8\) has a remainder \(3\) when it is divided by \(5\).

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

Condition 2)
We have \(m = 5b + 2\) from condition 1)
If we have \(m = 1\) and \(n = 2\), then \(mn = 2\) has a remainder \(2\) when it is divided by \(5.\)

If we have \(m = 4\) and \(n = 2,\) then \(mn = 8\) has a remainder \(3\) when it is divided by \(5.\)
Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, C is the answer.
Answer: C

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

Originally posted by MathRevolution on 18 Sep 2019, 00:21.
Last edited by MathRevolution on 25 Jun 2021, 02:59, edited 1 time in total.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(algebra) What is the value of \((x+y)(2x-y)-(x-y)(2x+y)\)?

\(1) x=\frac{3}{2}\)

\(2) xy=2\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(algebra) \(<x,y>\) denotes \(x + \frac{y}{2}.\) What is the value of \(x\)?

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

\(2) <2x,2y>+1=<y,x> - 2\)


=>

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.

When we simply condition 2), we have
\(<2x, 2y>+1=<y, x>-2\)

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

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

\(=> (\frac{3}{2})x = -3\)

\(=> x = -2.\)

Condition 2) is sufficient.

Condition 1)
Since we have \(<x,y> = y + \frac{x}{2} = x + \frac{y}{2}\), we have \(\frac{y}{2} = \frac{x}{2}\) or \(x = y.\)

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

Therefore, B is the answer.
Answer: B

Originally posted by MathRevolution on 19 Sep 2019, 01:12.
Last edited by MathRevolution on 13 Jul 2021, 00:25, edited 1 time in total.
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