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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) If two integers have no common factors other than $$1$$, they are called relatively prime. Are $$x$$ and $$z$$ relatively prime?

1) $$x$$ and $$y$$ are relatively prime.

2) $$y$$ and $$z$$ are relatively prime.

=>

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 $$x = 2, y = 3, z = 5$$, then $$x$$ and $$z$$ are relatively prime, and the answer is ‘yes’.

If $$x = 2, y = 3, z = 2,$$ then $$x$$ and $$z$$ are not relatively prime, and the answer is ‘no’.

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

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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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
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(absolute value) Is $$x<y<z$$?

$$1) |x+2|<y<z+2$$

$$2) |x-2|<y<z-2$$

=>

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.

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)
Now, by condition 1),$$x < x + 2 ≤ | x + 2 | < y < z + 2,$$ and $$x < y$$.
By condition 2), $$|x-2|<y<z-2 < z,$$ and so $$y < z$$.
Thus, both conditions together are sufficient since they yield $$x < y < z$$, and the unique answer is ‘yes’.

Since this question is an inequality 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)
If $$x = 1, y = 4$$, and $$z = 7$$, then the answer is ‘yes’.
If $$x = 1, y = 4,$$ and $$z = 3,$$ then the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
If $$x = 1, y = 4,$$ and $$z = 7$$, then the answer is ‘yes’.
If $$x = 5, y = 4,$$ and$$z = 7,$$ then the answer is ‘no’.
Condition 2) is not sufficient since it doesn’t yield a unique answer.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(function) If the symbol $$#$$ represents one of addition, subtraction, multiplication, or division, what is the value of $$(2 # 1)$$?

1) 2 # 2 = 1

2) (–1/2) # (-1) = $$\frac{1}{2}$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[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

=>

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.

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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(function) If the symbol $$#$$ represents one of addition, subtraction, multiplication, or division, what is the value of $$(2 # 1)$$?

1) 2 # 2 = 1

2) (–1/2) # (-1) = $$\frac{1}{2}$$

=>

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.

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

Condition 1)
$$#$$ is the division operation since $$\frac{2}{2} = 1.$$

So, $$2#1 = \frac{2}{1} = 2.$$

Condition 1) is sufficient since it yields a unique answer.

Condition 2)
$$#$$ could be the multiplication, subtraction or the division operation
since $$(-\frac{1}{2}) * (-1) = \frac{1}{2}, (-\frac{1}{2})-(-1) = (\frac{1}{2})$$ and (-$$\frac{1}{2})/(-1) = \frac{1}{2}.$$

Since $$2 * 1 = 2$$ and $$\frac{2}{1} = 2$$, but $$2-1 = 1$$, condition 2) does not yield a unique value for $$2#1$$.

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

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(inequality) $$0 < a < b < c$$. Is $$a < 3$$?

$$1) \frac{1}{c} > \frac{1}{3}$$

$$2) \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(inequality) Is $$x^3-x^2+x-1 > 0?$$

$$1) x^5 > x^2$$

$$2) x^3 + x > x^2 + 1$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(inequality) $$0 < a < b < c$$. Is $$a < 3$$?

$$1) \frac{1}{c} > \frac{1}{3}$$

$$2) \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$$

=>

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.

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 $$\frac{1}{a} > \frac{1}{b} > \frac{1}{c}$$, since we are given that $$0 < a < b < c.$$

Condition 1) implies that $$c < 3$$. Therefore $$a < 3$$, and Condition 1) is sufficient.

Condition 2)
Since $$\frac{1}{a} > \frac{1}{b} > \frac{1}{c} > 0$$, and $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$$, we must have $$\frac{1}{a} + \frac{1}{a} + \frac{1}{a} = \frac{3}{a} > 1$$. Therefore, $$a < 3$$, and the answer is ‘yes’.
Condition 2) is sufficient since it yields a unique answer.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(number properties) If $$x$$ and $$y$$ are positive integers, are $$x$$ and $$y$$ consecutive?

$$1) x+y=3$$

$$2) xy=2$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(inequality) Is $$x^3-x^2+x-1 > 0?$$

$$1) x^5 > x^2$$

$$2) x^3 + x > x^2 + 1$$

=>

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.

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

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.

$$x^3-x^2+x-1 > 0$$

$$=> x^2(x-1) + (x-1) > 0$$

$$=> (x^2+1)(x-1) > 0$$

$$=> (x-1) > 0$$, since $$x^2+1 > 0$$

$$=> x>1$$

The question asks if $$x > 1.$$

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)
$$x^5 > x^2$$

$$=> x^5 - x^2 > 0$$

$$=> x^2(x^3 – 1) > 0$$

$$=> x^2(x-1)(x^2+x+1) > 0$$

$$=> x – 1 > 0$$ since $$x^2+x+1 >0,$$ and $$x^2 > 0$$ if $$x ≠ 0$$
$$x > 1$$
This condition is equivalent to the question. Therefore, condition 1) is sufficient.

Condition 2)
$$x^3 + x > x^2 + 1$$

$$=> x^3 - x^2 + x – 1 > 0$$

$$=> x^2(x-1)+ (x-1) > 0$$

$$=> (x^2+1)(x-1) > 0$$

$$=> (x-1) > 0$$, since $$x^2+1 > 0$$

$$=> x > 1$$

This condition is equivalent to the question. Therefore, condition 2) is sufficient.

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).

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(number properties) If p and q are prime numbers, what is the number of the different factors of p^2q^3?

1) pq=143
2) p and q are different
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) If $$x$$ and $$y$$ are positive integers, are $$x$$ and $$y$$ consecutive?

$$1) x+y=3$$

$$2) xy=2$$

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.

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 x and y are positive integers, condition 1) tells us that either x = 1 and y = 2, or x = 2 and y= 1. Thus, x and y are consecutive integers.

Since x and y are positive integers, condition 2 tells us that either x = 1 and y = 2, or x = 2 and y= 1. Thus, x and y are consecutive integers.

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).
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(function) At how many points does the parabola y=f(x)=ax^2+bx+c (a≠0) intersect the x-axis?

1) x=f(y) intersects the y-axis at two points.
2) y=f(x-1) intersects the x-axis at two points.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) If p and q are prime numbers, what is the number of the different factors of p^2q^3?

1) pq=143
2) p and q are different

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.

Since p and q are prime numbers, condition 1) tells us that p = 11 and q = 13, or p = 13 and q = 11. Therefore, since p and q are different prime numbers, the number of different factors of p^2q^3 is (2+1)(3+1) = 12. Condition 1) is sufficient since it yields a unique solution.

Condition 2)
Since condition 2) tells us that p and q are different prime numbers, the number of factors of p^2q^3 is (2+1)(3+1) = 12.
Condition 2) is sufficient since it yields a unique solution.

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).
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(function) At how many points does the parabola y=f(x)=ax^2+bx+c (a≠0) intersect the x-axis?

1) x=f(y) intersects the y-axis at two points.
2) y=f(x-1) intersects the x-axis at two points.

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)
Condition 1) is equivalent to the statement that y=f(x) intersects the x-axis at two points, simply by switching the roles of x and y. This means that y=f(x) intersects the x-axis at two points.
Condition 1) is sufficient since it yields a unique solution.

Condition 2)
The graph of y = f(x-1) is the same as the graph of f(x), moved to the right by one unit. This horizontal movement does not change the number of roots of y. Therefore, y = f(x) also has two roots.
Condition 2) is sufficient since it yields a unique solution.

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).
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(inequality) Is x<0?

1) x^3+1<0
2) x^3+2x^2+x+2=0
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Joined: 13 Apr 2018
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Concentration: Finance, Real Estate
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MathRevolution wrote:

Hello, I am Max Lee, Founder and Lead Math Tutor at Math Revolution.

I have over 6,000 posts and almost 5,000 Kudos. This topic is a new feature on the DS Forum and a way for you to directly interact with me and ask anything about the DS, e.g. if you want a certain concept explained or have a particular you question you want me addressed, this is the place to post a link to it or your question. I intend to have this thread be as a "Everything You Need to Know about DS" type of thread. I will keep updating this post with links and resources that are helpful for the DS. Meanwhile, you can ask me anything My other discussions you may be interested in:

Thank you all - good luck on the GMAT and look forward to seeing you in the DS forum!
- Max Lee.

Hi,

Do you have a matrix with variables and equations? i.e. if there are 2 variables and 2 equations you can solve the equation etc

Best
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(absolute value) If |2x|>|3y|, is x >y?

1) x>0
2) y>0
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8437
GMAT 1: 760 Q51 V42
GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(inequality) Is x<0?

1) x^3+1<0
2) x^3+2x^2+x+2=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.

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)
x^3+1<0
⇔ (x+1)(x^2-x+1) < 0
⇔ x+1<0 since x^2-x+1 > 0
⇔ x < -1
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
Thus, since the solution set of the question, ‘x<0’, includes that of condition 1), ‘x<-1’, condition 1) is sufficient.

Condition 2)
x^3+2x^2+x+2=0
⇔ x^2(x+2)+(x+2)=0
⇔ (x^2+1)(x+2)=0
⇔ x+2=0 since x^2+1 > 0
⇔ x = -2
Thus, x < 0 and condition 2) is sufficient.

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.
_________________ Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 19 Jun 2019, 00:18

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# Math Revolution DS Expert - Ask Me Anything about GMAT DS  