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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) If $$k$$ is a positive integer and $$n=(k-1)k(k+1)$$, is $$n$$ a multiple of $$8$$?

1) $$k$$ is an odd number
2) $$k = 1$$

=>

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.

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

Condition 1)
Since $$k$$ is an odd number, $$k – 1$$ and $$k + 1$$ are consecutive even integers.
Any product of consecutive even integers is a multiple of $$8$$.
Thus, condition 1) is sufficient.

Condition 2)
Since $$k = 1$$, we have $$n = (k-1)k(k+1) = 0*1*2 = 0. 0$$ is a multiple of any number, so $$n = 0$$ is a multiple of $$8$$.
Thus, condition 2) is sufficient.

Since this question is a CMT4(B) question. Condition 2) is easy to understand and condition 1) is hard. When one condition is easy to understand, and the other is hard, D is most likely to be the 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.
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Joined: 18 Jul 2018
Posts: 1061
Location: India
Concentration: Finance, Marketing
GMAT 1: 590 Q46 V25 GMAT 2: 690 Q49 V34 WE: Engineering (Energy and Utilities)

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(inequality) Is $$x + \frac{1}{x} > 2$$?

$$1) x > 0$$
$$2) x ≠ 1$$

From statement 1:

x>0. If x = 1. then x+1/x = 2. And 2 is not greater than 2.
If x = 2. Then x+1/x = 2.5>2. hence insufficiennt.

From statement 2:

x not equal to 1.
If x = -1. Then x+1/x = -2, which is not greater than 2.
If x = 2. Then x+1/x = 2.5>2. hence insufficient.

Combining both tell that x>0 and x not equal to 0. Then x+1/x is always greater than 2.
GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(inequality) Is $$x + \frac{1}{x} > 2$$?

$$1) x > 0$$
$$2) x ≠ 1$$

Excellent problem, Max. Congrats! (kudos!)

$$x + {1 \over x}\,\,\mathop > \limits^? \,\,2$$

$$\left( 1 \right)\,\,x > 0\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,x \ne 1\,\,\left\{ \matrix{ \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,x = - 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,0\,\,\mathop < \limits^{x\, \ne \,1} \,\,{\left( {x - 1} \right)^2}\,\, = \,\,\,{x^2} - 2x + 1\,\,\,\,\mathop \Leftrightarrow \limits^{\,x\, > \,\,0} \,\,\,\,0 < {{{x^2} - 2x + 1} \over x} = x - 2 + {1 \over x}\,\,\,\,\, \Leftrightarrow \,\,\,\,x + {1 \over x} > 2\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) $$n$$ is a positive integer. Is $$n$$ divisible by $$3$$?

1) $$\frac{36}{n}$$ is divisible by $$3$$
2) $$\frac{27}{n}$$ is divisible by $$3$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(inequality) Is $$x + \frac{1}{x} > 2$$?

$$1) x > 0$$
$$2) x ≠ 1$$

=>

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.

$$x + \frac{1}{x} > 2$$
$$=> x^3 + x > 2x^2$$ after multiplying both sides by $$x^2$$
$$=> x^3 - 2x^2 + x > 0$$
$$=> x^3 - 2x^2 + x > 0$$
$$=> x(x^2 - 2x + 1) > 0$$
$$=> x(x-1)^2 > 0$$
$$=> x > 0$$ and $$x ≠ 1$$

Thus, we need both conditions together for sufficiency.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(statistics) $$x$$ is a positive number. What is the median of $$x, √x$$ and $$x^2$$?

$$1) x^2=x$$
$$2) x^2+x+1=3x$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) $$n$$ is a positive integer. Is $$n$$ divisible by $$3$$?

1) $$\frac{36}{n}$$ is divisible by $$3$$
2) $$\frac{27}{n}$$ is divisible by $$3$$

=>

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 in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first. It is suggested we plug in numbers when solving remainder problems.

Condition 1)
If $$n = 3$$, then $$\frac{36}{3} = 12$$ is divisible by $$3$$, and $$n$$ is divisible by $$3$$. The answer is ‘yes’.
If $$n = 1$$, then $$\frac{36}{1} = 36$$ is divisible by $$3$$, but $$n$$ is not divisible by $$3$$. The answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
If $$n = 3$$, then $$\frac{27}{3} = 9$$ and $$n$$ is divisible by $$3$$. The answer is ‘yes’.
If $$n = 1$$, then $$\frac{27}{1} = 27$$ and $$n$$ is not divisible by $$3$$. The answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Even if we consider both conditions together, we still have two possible values of $$n: n = 1$$ and $$3$$.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

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: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) If $$a$$ and $$b$$ are integers, is $$a-b$$ an even number?

1) $$a^2b^2$$ is an even number
2) $$a^2+2b^2$$ is an even number
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(statistics) $$x$$ is a positive number. What is the median of $$x, √x$$ and $$x^2$$?

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

=>

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.

If $$x >1$$, then $$√x < x < x^2$$ and $$x$$ is their median.
If $$0 < x <1$$, then $$√x > x > x^2$$ and $$x$$ is their median.
If $$x = 1$$, then $$√x = x = x^2$$ and $$x$$ is their median.
Thus, the question asks for the value of $$x$$.

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)
$$x^2=x$$
$$=> x^2-x=0$$
$$=> x(x-1)=0$$
$$=> x = 0$$ or $$x = 1$$
Since $$x$$ is positive, $$x = 1.$$
Condition 1) is sufficient.

Condition 2)
$$x^2+x+1=3x$$
$$=> x^2-2x+1=0$$
$$=> (x-1)^2=0$$
$$=> x = 1$$.
Condition 2) is sufficient.

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.

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: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(inequality) Is $$x>y$$?

$$1) x+y>2$$
$$2) x^2<2y$$
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Intern  B
Joined: 02 Nov 2017
Posts: 32
Location: India
GPA: 3.87

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(ex 1) A certain store, books are sold. Books are hard cover or soft cover and hard cover books sold $10 each and soft cover books sold$6. Is the number of hard cover books sold greater than that of soft cover books sold?
1) The average price sold of total books is $9 2) The number of hard cover books sold is 100 Math Revolution GMAT Instructor V Joined: 16 Aug 2015 Posts: 8594 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] ### Show Tags MathRevolution wrote: [Math Revolution GMAT math practice question] (number properties) If $$a$$ and $$b$$ are integers, is $$a-b$$ an even number? 1) $$a^2b^2$$ is an even number 2) $$a^2+2b^2$$ is an even number => 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 question: For $$a – b$$ to be an even number, either both a and b must be even numbers or both a and b must be odd numbers. Since we have $$2$$ variables ($$a$$ and $$b$$) 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) From condition 2), a is an even number. From condition 1), b might either be even or odd. Thus, both conditions together are not sufficient, since they do not yield a unique solution. Therefore, the correct answer is E. 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. _________________ Math Revolution GMAT Instructor V Joined: 16 Aug 2015 Posts: 8594 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] ### Show Tags MathRevolution wrote: [Math Revolution GMAT math practice question] (inequality) Is $$x>y$$? $$1) x+y>2$$ $$2) x^2<2y$$ => 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 $$x = \frac{4}{3}$$ and $$y = 1$$, then $$x + y > 2, x^2<2y$$ and $$x > y$$. The answer is ‘yes’. If $$x = 1$$ and $$y =\frac{3}{2},$$ then $$x + y > 2, x^2<2y$$ and $$x < y$$. The answer is ‘no’. Thus, both conditions together are not sufficient, since they do not yield a unique solution. Therefore, the correct answer is E. 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. _________________ Math Revolution GMAT Instructor V Joined: 16 Aug 2015 Posts: 8594 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] ### Show Tags [Math Revolution GMAT math practice question] (number property) [$$x$$] is the greatest integer less than or equal to $$x$$. What is the value of $$x$$? $$1) [x] = 2$$ $$2) x$$ is an integer _________________ Math Revolution GMAT Instructor V Joined: 16 Aug 2015 Posts: 8594 GMAT 1: 760 Q51 V42 GPA: 3.82 Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] ### Show Tags indu1954 wrote: (ex 1) A certain store, books are sold. Books are hard cover or soft cover and hard cover books sold$10 each and soft cover books sold $6. Is the number of hard cover books sold greater than that of soft cover books sold? 1) The average price sold of total books is$9
2) The number of hard cover books sold is 100

Hello, indu1954
We can use 2x2 matrix for this question

________| Hard | Soft
------------------------------
Num. Books | h | s
------------------------------
__Unit Price | 10 | 6

Q: h > s?
⇔ h/s > 1

1) ( 10h + 6s ) / ( h + s ) = 9
2) h = 100

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 A is most likely to be the answer to this question.

Condition 1)
( 10h + 6s ) / ( h + s ) = 9
( 10h + 6s ) = 9( h + s )
10h + 6s = 9h + 9s
h = 3s
h = 3s > s since s > 0 and s is a number of books.
Condition 1) is sufficient.

Condition 2) is not sufficient, since we don't have any information about s.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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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 $$y$$=?

$$1) x>1$$
$$2) y<2$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) [$$x$$] is the greatest integer less than or equal to $$x$$. What is the value of $$x$$?

$$1) [x] = 2$$
$$2) x$$ is an integer

=>

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 $$[x] = n$$.

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)
Since there are a lot of integers, condition 2) does not yield a unique solution. This condition is not sufficient.

Conditions 1) & 2)
$$x = 2$$ is the unique integer such that $$2 ≤ x < 3.$$
Thus, both conditions together are 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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number property) If $$n$$ is a positive integer, is $$\sqrt{n+1}$$ an integer?

1) $$n$$ is a multiple of $$8$$
2) $$n$$ is the product of $$2$$ consecutive even numbers
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

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

$$1) x>1$$
$$2) y<2$$

=>

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.

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: 8594
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) If $$m$$ and $$n$$ are positive integers, is $$mn$$ an even number?

1) $$\frac{m}{n}$$ is an even number.
2) $$m + n$$ is an even number.
_________________ Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 13 Dec 2018, 05:08

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