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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Equation) For a quadratic equation $$x^2 + px + q = 0$$, what is the value of $$p + q$$?

1) The roots of $$x^2 + px + q = 0$$ are consecutive positive integers.
2) The difference between the squares of the two roots of $$x^2 + px + q = 0$$ is $$25.$$

=>

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.

Thus, look at condition 1).
Assume $$r$$ and $$r+1$$ are roots of the equation $$x^2 + px + q = 0.$$ It tells us that $$p = -25$$ and $$q = 156$$ for the following reason, which is exactly what we are looking for.

$$(r + 1)^2 – r^2 = r^2 + 2r + 1 – r^2 = 2r + 1 = 25$$ or $$r = 12.$$

Then we have $$x^2 + px + q = (x - r)(x -(r + 1)) = x^2 – (r + r + 1)x + r(r + 1) = x^2 – (2r+1)x + r(r+1)$$ and we have $$p = -2r-1 = -25$$ and $$q = r(r+1) = 12*13 = 156.$$

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.

Condition 2)
If $$1$$ and $$2$$ are roots of the equation, then we have $$(x - 1)(x - 2) = x^2 - 3x + 2 = x^2 + px + q, p = -3$$ and $$q = 2.$$

If $$2$$ and $$3$$ are roots of the equation, then we have $$(x - 2)(x - 3) = x^2 - 5x + 6 = x^2 + px + q, p = -5$$ and $$q = 6.$$

The answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, A is the correct answer.
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(Coordinate Geometry) What is the value of $$a + b + c$$?

1) One of roots of the quadratic equation $$ax^2 + bx + c = 0$$ is $$2.$$

2) The intersection of two functions $$y = ax^2$$ and $$y = -bx - c$$ is ($$-1, 2$$).
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Number Property) $$A, B,$$ and $$n$$ are positive integers. What is the value of $$n$$?

1) $$x^4+x^2-n$$ can be factored to $$(x^2+A)(x^2-B)$$

2) $$1 ≤ n ≤ 10$$

=>

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.

Conditions 1) & 2)
$$(x^2+A)(x^2-B) = x4^+(A-B)x^2-AB = x^4+x^2-n$$
Then we have $$A – B = 1$$ and $$AB = n$$
If $$A = 2$$ and $$B = 1$$, we have $$n = AB = 2.$$
If $$A = 3$$ and $$B = 2$$, we have $$n = AB = 6.$$

The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together are not sufficient.

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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Coordinate Geometry) What is the value of $$a + b + c$$?

1) One of roots of the quadratic equation $$ax^2 + bx + c = 0$$ is $$2.$$

2) The intersection of two functions $$y = ax^2$$ and $$y = -bx - c$$ is ($$-1, 2$$).

=>

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.
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 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)

When we replace $$x$$ in the equation $$ax^2 + bx + c = 0$$ with $$2$$ from condition 1), we have $$4a + 2b + c = 0.$$

When we substitute $$x$$ and $$y$$ in the equation $$y = ax^2$$ with $$-1$$ and $$2$$, respectively, we have $$2 = a(-1)^2$$, and $$a = 2$$.

When we substitute $$x$$ and $$y$$ in the equation $$y = -bx - c$$ with $$-1$$ and $$2$$, respectively, we have $$2 = -b(-1) – c,$$ and $$b – c = 2.$$

If we replace a in the equation $$4a + 2b + c = 0$$ with $$2$$, we have $$4(2) +2b +c = 0$$, and $$2b + c = -8.$$

When we add the last two equations, we get $$b – c + 2b + c = 2 – 8, 3b = -6$$ or $$b = -2.$$

If we replace $$b$$ in the equation $$b – c = 2$$ with $$-2$$, we have $$-2 – c = 2, -c = 4$$, and $$c = -4.$$

Thus, we have $$a + b + c = 2 +(-2) + (-4) = -4.$$

The answer is unique, and both conditions combined are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, C is the correct 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: 9164
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(Statistics) The table shows 5 people (Adam, Betty, Carol, David, and Eddy) and the difference between each weight and the average weight of the 5 people. What is the weight of the lightest person among those five people?

Attachment: 6.1 DS.png [ 3.19 KiB | Viewed 269 times ]

1) If another person, Fred, who is 8 kg heavier than Adam, is included, the average is increased by 4%.
2) x is an even prime number.
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Math Revolution GMAT Instructor V
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[GMAT math practice question]

(Number Properties) $$x$$ and $$y$$ are positive integers. $$\sqrt{\frac{y}{x}}$$ is rounded to the closest integer to $$3$$. What is the value of $$x + y$$?

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

2) $$|x - y| = 50.$$
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
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MathRevolution wrote:
[GMAT math practice question]

(Statistics) The table shows 5 people (Adam, Betty, Carol, David, and Eddy) and the difference between each weight and the average weight of the 5 people. What is the weight of the lightest person among those five people?

Attachment:
6.1 DS.png

1) If another person, Fred, who is 8 kg heavier than Adam, is included, the average is increased by 4%.
2) x is an even prime 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.
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 sum of all the differences between the weights and their average is zero, we have $$4 + (-7) + 6 + (-5) + x = 0$$ or $$x = 2.$$

Thus, we don’t need condition 2), since the table already tells us that $$x = 2.$$

Let’s look at the condition 1).

Assume the new average, including Fred, is m’.
We have $$m’ = [5m + \frac{(m + 12)] }{ 6}, m’ = \frac{(6m + 12) }{ 6}, m’ = m + 2.$$ We also $$m’ = 1.04m$$, since we are told the average increases by 4%.
Putting the 2 equations together gives us $$m + 2 = 1.04m.$$
Then we have $$0.04m = 2$$ or $$m = 50.$$
Thus, the lightest weight is Betty’s weight $$m – 7 = 43.$$

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.
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Math Revolution GMAT Instructor V
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Posts: 9164
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[GMAT math practice question]

(Algebra) $$s$$ and $$t$$ are the roots of $$(a^2 + 1)x^2 - 4ax + 2 = 0$$. What is the value of $$a$$?

1) $$s = t$$

2) $$st > 0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) $$x$$ and $$y$$ are positive integers. $$\sqrt{\frac{y}{x}}$$ is rounded to the closest integer to $$3$$. What is the value of $$x + y$$?

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

2) $$|x - y| = 50.$$

=>

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 $$2$$ variables ($$p$$ and $$q$$) and $$0$$ equations, C 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.

Conditions 1) & 2) together give us that:

Since $$\sqrt{\frac{y}{x}}$$ is rounded to the closest integer to $$3$$, we have $$2.5 ≤ \sqrt{\frac{y}{x}} < 3.5 or 6.25 ≤ \frac{y}{x} < 12.25$$ (by squaring all parts of the equation).

Since $$\frac{y}{x} > 1$$, we have $$y > x$$ and $$y = x + 50.$$

$$6.25 ≤ \frac{y}{x} < 12.25$$

⇔ $$6.25 ≤ \frac{[x + 50] }{ x} < 12.25$$

⇔ $$6 .25 ≤ 1 +\frac{ 50}{x} < 12.25$$ (dividing the middle portion by $$x$$)

⇔ $$5.25 ≤ \frac{50}{x} < 11.25$$ (subtracting 1)

⇔ $$\frac{1}{11.25}< \frac{x}{50} ≤ \frac{1}{5.25}$$ (inversing all portions)

⇔ $$\frac{50}{11.25}< x ≤ \frac{50}{5.25}$$ (multiplying by $$50$$)

⇔ $$4.444… < x ≤ 9.523$$ (simplifying)

⇔ $$5 ≤ x ≤ 9.$$ (rounding_

The possible pairs of $$(x, y)$$ are $$(5, 55), (6, 56), (7, 57), (8, 58)$$ and $$(9, 59)$$ since $$y = x + 50$$ from condition 2).

When we consider condition 1), we have two pairs of $$(x, y)$$ which are $$x = 7, y = 57$$ and $$x = 9, y = 59.$$

The answer is not unique, and the conditions are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) and 2) together are not sufficient.

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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Math Revolution GMAT Instructor V
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[GMAT math practice question]

(Equation) If $$A = {x|x^2 - 2(k - 1)x + 4 = 0}$$, what is set $$A$$?

1) $$k$$ is a positive integer.

2) The is $$1$$ element in set $$A$$.
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Joined: 18 Jan 2020
Posts: 1136
Location: India
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MathRevolution wrote:
[GMAT math practice question]

(Equation) If $$A = {x|x^2 - 2(k - 1)x + 4 = 0}$$, what is set $$A$$?

1) $$k$$ is a positive integer.

2) The is $$1$$ element in set $$A$$.

Sir can you confirm whether the x in starting of equation is in multiplication or not.
Also the statement 2.
Thank You.

Posted from my mobile device
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Algebra) $$s$$ and $$t$$ are the roots of $$(a^2 + 1)x^2 - 4ax + 2 = 0$$. What is the value of $$a$$?

1) $$s = t$$

2) $$st > 0$$

=>

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 can figure $$s$$ and $$t$$ out if we know the value of $$a$$, we have $$1$$ variable $$1$$ and $$0$$ equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)
Since two roots are equal, its discriminant is $$0$$. Then, we have $$(-4a)^2 – 4·2(a^2 + 1) = 0 , 16a^2 – 8a^2 – 8 = 0, 8a^2 – 8 = 0, 8(a^2 - 1) = 0, 8(a + 1)(a - 1) = 0.$$

Thus, we have $$a = -1$$ or $$a = 1.$$

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2):
$$a = s = t = 1$$ and $$a = s = t = -1$$ are solutions to the question.
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2)
We have two solutions, $$a = s = t = 1$$ and $$a = s = t = -1$$ even when we consider both conditions together.
The answer is not unique, and the conditions are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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[GMAT math practice question]

(Algebra) What is the value of $$m$$?

1) The difference between the two roots of $$x^2 + (1 + m)x + 20 = 0$$ is $$1.$$

2) $$m > 0.$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Equation) If $$A = {x|x^2 - 2(k - 1)x + 4 = 0}$$, what is set $$A$$?

1) $$k$$ is a positive integer.

2) The is $$1$$ element in set $$A$$.

=>

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

Condition 1) tells us that $$k$$ is a positive integer, from which we cannot determine the unique set of $$A$$. If $$k = 3$$, then $$x^2 - 2(k - 1)x + 4 = x^2 – 2(3 – 1)x + 4 = x^2 - 4x + 4 = (x - 2)^2 = 0$$ and $$A = {2}.$$ However, if $$k = 1$$, then $$x^2 - 2(k - 1)x + 4 = x^2 - 2(1 – 1) + 4 = x^2 + 4 = 0$$, which does not have a root and $$A$$ is an empty set.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that the number of elements in $$A$$ is $$1$$, from which we get that the discriminant of the quadratic equation is 0.

Since the number of roots of the equation is $$1$$, its discriminant
$$(b^2 – 4ac)$$

$$= [2(k – 1)]^2 - 4·1·4$$

$$= 4(k - 1)^2 - 16$$

$$= 4(k^2 - 2k + 1) - 4·4$$

$$= 4(k^2 – 2k + 1 - ) 4$$ (taking out a common factor of $$4$$)

$$= 4(k^2 - 2k - 3)$$

$$= 4(k + 1)(k - 3)$$ is zero.

Then, we have $$4(k + 1)(k - 3) = 0$$ and $$k = -1$$or $$k = 3.$$

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2)
Then only $$k = 3$$ is the unique answer.

The answer is unique, and both conditions together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) 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: 9164
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(Algebra) What is the value of $$m$$?

1) The difference between the two roots of $$x^2 + (1 + m)x + 20 = 0$$ is $$1.$$

2) $$m > 0.$$

=>

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

Condition 1) tells us that the difference between the two roots of $$x^2 + (1 + m)x + 20 = 0$$ is $$1.$$

We can assume $$p$$ and $$p+1$$ are the roots of the equation $$x^2 + (1 + m)x + 20 = 0.$$

We have $$(x - p)(x - (p + 1)) = x^2 - (2p + 1)x + p(p + 1) = x^2 + (1 + m)x + 20.$$

Then, we have $$1 + m = -2p – 1$$ or $$m = -2p – 2$$. We also have $$p(p + 1) = 20$$ or $$p^2 + p - 20 = (p - 4)(p + 5) = 0.$$

Thus $$p = 4$$ or $$p = -5$$, which we can substitute into the first equation giving us $$m = -2p – 2 = -2·4 – 2 = -10$$, or $$m = -2p – 2 = -2·(-5) – 2 = 8 .$$

Then we have $$m = -10$$ or $$m = 8.$$

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that $$m > 0$$, from which we cannot determine the value of $$m$$. For example, $$m$$ can be $$2$$ or $$3.$$

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2) together tell us that the answer, m = 8 is unique, and both conditions are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) 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
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[GMAT math practice question]

(Algebra) $$x$$ and $$y$$ are real numbers. What is the value of $$x$$?

1) $$x^2 - 6xy + 9y^2 + x - 3y = 6$$

2) $$y = 1 + 2√3$$
_________________
Math Revolution GMAT Instructor V
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[GMAT math practice question]

(Geometry) What is the ratio of the area of $$A$$ to the area of $$B$$ in the figure?

Attachment: 6.9DS.png [ 20.87 KiB | Viewed 105 times ]

1) The biggest triangle consists of $$6$$ different isosceles right triangles.

2) $$PQ = 4$$
_________________
Math Revolution GMAT Instructor V
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MathRevolution wrote:
[GMAT math practice question]

(Algebra) $$x$$ and $$y$$ are real numbers. What is the value of $$x$$?

1) $$x^2 - 6xy + 9y^2 + x - 3y = 6$$

2) $$y = 1 + 2√3$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have $$2$$ variables ($$x$$ and $$y)$$ and $$0$$ equations, C 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.

Conditions 1) & 2) together give us:
$$x^2 - 6xy + 9y^2 + x - 3y = 6$$ (from Condition 1)

⇔ $$x^2 - 6xy + 9y2 + x - 3y - 6 = 0$$ (subtracting $$6$$ from both sides)

⇔ $$(x - 3y)^2 + (x - 3y) - 6 = 0$$ (factoring the first $$3$$ terms)

⇔ $$m^2 + m – 6 = 0$$ (substituting m for ($$x - 3y$$))

⇔ $$(m – 2)(m + 3) = 0$$ (trinomial factoring)

⇔ $$(x - 3y - 2)(x - 3y + 3) = 0$$ (substituting ($$x – 3y$$) for $$m$$)

⇔ $$x – 3y – 2 = 0$$ or $$x – 3y + 3 = 0$$

⇔ $$x - 3y = 2$$ or $$x - 3y = -3$$

⇔ $$x = 3y + 2$$ or $$x = 3y – 3$$

⇔ $$x = 3(1 + 2√3) + 2$$ or $$x = 3(1 + 2√3) – 3$$ (substituting ($$1 + 2√3$$) from condition $$2$$ for $$y$$)

⇔ $$x = 5 + 6√3$$ or $$x = 2√3$$ (simplifying)

The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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]

(Function) $$f(x)$$ is a function. What is the value of the non-zero solution($$s$$) of $$f(x) = f(-x)$$?

1) $$f(x)+2f(\frac{1}{x})=3x$$

2) $$x$$ is an irrational number.
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MathRevolution wrote:
[GMAT math practice question]

(Geometry) What is the ratio of the area of $$A$$ to the area of $$B$$ in the figure?

Attachment:
The attachment 6.9DS.png is no longer available

1) The biggest triangle consists of $$6$$ different isosceles right triangles.

2) $$PQ = 4$$

Attachment: 6.9DS(A).png [ 22.63 KiB | Viewed 70 times ]

=>

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.

Let’s look at the condition 1). It tells us that the ratio of the areas of triangles $$A$$ and $$B$$ is $$3:8$$, as shown below:

Assume $$PQ = x.$$

Then $$PQ = PR = x$$ and $$SR = SP = \frac{x}{√2}.$$

We have $$TS = TP = \frac{SP}{√2} = \frac{x}{2}.$$

$$US = UT = \frac{ST}{√2} = \frac{x}{2√2}.$$

Then $$UR = US + SP = \frac{x}{2√2} + \frac{x}{√2} = \frac{3x}{2√2}.$$

The area of triangle $$A$$ is $$(\frac{1}{2})(\frac{3x}{2√2})^2 = \frac{3x^2}{16}.$$

The area of triangle $$B$$ is $$(\frac{1}{2})x^2 = \frac{x^2}{2}.$$

Thus, the ratio of the areas of $$A$$ to $$B$$ is $$\frac{3x^2}{16}$$ to $$\frac{x^2}{2}$$or $$3:8.$$

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2)
We don’t assume anything about $$PR, RU$$, and so on.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition 1) ALONE is sufficient

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