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GMATH Teacher
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11 Jan 2019, 06:28
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(function) In the $$xy$$-plane, a circle has center $$(0,0)$$ and radius $$5$$. Is the point $$(r,s)$$ inside or on the circle?

$$1) -3 < r < 3$$
$$2) -4 < s < 4$$

$$\left( {r,s} \right)\,\,\,\mathop \in \limits^? \,\,\,\left\{ {\,\left( {x,y} \right)\,\,:\,\,{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2} \leqslant {5^2}\,} \right\}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,\,\,{r^2} + {s^2}\,\,\,\mathop \leqslant \limits^? \,\,\,25\,}\,\,$$

$$\left( 1 \right)\,\,\,\,\left| r \right| < 3\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {r,s} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {r,s} \right) = \left( {0,6} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,\left| s \right| < 4\,\,\,\,\left\{ \matrix{ \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {r,s} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {r,s} \right) = \left( {6,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\left\{ \matrix{ \,{r^2} = {\left| r \right|^2} < {3^2} \hfill \cr \,{s^2} = {\left| s \right|^2} < {4^2} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{r^2} + {s^2} < 25\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

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

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)

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11 Jan 2019, 07:09
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(geometry) $$x, y$$ and $$z$$ are the sides of the triangle shown and $$h$$ is its height. Is the perimeter, $$x + y + z$$ of the triangle greater than $$1$$?

$$1) h = \frac{1}{2}$$
$$2) x = y = \frac{1}{3}$$

$$x + y + z\,\,\mathop > \limits^? \,\,\,1$$

$$\left( 1 \right)\,\,\,\left\{ \matrix{ \,x\mathop > \limits^{\left( * \right)} \,\,\,h = {1 \over 2} \hfill \cr \,z\mathop > \limits^{\left( * \right)} \,\,\,h = {1 \over 2} \hfill \cr y > 0 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,x + y + z > 1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

$$\left( 2 \right)\,\,\left\{ \matrix{ \,{\rm{figure}}\,\,{\rm{on}}\,\,{\rm{the}}\,\,{\rm{left}}\,\,\, \Rightarrow \,\,\,\,\,x + y + z\,\,\,\, < \,\,\,\,3\left( {{1 \over 3}} \right)\,\, = \,\,1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr \,{\rm{figure}}\,\,{\rm{on}}\,\,{\rm{the}}\,\,{\rm{right}}\,\,\, \Rightarrow \,\,\,\,\,x + y + z\,\,\,\,\mathop < \limits^{{\rm{as}}\,{\rm{near}}\,\,{\rm{as}}\,\,{\rm{desired}}!} \,\,\,\,{2 \over 3} + {1 \over 3}\left( {\sqrt 2 } \right)\,\, = \,\,{{\sqrt 2 + 2} \over 3}\,\,\,\,\,\left[ { > \,\,\,1} \right]\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$

The correct answer is therefore (A).

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

Regards,
Fabio.

P.S.: it is interesting to study the case in which D does not belong to the side BC!
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)

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Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
GPA: 3.82

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13 Jan 2019, 16:52
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(geometry) $$x, y$$ and $$z$$ are the sides of the triangle shown and $$h$$ is its height. Is the perimeter, $$x + y + z$$ of the triangle greater than $$1$$?

$$1) h = \frac{1}{2}$$
$$2) x = y = \frac{1}{3}$$

Attachment:
1.10.png

=>

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 a triangle, the sum of the lengths of two sides is always greater than the length of the other side.
Thus, from triangle $$ABD$$, we must have $$AB + BD > AD = h$$, and from triangle $$ABD$$, we must have $$AC + CD > AD = h$$. If $$h = \frac{1}{2}$$, then $$AB + BC + CA = AB + BD + DC + CA > \frac{1}{2} + \frac{1}{2} = 1.$$
Condition 1) is sufficient.

Condition 2):
If $$AC = \frac{1}{2}$$, then $$AB + BC + CA = \frac{1}{3} + \frac{1}{3} + \frac{1}{2} = \frac{7}{6} > 1$$ and the answer is ‘yes’.
If $$AC = \frac{1}{4}$$, then $$AB + BC + CA = \frac{1}{3} + \frac{1}{3} + \frac{1}{4} = \frac{11}{12} < 1$$ and the answer is ‘no’.
Since it does not yield a unique solution, condition 2) is not sufficient.

Therefore, the correct answer is A.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6804 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 13 Jan 2019, 17:00 Attachment: 1.14.png [ 22.82 KiB | Viewed 74 times ] MathRevolution wrote: [Math Revolution GMAT math practice question] (function) In the $$xy$$-plane, a circle has center $$(0,0)$$ and radius $$5$$. Is the point $$(r,s)$$ inside or on the circle? $$1) -3 < r < 3$$ $$2) -4 < s < 4$$ => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. The inequality satisfied by points inside or on the circle is $$r^2+s^2≤5^2=25$$. Since we have $$2$$ variables ($$r$$ and $$s$$) and $$0$$ equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. Conditions 1) & 2): Since $$-3<r<3$$ and $$-4<s<4$$, we have $$0≤r^2<3^2=9$$ and $$0≤s^2<4^2=16$$. Thus, $$0≤r^2+s^2<9+16=25$$ and both conditions together are sufficient. Attachment: 1.14.png [ 22.82 KiB | Viewed 74 times ] Therefore, the answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6804
GMAT 1: 760 Q51 V42
GPA: 3.82

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14 Jan 2019, 00:04
[Math Revolution GMAT math practice question]

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

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

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 610 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 14 Jan 2019, 04:51 MathRevolution wrote: [Math Revolution GMAT math practice question] (absolute value, geometry) Is $$a > b - c$$? $$1) |c-b| < a$$ $$2) a, b$$, and $$c$$ are the lengths of the $$3$$ sides of a triangle. $$a\,\,\mathop > \limits^? \,\,b - c$$ $$\left( 1 \right)\,\,\,a\,\, > \,\,\left| {c - b} \right|\,\,\, = \,\,\,\left| {b - c} \right|\,\,\,\, \ge \,\,\,b - c\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,$$ $$\left( 2 \right)\,\, \Rightarrow \,\,\left( 1 \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,$$ The correct answer is therefore (D). This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6804 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 14 Jan 2019, 23:58 [Math Revolution GMAT math practice question] (number properties) If $$x, y$$ are integers, is $$x^2+x+y$$ an odd integer? 1) $$x$$ is an odd integer 2) $$y$$ is an odd integer _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 610

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15 Jan 2019, 05:51
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$x, y$$ are integers, is $$x^2+x+y$$ an odd integer?

1) $$x$$ is an odd integer
2) $$y$$ is an odd integer

$$x,y\,\,{\text{ints}}\,\,\,\left( * \right)$$

$${x^{\text{2}}} + x + y\,\, = \,\,\underbrace {x\left( {x + 1} \right)}_{\left( * \right)\,\,{\text{even}}} + y\,\,\mathop = \limits^? \,\,\,{\text{odd}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,\,y\,\,\mathop = \limits^? \,\,\,{\text{odd}}\,\,}$$

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

$$\left( 2 \right)\,\,y\,\,{\text{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle$$

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

Regards,
Fabio.
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6804
GMAT 1: 760 Q51 V42
GPA: 3.82

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16 Jan 2019, 00:14
MathRevolution wrote:
[Math Revolution GMAT math practice question]

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

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

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

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

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

Condition 2)
Since the sum of the lengths of two sides of a triangle is greater than the length of the third side, we must have $$a + c > b$$. Thus, condition 2) is sufficient.

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.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6804 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 16 Jan 2019, 00:15 [Math Revolution GMAT math practice question] (number properties) If $$x$$ and $$y$$ are integers, is $$x^2-y^2$$ an even integer? 1) $$x^3-y^3$$ is an even integer 2) $$x+y$$ is an even integer _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS &nbs [#permalink] 16 Jan 2019, 00:15

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