Perimeter of rectangle and equilateral triangle. Please help

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 12:52 am

If the perimeter of a rectangle is less than the perimeter of equilateral triangle then which one of the following is true?

I. The larger side of the rectangle must be greater than the side of the triangle.

II. The smaller side of the rectangle must be less than the side of the triangle.

III. The small side of the rectangle may be equal to the side of the triangle.

IV. The triangle and the rectangle can share one side.

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Source: Beat The GMAT — Problem Solving | Thu Jun 04, 2015 12:52 am

theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Thu Jun 04, 2015 1:40 am

nikhilgmat31 wrote:If the perimeter of a rectangle is less than the perimeter of equilateral triangle then which one of the following is true?

I. The larger side of the rectangle must be greater than the side of the triangle.

II. The smaller side of the rectangle must be less than the side of the triangle.

III. The small side of the rectangle may be equal to the side of the triangle.

IV. The triangle and the rectangle can share one side.

Here is the start:

perimeter of rectangle < perimeter of triangle
by assigning variables to the sides
2ra+2rb <3t
2(ra+rb)<3t
ra+rb < 1.5t

Process of elimination
I. The larger side of the rectangle must be greater than the side of the triangle.
ra+rb < 1.5t
2 +1 < 1.5*4=6 -> FALSE

III. The small side of the rectangle may be equal to the side of the triangle.
ra+rb < 1.5t
2 +1 < 1.5*1=1.5 - > FALSE

Last edited by theCEO on Thu Jun 04, 2015 2:22 am, edited 2 times in total.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 1:58 am

Options III and IV seems same to me

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 2:11 am

Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Thu Jun 04, 2015 2:34 am

nikhilgmat31 wrote:Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

Thanks for the question nikhilgmat31

II. The smaller side of the rectangle must be less than the side of the triangle
ra+rb < 1.5t

if we set both sides to being equal,
ra+ra = 1.5t
2 ra = 1.5t
ra = 0.75 t
this shows that one of the sides of rect. is always going to be smaller than side of triangle

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 2:36 am

so II is the correct answer ?

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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Thu Jun 04, 2015 3:10 am

nikhilgmat31 wrote:Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

I think IV is possible

IV. The triangle and the rectangle can share one side
ra+rb < 1.5t
1+0.4<1.5*1 -> true

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 3:12 am

introduces confusion.

I thought we finished question

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 3:31 am

1 & 0.4 is basically same as Option II. side of triangle is 1

II. The smaller side of the rectangle must be less than the side of the triangle.

so answer is again II

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Thu Jun 04, 2015 6:03 am

Say the equilateral triangle has sides of length x. Its perimeter is 3x.

Say the rectangle has dimensions L and W, where W < L. Its perimeter is 2W + 2L. If we've assumed W < L, then the rectangle's perimeter is thus greater than 2W + 2W = 4W.

We know that the triangle's perimeter exceeds the rectangle's perimeter, which, as we just saw, exceeds 4W. So 3x > 4W, and x > (4/3)W. Since W is positive, (4/3)W is definitely bigger than W, so if x > (4/3)W, then x > W is true, which is what II says.

It is also possible for the rectangle and triangle to share one side, though that side needs to be the length of the rectangle. If, say the sides of the triangle are 1, 1, 1, then the length of the rectangle can certainly be 1. The perimeter of the rectangle will then be less than that of the triangle as long as the width of the rectangle is less than 0.5. So IV can also be true.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Thu Jun 04, 2015 9:27 am

Hi nikhilgmat31,

What is the source of this question? Is it from a GMAT book or a "math" book?

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Thu Jun 04, 2015 8:43 pm

I took a mock test from egamt.com & got this question there.

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