Geometry question

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w0283767: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Wed Apr 30, 2008 12:12 pm; Thanked: 5 times

by w0283767 » Mon Aug 25, 2008 5:23 pm

This is from MGMAT geometry page 22.

If the perimeter of a rectangular flower bed is 30 feet, and its area is 44 square feet, what is the length of each of its shorter sides?

I know how to solve this problem, but i am trying to learn the way MGMAT solves this problem by using the substitution method.

44= l x w => l = (44/w)

30= 2(l + w)

30 = 2 (44/w + w)

I understand up until this part, but then they say to multiply the entire equation by w/2. I don't understand why.

Once they do that they get

15w = 44 + w^2

w^2 - 15w + 44 = 0 I get the quadratic part but how do they make
the jump from the original equation to get the
(w - 11) (w - 4) = 0 15w = 44 + w^2?

Thanks in advance, any input would be greatly appreciated.
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Source: Beat The GMAT — Problem Solving | Mon Aug 25, 2008 5:23 pm

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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

Re: Geometry question

by Ian Stewart » Mon Aug 25, 2008 6:50 pm

w0283767 wrote:
44= l x w
30= 2(l + w)

I find that solution unnecessarily complicated. From the above, we know

lw = 44
l + w = 15

That is, we are looking for two numbers that multiply to 44 and add to 15. We look for pairs of numbers like this every time we factor a quadratic, and in this case the numbers must be 4 and 11.

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

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w0283767: Junior | Next Rank: 30 Posts; Posts: 22; Joined: Wed Apr 30, 2008 12:12 pm; Thanked: 5 times

thanks

by w0283767 » Mon Aug 25, 2008 6:55 pm

Thank you, but the object of my question is to enhance my math not find the easy solution. I am also shooting for a 700 and I believe knowing this substitution trick can help out on more difficult questions.

Do you agree that it would be a useful skill or should i not worry about it.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

Re: thanks

by Stuart@KaplanGMAT » Mon Aug 25, 2008 11:25 pm

w0283767 wrote:Thank you, but the object of my question is to enhance my math not find the easy solution. I am also shooting for a 700 and I believe knowing this substitution trick can help out on more difficult questions.

Do you agree that it would be a useful skill or should i not worry about it.

Knowing how to substitute to solve equations is a key skill for the GMAT. However, the question you posted seems overly complicated, at least the way it's explained.

If the perimeter of a rectangular flower bed is 30 feet, and its area is 44 square feet, what is the length of each of its shorter sides?

So:

lw = 44
l + w = 30
l = 30 - w

(30 - w)w = 44
30w - w^2 = 44
0 = w^2 -30w + 44

is a MUCH simpler way to get to the same equation. By starting with the addition equation, we keep the whole thing much simpler.

As an aside, I 100% agree with Ian - no matter how good you are at math, either using common sense or backsolving would be a far far quicker way of solving this question. From the answer choices, this question could be backsolved in under 30 seconds.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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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

Re: thanks

by Ian Stewart » Tue Aug 26, 2008 1:03 am

Stuart's approach to substitution here is much better than that taken in the original solution to the problem. I'd just point out one typo, in case anyone finds it confusing-

Stuart Kovinsky wrote: l + w = 30

should read l + w = 15 (the perimeter is 30, so the length and width sum to 15), and the remaining 30s should all be 15s.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

Re: thanks

by Stuart@KaplanGMAT » Tue Aug 26, 2008 12:02 pm

Ian Stewart wrote:Stuart's approach to substitution here is much better than that taken in the original solution to the problem. I'd just point out one typo, in case anyone finds it confusing-

Stuart Kovinsky wrote: l + w = 30
should read l + w = 15 (the perimeter is 30, so the length and width sum to 15), and the remaining 30s should all be 15s.

Thanks!

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kandyhot27: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Sun Apr 20, 2008 3:23 pm

Re: Geometry question

by kandyhot27 » Wed Aug 27, 2008 4:30 am

w0283767 wrote:This is from MGMAT geometry page 22.

If the perimeter of a rectangular flower bed is 30 feet, and its area is 44 square feet, what is the length of each of its shorter sides?

I know how to solve this problem, but i am trying to learn the way MGMAT solves this problem by using the substitution method.

44= l x w => l = (44/w)

30= 2(l + w)

30 = 2 (44/w + w)

I understand up until this part, but then they say to multiply the entire equation by w/2. I don't understand why.

Once they do that they get

15w = 44 + w^2

w^2 - 15w + 44 = 0 I get the quadratic part but how do they make
the jump from the original equation to get the
(w - 11) (w - 4) = 0 15w = 44 + w^2?

Thanks in advance, any input would be greatly appreciated.
[/list]

they are complicating things by stating the obvious.
For solving
30 = 2 (44/w + w)
indirectly you are multiplying w/2 on both sides anyways.
Take it in steps you would first do the following:

15 = 44/w + w

and then multiply by w on both sides to get the quadratic equation
Jumping from there to (w-11) (w-4) is just skipping two steps to solve the quadratic equation. Just matter of practice

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