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A photograph rectangular in shape is surrounded by a border that is 2

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Bunuel
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A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   16 Apr 2021, 05:30
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A photograph rectangular in shape is surrounded by a border that is 2 centimeters wide on each side. The combined area of the photograph and the border is a square centimeters . Had the border been 4 centimeters wide on each side, the total area would have been (a + 100) square centimeters. What is the perimeter, in centimeters, of the photograph?

(A) 18
(B) 20
(C) 24
(D) 26
(E) 52
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D
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stne
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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   23 Apr 2021, 22:33
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Bunuel wrote:
A photograph rectangular in shape is surrounded by a border that is 2 centimeters wide on each side. The combined area of the photograph and the border is a square inches. Had the border been 4 centimeters wide on each side, the total area would have been (a + 100) square centimeters. What is the perimeter, in centimeters, of the photograph?

(A) 18
(B) 20
(C) 24
(D) 26
(E) 52



Please look at the figure below :

The green area is the border

Attachment:
GMAT photograph.png
GMAT photograph.png [ 7.1 KiB | Viewed 2956 times ]


The length of the photograph is \(a\) and width is \(b\).

Hence perimeter will be \(2(a+b)\)

When the border is \(2\) cms :
The combined length becomes \(a+4\) and combined width becomes \(b+4\)

Hence area

\((a+4 )(b+4) =a\)
\(ab +4a +4b+16 =a (i)\)

When the border is \(4\) cms :
The combined length becomes \(a+8\) and combined width becomes \(b+8\)

Hence area:

\((a+8 )(b+8 ) =( a+100)\)
\(ab+8a+8b+64 = a+100 (ii)\)

\((ii)-(i) \)

\(4a+4b +48=100\)
\(4a+4b=52\)
\(a+b=13\)

\(2(a+b)=26 =\) Perimeter

Ans=D

Hope it's clear.
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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   29 Apr 2021, 05:13
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ansarose wrote:
In the question it's given
'The combined area of the photograph and the border is a square inches.'
Whereas all others are in cm ...I was baffled about whether this was intentional or an error. Could you confirm this? Bunuel

Posted from my mobile device


ansarose,

You are correct, it should have been square cms, letting the moderator know, hope it will soon be corrected. Thank you.
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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   24 Apr 2021, 14:10
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Bunuel wrote:
A photograph rectangular in shape is surrounded by a border that is 2 centimeters wide on each side. The combined area of the photograph and the border is \(a\) square inches. Had the border been 4 centimeters wide on each side, the total area would have been (a + 100) square centimeters. What is the perimeter, in centimeters, of the photograph?

(A) 18
(B) 20
(C) 24
(D) 26
(E) 52


To summarize, increasing the border by 4 centimeters would increase the total area by 100 square centimeters. (We increase by 2 cm on both sides so the total increase is 4 cm)

Let the current length and width be \(l\) and \(w\).

\((l + 4)(w + 4) = lw + 100\)

\(4L + 4W + 16 = 100\)

\(4L + 4W = 84\)

\(2L + 2W = 42\).

We want the perimeter of the photograph however, so we want \(2(L - 4) + 2(W - 4) = 2L + 2W - 16 = 42 - 16 = 26\).

Ans: D
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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   27 Apr 2021, 05:17
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In the question it's given
'The combined area of the photograph and the border is a square inches.'
Whereas all others are in cm ...I was baffled about whether this was intentional or an error. Could you confirm this? Bunuel

Posted from my mobile device
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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   28 Apr 2021, 00:29
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(L + 4) (W + 4) = A

(L + 8) (W + 8) = A + 100

Expand the quadratics, then subtract equation 1 from equation 2:


4L + 4W + 48 = 100

Perimeter of rectangle = 2L + 2W = 26

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Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink] New post   29 Apr 2021, 07:58
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stne wrote:
ansarose wrote:
In the question it's given
'The combined area of the photograph and the border is a square inches.'
Whereas all others are in cm ...I was baffled about whether this was intentional or an error. Could you confirm this? Bunuel

Posted from my mobile device


ansarose,

You are correct, it should have been square cms, letting the moderator know, hope it will soon be corrected. Thank you.


Fixed the issue. Thank you both!
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gmatclubot
Re: A photograph rectangular in shape is surrounded by a border that is 2 [#permalink]
29 Apr 2021, 07:58
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