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# A rectangular photo has a 2mm thick frame uniformly around it. If the

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A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 10:08
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GMATbuster's Weekly Quant Quiz#10 Ques #4

A rectangular photo has a 2mm thick frame uniformly around it. If the frame size is reduced to 1 mm, the total area of the photograph with the frame is reduced by 20 mm^2. If the length and breadth of the photograph are integers and if the length is greater than breadth, find the ratio of the length and breadth of the photograph including the original frame.
A) 6/5
B) 7/5
C) 5/3
D) 2/1
E) 3/1

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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 10:16
Answer E, from given info l+b = 4, and since l>b, l=3, b=1
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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 10:25
1
Let L be length and B be breadth, given that L and B are integers and L > B.
Original frame will have length = L + 2 + 2 = L + 4 and breadth = B + 2 + 2 = B + 4
Smaller frame will have length = L + 1 + 1 = L + 2 and breadth = B + 1 + 1 = B + 2

So, as per condition given in question:
(L + 4)*(B + 4) - (L + 2)*(B + 2) = 20
LB + 4B + 4L + 16 - LB - 2B - 2L - 4 = 20
2*(L + B) + 12 = 20
L + B = 4
Now, possible combinations of integers for L and B can be 1 + 3 = 4, 2 + 2 = 4 or 3 + 1 = 4.
But it is given that L > B, so only case possible is 3 + 1 = 4. Hence L = 3 and B = 1

Ratio asked = (L + 4) / (B + 4) = (3+4)/(1+4) = 7/5

IMO Option B! Is there a shortcut though?
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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 10:45
E.
Original Area =(X+4)*(Y+4)
since thickness of frame will add on to both sides.

New thickness = (X+2)*(Y+2)

subtracting both :
2X + 2Y +12 =20
X+Y =4

Now since its given length is more so L=3 and B=1

So option E
X+Y

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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 11:03
Let photograph be LxB
Thus
(L+4)(B+4)-(L+2)(B+2)=20
2(L+B)+12= 20
L+B= 4
Since both are integers and L>B thus L=3, B= 1
Hence ratio=3:1

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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 11:25
gmatbusters wrote:

GMATbuster&#39;s Weekly Quant Quiz#10 Ques #4

A rectangular photo has a 2mm thick frame uniformly around it. If the frame size is reduced to 1 mm, the total area of the photograph with the frame is reduced by 20 mm^2. If the length and breadth of the photograph are integers and if the length is greater than breadth, find the ratio of the length and breadth of the photograph including the original frame.
A) 6/5
B) 7/5
C) 5/3
D) 2/1
E) 3/1

Let the length of photo be x
Let the breadth of photo be y
Area of photo -xy
Photo has a frame of 2 mm thick around it ,making the length as (x+4) and breadth (y+4)
Area of photo with frame-(x+4)(y+4)
Now the frame size is reduced to 1mm ,making the new area of photo with frame as (x+2)(y+2)
The total area of the photo with frame has been reduced to 20mm2
(x+4)(y+4)-20=(x+2)(y+2)
Solving tbe above equation will refuce to 2x+2y=8
x+y=4
y= 4-x
The ration of length to breadth becomes x/4-x
Checking all the options ,only option E satisty the above equation

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Re: A rectangular photo has a 2mm thick frame uniformly around it. If the  [#permalink]

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24 Nov 2018, 14:41
Area of uniform rectangular photo frame with length l, breadth b and width w is given by (l+2w)(b+2w) ............... (I)

Difference in area of photo of width 2mm and photo of width 1mm = 20 mm^2 ............. Given ................ (II)

Using (I) and (II), (l+4)(b+4) - (l+2)(b+2) = 20
that is, lb + 4l + 4b + 16 - lb - 2l - 2b - 4 = 20
that is, 2l + 2b = 8
that is, l + b = 4 ........... (III)

Length and breadth both are integers and length is greater than breadth ............. Given ........ (IV)

Using (III) and (IV), only possible values of length and breadth are 3 and 1 respectively.

We need to find ratio of length and breadth along with original frame,

So, Ratio = (3+2*2)/(1+2*2) = 7/5.

Hence, B is the correct answer.
Re: A rectangular photo has a 2mm thick frame uniformly around it. If the   [#permalink] 24 Nov 2018, 14:41
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