Latest CAT Aptitude Question SOLUTION: How many distinct rectangles having integral sides and a perimeter less than equal to 150 cm are possible

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How many distinct rectangles having integral sides and a perimeter less than equal to 150 cm are possible

Asked In CAT arjun (12 years ago)

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Dr.Dipin Singh  (12 years ago)

L can have values 1 ,2,3..........38
B can have values 1.,2,3,4.......37

so total 38*37=1406

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Dimple  (12 years ago)

L+BB otherwise rectangle with dimensions 37x36 and 36x37 are same. here we are interested in distinct rectangles.
For L=35, 35 rectangles are possible.
..
..
For L=1, only one rectangle 1x1 is possible.

so total rectangles will be

2*(1+2+...37)= 2*37*38/2 = 1406 .. same as given by Dipin.

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Dimple  (12 years ago)

L+BB so rectangle with L=37 and B=38 is not considered
otherwise rectangle with dimensions 37x38 and 38x37 are same. here we are interested in distinct rectangles.
For L=36, 36 rectangles are possible.
For L=35, 35 rectangles are possible.
..
..
For L=1, only one rectangle 1x1 is possible.

so total rectangles will be

2*(1+2+...37)= 2*37*38/2 = 1406 .. same as given by Dipin.

Is this solution Helpfull? Yes (1) | No   

Aashu  (12 years ago)

If length = 1 then breadth = 1
if length = 2 then breadth = 1,2
similarly for length = 37, breadth = 1,2,3,4........,37
after that for length = 38, breadth = 1,2,3,4.......,37
for length = 39, breadth = 1,2,3,4,5......36
and for length = 74, breadth = 1
so total possibilities = 1+2+3+4+........+37+37+36+35+.........+2+1 = 2*37*38/2
= 37*38 = 1406

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Dr.Dipin Singh  (12 years ago)

If L is length and B is breadth, then
as per condition
L+ B

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Dr.Dipin Singh  (12 years ago)

38*37 = 1406

If L is length and B is breadth, then
as per condition
L+ B

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arjun  (12 years ago)

f length=1 then (1,1) (1,2) (1,3) ......(1,74)==74ractangles
if length=2 then (2,1) (2,2) (2,3) ......(2,73)==73 ractangles
similarly 72,71,70....upto1
now if i m doing mistak then please mention it your reply
it should be 2755 ractangles

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Dimple  (12 years ago)

Answer not getting posted completely.
L+B

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arjun  (12 years ago)

thanks to dipin and dimple. i was confuced that should i treat (1,2) and (2,1) dimention ractangles as different or as same bt acording to u it should be same.........

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Jai  (12 years ago)

if one side of rectangle is 1 ... other side can be between 1 and 73 ... total no. of possibilities = 73

if one side is 2 .. other side can be anything between 2 and 72 ... total no. of possibilities = 71

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similalry it will go till combination of 37 and 37 where total possibilities = 1

thus total no of rectangles that can be formed are 1+3+5+.....+73 (Sum of odd nos.)

using geometric progression the sum comes out to be 1369 ...........ANS

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