the ratio of the length and breadth of a rectangle is 3:4. when the breadth is decreased by 3 cm and length increased by 2 cm, the ratio becomes 1:2 find the length and breadth of the original rectangle
Answers
Let the length be 'l' and the breadth be 'b'
Given
l : b = 2 : 1
Now Let the new length be 'L' and 'B'
Given
L = l + 2
B = b +3
Formula: Perimeter of a rectangle = 2 ( Length + Breadth)
Now
Perimeter of old rectangle say 'p' = 2 ( l + b )
Perimeter of new rectangle say 'P' = 2 ( L + B )
Given
p : P = 4 : 3
=> 2 ( l + b ) : 2 ( L + B ) = 4 : 3
=> ( l + b ) : ( L + B ) = 4 : 3
=> ( l + b ) : ( l + 2 + b +3 ) = 4 : 3
=> ( l +b ) : ( l + b + 5 ) = 4 : 3
\begin{gathered}= > \frac{l+b}{l+b+5}=\frac{4}{3}\\\\= > 3(l+b)=4(l+b+5)\\\\= > l+b=-20\\which\; is\; not \;possible\\\\Hence \;the\;given\;information\;is\;wrong\\\end{gathered}
=>
l+b+5
l+b
=
3
4
=>3(l+b)=4(l+b+5)
=>l+b=−20
whichisnotpossible
Hencethegiveninformationiswrong
Now
Three cases may arise
(i) length and breadth have been decreased instead of increment
(ii) breadth was decreased instead of increment
(iii) ratio 4:3 is P : p rather than p : P
Case i:
Length and breadth have been decreased then
L = l - 2
B = b - 3
P = 2 ( l + b - 5)
Now
p : P = 4 : 3
=>( l + b ) : ( l + b - 5 ) = 4 : 3
\begin{gathered}= > \frac{l+b}{l+b-5}=\frac{4}{3}\\\\= > 3(l+b)=4(l+b-5)\\\\= > l+b=20\\\\Given\\l:b=2:1\\\\= > \frac{l}{b}=2\\\\= > l=2b\\\\Sub\;above\\\\= > 2b+b=20\\\\= > 3b=20\\\\= > b=\frac{20}{3}\\\\= > l=\frac{40}{3}\\\\\end{gathered}
=>
l+b−5
l+b
=
3
4
=>3(l+b)=4(l+b−5)
=>l+b=20
Given
l:b=2:1
=>
b
l
=2
=>l=2b
Subabove
=>2b+b=20
=>3b=20
=>b=
3
20
=>l=
3
40
Hence in this case length = 40/3 m breadth is 20/3 m
Case ii
Breadth was decreased instead of increment then
L = l + 2
B = b - 3
P = 2 ( l + b - 1 )
Now
p : P = 4 : 3
=>( l + b ) : ( l + b - 1 ) = 4 : 3
\begin{gathered}\frac{l+b}{l+b-1}=\frac{4}{3}\\\\by\;observation\\\\l+b=4\\\\Now\; Since \;l=2b\\\\2b+b=4\\\\= > 3b=4\\\\= > b=\frac{4}{3}\\\\= > l=\frac{8}{3}\end{gathered}
l+b−1
l+b
=
3
4
byobservation
l+b=4
NowSincel=2b
2b+b=4
=>3b=4
=>b=
3
4
=>l=
3
8
In this case length is 8/3 m breadth is 4/3 m
Case iii
P : p = 4 : 3
=> ( l + b + 5 ) : ( l + b ) = 4 : 3
\begin{gathered}\frac{l+b+5}{l+b}=\frac{4}{3}\\\\= > 3(l+b+5)=4(l+b)\\\\= > l+b=15\\\\Since\;\;l=2b\\substitute\; above\\\\= > 2b +b = 15\\\\= > 3b=15\\\\= > b=5\\\\= > l=10\\\\\end{gathered}
l+b
l+b+5
=
3
4
=>3(l+b+5)=4(l+b)
=>l+b=15
Sincel=2b
substituteabove
=>2b+b=15
=>3b=15
=>b=5
=>l=10
In this case the length is 10 m and the breadth is 5m