Question 103990
Let L=length of the rectangle, w=width of the rectangle, {{{P[s]}}}= perimeter of the square, and {{{P[r]}}}= perimeter of the rectangle



Let's start with what we know and what we are given



This is what we are given:

{{{L=4+S}}} "The length of a rectangle is 4m more than the side of the square"


{{{W=S-2}}} "the width of the rectangle is 2m less than the side of the square




And this is what we know :

Perimeter of the square:
{{{P[s]=4S}}}  

 Perimeter of the rectangle
{{{P[r]=2L+2W}}}



Remember, the problem says: "The perimeter of the rectangle is 24m less than twice the perimeter of the square". So this means the perimeter of the rectangle is 



{{{P[r]=2P[s]-24}}}


since {{{P[s]=4S}}}, we can substitute that into {{{P[r]=2P[s]-24}}}. 




{{{P[r]=2(4S)-24}}} Plug in {{{P[s]=4S}}}



{{{P[r]=8S-24}}} Multiply 



Since {{{P[r]=8S-24}}} and {{{P[r]=2L+2W}}}, set them equal to each other


{{{8S-24=2L+2W}}}


{{{8S-24=2(4+S)+2(S-2)}}} Now plug in {{{L=4+S}}} and {{{W=S-2}}}



{{{8S-24=8+2S+2S-4}}} Distribute





{{{8S-24=4S+4}}} Combine like terms on the right side



{{{8S=4S+4+24}}}Add 24 to both sides



{{{8S-4S=4+24}}} Subtract 4S from both sides



{{{4S=4+24}}} Combine like terms on the left side



{{{4S=28}}} Combine like terms on the right side



{{{S=(28)/(4)}}} Divide both sides by 4 to isolate S




{{{S=7}}} Divide


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Answer:

So our answer is {{{S=7}}}




So the side length of the square is 7m. This means the length of the rectangle is:


{{{L=4+S=4+7=11}}}


So the length is 11m



This also means the width of the rectangle is:


{{{W=S-2=7-2=5}}}


So the width is 5m




Check:


First find the perimeter of the square:

{{{P=4S=4(7)=28}}}


Now multiply 28 by 2 and subtract 24 to get...


{{{28*2-24=32}}}


Now find the perimeter of the rectangle


{{{P=2L+2W=2(11)+2(5)=22+10=32}}}


Since the perimeter of the rectangle is 24 less than twice the perimeter of the square, our answer is verified