Question 1187561
let A = the length of the living room.
let B = the width of the living room.
the area of the living room is equal to A * B.


let C = the length of the rug.
let D = the width of the rug.
the area of the rug is equal to C * D.


you are given that the length of the rug is half the length of the room.
you are given that the width of the rug is half the width of the room.


this makes C = 1/2 * A
this makes D = 1/2 * B


since the area of the rug is equal to C * D, then the area of the rug must also be equal to A/2 * B/2 = AB/4.


this makes me believe that the area of the rug is 1/4 * the area of the room.


the ratio can be expressed as:


area of rug / area of living room = 1/4.


to see if this holds true, then make the length of the room equal to a random number that looks reasonable.


i chose 12 feet.


since the length is twice the width, then the width has to be 6 feet.


the area of the room is equal to 12 * 6 = 72 square feet.


the length of the rug is half the lengh of the room, so the length of the rug is 6 feet.


the width of the rug is half the width of the room, so the width of the rug is 3 feet.


the area of the rug is 6 * 3 = 18 square feet


18 square feet / 72 square feet is equal to  1/4.


the ratio of the area of the rug to the area of the room is 1/4.


i don't believe the fact that the width of the room is half the length of the room is important.


i think the ratio would have been the same regardless, as long as the width of the rug was half the width of the room as well as half the length of the room.


for example:


assume the room was square.
then the length of the room is equal to the width of the room.
the area of the room is L * W
if the length of the rug is half the length of the room and half the width of the room, then the room and the rug are in proportion, as before.


the area of the rug will be 1/2 * L * 1/2 * W = 1/4 * L * W.


this will hold for any values of L and W of the room as long as the dimensions of the rug are both half the dimensions of the room.


assuming the length of the room is 12 and the width of the room is 12, then the area of the room is 144 square.


when the length of the rug is 6 and the width of the rug is 6, then the area of the rug is 36 square.


36 / 144 = 1/4 once again.