Question 1187561: Can someone please help me with this problem? I’ve been struggling with it for a while now and my Geometry textbook doesn’t explain it very well.
Here is the problem:
Brett is buying a rug for his living room. His living room is a rectangle with a length that is twice as long as its width. Brett decides that the rug will also be a rectangle, but it will be only one half the length of the room and one half as wide. What will be the ratio of the rug’s area to the living room’s area?
Found 2 solutions by greenestamps, Theo:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
If the rug is half the length of the room and also half the width, then the rug and the room are similar figures (scale models of each other) with a scale ratio 1:2.
In any similar figures, if the ratio of linear measurements (e.g., length and width) is a:b, then the ratio of area measurements is a^2:b^2.
In your problem, since the scale factor is 1:2, the ratio of area measurements is 1:4.
ANSWER: The ratio of the rug's area to the area of the room is 1:4
It will help to understand this concept if you put some example numbers to the problem.
room 10*16; area 160; rug 5*8; area 40; 40 is 1/4 of 160
room 12*20; area 240; rug 6*10; area 60; 60 is 1/4 of 240
etc., etc....
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! 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.
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