In a residential plan, in scale, when using a conventional ruler, it is noticed that the sides of the rectangular room measure, exactly, 16 centimeters and 9 centimeters.
If the actual area of the room in question is equal to 36 square meters, then the actual perimeter of the room is equal to:
a) 21 meters b) 19 meters c) 20 meters d) 25 meters e) 22 meters
The answer is 25 meters. Alternative "d)"
I tried the following: I went by the trial method, that is, "playing" with the alternatives and seeing which one "fits" the question. I saw that it was the alternative "d" because the rectangle could be formed like this: 4.5 in base and 8 in height. So the perimeter is 4.5 + 4.5 + 8 + 8 = 25 meters and the area would be: 4.5 x 8 = 36 square meters
I want to know how would you guys do it? What would the traditional method look like?
I would proceed as follows. Let $x$ be the scaling factor of the plan, then the true side lengths of the room would be $16x$ and $9x$ respectively. Thus the actual area is $$16x\times 9x=36\iff x^2=\frac14\implies x=\frac12,$$ from which one deduces that the perimeter is simply $2\times(9x+16x)=50x=25$.