Rug Sample Question from "Fractions, Decimals, Percents, and Ratios Part 2"

GRE Quant Problem Solving
1

I just started studying for the GRE using the 2 month study guide. Around the 57 minute mark of the video “Fractions, Decimals, Percents, and Ratios Part 2”, the last sample problem asks for the length of an area rug that takes up exactly half the area of a room with a width (w) = 9 ft. and a length (l) = 12 ft.

I understand how we’re able to determine that the area of the rug is 54 sq. ft. (9*12 = 108/2 = 54), and I understand how 9/12 is the ratio of the width to the length of the room, which simplifies to 3/4. What I’m stuck on is why the length is squared. Specifically, Greg writes 54 = (3/4)(l^2). If possible, could someone explain why the length is squared in this solution?

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2

The area of the floor is 54 sq ft. Let its length and width be l and w respectively. We have

lw = 54

Now, the point is that we are trying to get a relationship between w and l, which we’ve found out to be

w = \frac{3}{4} l

Put it back in the first equation, and we get

l \left(\frac{3}{4} l\right) = 54

Can you follow from there?

3

Ahhh, that makes sense now! Thank you so much.