The pattern shown above is composed of rectangles. This pattern is used repeatedly to completely cover a rectangular region 12L units long and 10L units wide. How many rectangles of dimension L by W are needed?

answer choices:
A) 30
36
C) 100
D) 150
E) 180 << that's the answer.

i don't get how to arrive at this solution, thanks!

we can see the dimensions of the pattern are 2L x (L+W)= 2L x (5/3 L). luckily, if you use 6 of these as the length, and 6 as the width (creating a 6x6 array of this pattern), the dimensions would be (2L * 6) x (5/3 L * 6) = 12L x 10L. thus there would be 36 of the pattern shown, and since there are 5 L x W rectangles in each pattern, there are 36 * 5 =180 L x W rectangles needed.

Bigb14: Your reply to Urc is incorrect. Urc VERIFIED that there was a solution. You did not. Urc's answer is better (proved) as a result. pckeller has the same problem. 180 means nothing without the Urc's verification.

I substituted.
2L = 3w
L = 3/2w
Area of Rectangle = L x w = 3/2w x w = 3/2w^2
Area of Rectangular Region = 12L x 10L = 18w x 15w = 270w^2
(270w^2)/(1.5w^2) = 180.

so theres 12 rectangles going down vertically and then 18 going down horizontally in each line. So I just did (28/3)*18=168 and +12 = 180. Different method probably not as efficient as those mentioned above.

This thread offers a perfect example of the benefits of using this site. Some will view this problem as very basic and others as very hard. This thread shows 3-4 various ways to solve the problem through simple (and correct) approaches. This is not unusual for the SAT and students who learn to balance straight high school "techniques" with reasoning (or graphical) approaches will really help themselves.

## Replies to: sat math problem from new BB second ed.

therefore, each rectangle is L x 2/3L = 2/3 L^2

Total area is 120L^2

120L^2/ (2/3)L^2 = 180

we can see the dimensions of the pattern are 2L x (L+W)= 2L x (5/3 L). luckily, if you use 6 of these as the length, and 6 as the width (creating a 6x6 array of this pattern), the dimensions would be (2L * 6) x (5/3 L * 6) = 12L x 10L. thus there would be 36 of the pattern shown, and since there are 5 L x W rectangles in each pattern, there are 36 * 5 =180 L x W rectangles needed.

Say you use L=6 and W=4. That makes the areal of the big region 12L x 10L = 72 x 60 = 4320.

Then, since each little rectangle has area 4 x 6 = 24, then the number of them you need is 4320 / 24 = 180

2L = 3w

L = 3/2w

Area of Rectangle = L x w = 3/2w x w = 3/2w^2

Area of Rectangular Region = 12L x 10L = 18w x 15w = 270w^2

(270w^2)/(1.5w^2) = 180.

so

W+x(1.5W)=15W

x(1.5W)=14W

x=28/3

so theres 12 rectangles going down vertically and then 18 going down horizontally in each line. So I just did (28/3)*18=168 and +12 = 180. Different method probably not as efficient as those mentioned above.

Now just apply the simple strategy "to see how many 2-dimensional objects fit inside another 2-dimensional object, divide the areas."

So the answer is (12L*10L)/6 = (12*3*10*3)/6 = 180.