# Two rectangles

I cut out two rectangles with 54 cm², 90 cm². Their sides are expressed in whole centimeters. If I put these rectangles together I get a rectangle with an area of 144 cm2. What dimensions can this large rectangle have? Write all options. Explain your calculation.

Correct result:

n =  6

#### Solution:

$54=ab \ \\ 90=bc \ \\ \ \\ b=90/c \ \\ \ \\ 54c=90a \ \\ 3c=5a \ \\ \ \\ a=3, b=18, c=5 \ \\ a=6, b=9, c=10 \ \\ a=9, b=6, c=15 \ \\ a=18, b=3, c=30 \ \\ a=27, b=2, c=45 \ \\ a=54, b=1, c=90 \ \\ \ \\ S=b \cdot \ (a+c) \ \\ S_{1}=18 \cdot \ 8=144 \ \text{cm}^2 \ \\ S_{2}=9 \cdot \ 16=144 \ \text{cm}^2 \ \\ S_{3}=6 \cdot \ 24=144 \ \text{cm}^2 \ \\ S_{4}=3 \cdot \ 48=144 \ \text{cm}^2 \ \\ S_{5}=2 \cdot \ 72=144 \ \text{cm}^2 \ \\ S_{6}=1 \cdot \ 144=144 \ \text{cm}^2 \ \\ \ \\ n=6$

The equations have the following integer solutions:
54=ab
90=bc

Number of solutions found: 6
##### a1=3, b1=18, c1=5a2=6, b2=9, c2=10a3=9, b3=6, c3=15a4=18, b4=3, c4=30a5=27, b5=2, c5=45a6=54, b6=1, c6=90

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