An automated machine takes any cardboard rectangle and cuts off a square with a side length that is equal to the shorter side length of the rectangle. Julia had a rectangle and, using only the machine, got 3 large squares, 2 medium squares, and 7 small squares with side length 5 cm. What were the dimensions of Julia's rectangle?

Since the rectangle with sides A and B , with B shorter than A , makes a square with sides B . Since the machine can only make squares we know that the machine will cut equal squares until the remaining piece is smaller than a square , then it will cut from the smaller size into more squares. Thus

1°) starting rectangle → 3 large squares and a smaller remainder

2°) length of remainder of the starting rectangle → 2 medium squares and a smaller remainder

2°) length of remainder of the medium rectangle → 7 small squares

then assuming that is no reminder after cutting the 7 squares , we have

medium square side = 7*small squares side = 7* 5 cm = 35 cm

length of the remainder of the starting rectangle = 2* 35 cm + 5 cm (smaller reminder) = 75 cm = length of the large square

thus

length of starting rectangle = 3* 75 cm + 35 cm ( reminder ) = 260 cm

thus the rectangle has dimensions of 260 cm of length and 75 cm height

to verify it, since the machine cuts the cardboard but does not remove any piece , the total area should be conserved . Thus we should verify that

area of starting square = 3* area of large squares + 2* area of medium squares + 7 * area of smaller squares

260 cm * 75 cm = 3*(75 cm)² + 2*(35 cm)² + 7*(5 cm)²

19500 cm² = 19500 cm²

thus our result is correct