Problem 4 · The two cut rectangles
Two different cuts, two equations: adding them solves everything.
Integer answer, at most 4 digitsAnna and Biel each have a rectangle, and the two rectangles are identical. Both cut their rectangle in half; Anna gets two identical rectangles of perimeter 40 cm, and Biel two identical rectangles of perimeter 50 cm. What was the perimeter of the original rectangle?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: the two perimeters differ because one cut is parallel to the base and the other to the height.
Let the rectangle be $a \times b$. Anna halves side $a$: pieces of size $\tfrac{a}{2} \times b$. Biel halves $b$: pieces $a \times \tfrac{b}{2}$.
$$2\left(\tfrac{a}{2} + b\right) = 40, \qquad 2\left(a + \tfrac{b}{2}\right) = 50.$$
That is $\tfrac{a}{2} + b = 20$ and $a + \tfrac{b}{2} = 25$. Adding: $\tfrac{3}{2}(a + b) = 45$, so $a + b = 30$.
$$P = 2(a+b) = 60\ \text{cm}. \quad (a = 20,\; b = 10.)$$
Answer: 60