Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.2 Junior Find the answer

What is the value of $k$ if the side lengths of four squares are shown, and the area of the fifth square is $k$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $s$ be the side length of the square with area $k$ .
The sum of the heights of the squares on the right side is $3+8=11$ .
The sum of the heights of the squares on the left side is $1+s+4=s+5$ .
Since the two sums are equal, then $s+5=11$ , and so $s=6$ .
Therefore, the square with area $k$ has side length 6, and so its area is $6^{2}=36$ .
In other words, $k=36$ .

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