Problem of fifty - cent coins

Geometry Level pending

A certain number of fifty-cent coins is to form an equilateral triangle. The same number of fifty-cent coins can also be used to form a square . The number of fifty-cent coins on each side of the square is 6 fewer than the number fifty-cent coins on each side of the equilateral triangle. How many fifty-cent coins are there altogether?

62 75 36 72 52

Sin Ndt by SIN NDT , 14, Vietnam

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1 solution

If n n coins are there between the two coins at two vertices of the triangle, then the total number of coins in the triangle is 3 n + 3 3n+3 . There are n − 6 n-6 coins between two coins at adjacent vertices of the square. So the total number of coins forming the square is 4 ( n − 6 ) + 4 4(n-6)+4 . Therefore we have 3 n + 3 = 4 ( n − 6 ) + 4 ⟹ n = 23 3n+3=4(n-6)+4\implies n=23 . Hence the total number of coins is 3 × 23 + 3 = 72 3\times 23+3=\boxed {72} .

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