Maths Olympiad Prep

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Algebra Difficulty 2.4 Junior Find the answer

In a magic square, the numbers in each row, the numbers in each column, and the numbers on each diagonal have the same sum. In the magic square shown, what is the value of xx?

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

Solution

Using the second row, we see that the sum of the numbers in each row, column and diagonal must be 3.6+3+2.4=93.6 + 3 + 2.4 = 9. Since the sum of the numbers in the first column must be 9, then the bottom left number must be 92.33.6=95.9=3.19 - 2.3 - 3.6 = 9 - 5.9 = 3.1. Since the sum of the numbers in the top left to bottom right diagonal must be 9, then the bottom right number must be 92.33=95.3=3.79 - 2.3 - 3 = 9 - 5.3 = 3.7. Since the sum of the numbers in the bottom row must be 9, then 3.1+x+3.7=93.1 + x + 3.7 = 9 and so x+6.8=9x + 6.8 = 9 or x=96.8=2.2x = 9 - 6.8 = 2.2. We can complete the magic square as shown: \begin{tabular}{|c|c|c|} \hline 2.3 & 3.8 & 2.9 \\ \hline 3.6 & 3 & 2.4 \\ \hline 3.1 & 2.2 & 3.7 \\ \hline \end{tabular}

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