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70.a complete the magic square. -5.6 -1.3 3 0.02 answer -5.6 -1.3 3 0.02

Question

70.a
complete the magic square.
-5.6
-1.3
3 0.02
answer
-5.6
-1.3
3 0.02

Explanation:

Step1: Find the magic constant (sum of third row)

The third row has values 3, \( x \) (let the middle cell of third row be \( x \)), and 0.02. First, we can find the sum of the main diagonal or use the property that in a 3x3 magic square, the sum of each row, column, and diagonal is equal. Let's first find the sum of the third row once we know the middle cell? Wait, no, let's use the center cell. In a magic square, the magic constant \( S = 3 \times \) center cell. Wait, is that true for any magic square? Wait, no, that's for a magic square with distinct numbers, but here we have decimals and integers. Wait, actually, the center cell is -1.3, so the magic constant \( S = 3 \times (-1.3) = -3.9 \). Let's verify.

So each row, column, diagonal should sum to -3.9.

Let's check the third row: 3 + \( x \) + 0.02 = -3.9. So \( x = -3.9 - 3 - 0.02 = -6.92 \). Wait, no, wait: 3 + \( x \) + 0.02 = S, and S = 3*(-1.3) = -3.9. So 3 + x + 0.02 = -3.9 → x = -3.9 - 3 - 0.02 = -6.92? Wait, that can't be right. Wait, maybe I made a mistake. Wait, let's recalculate.

Wait, center cell is -1.3, so magic constant S = 3*(-1.3) = -3.9. So each row, column, diagonal sums to -3.9.

Third row: 3 + (middle of third row) + 0.02 = -3.9 → middle of third row = -3.9 - 3 - 0.02 = -6.92. Wait, that seems odd, but let's proceed.

Now, first column: (top left) + (middle left) + 3 = -3.9.

Second column: (top middle) + (-1.3) + (middle of third row) = -3.9. So (top middle) + (-1.3) + (-6.92) = -3.9 → top middle = -3.9 + 1.3 + 6.92 = 4.32.

Third column: -5.6 + (middle right) + 0.02 = -3.9 → middle right = -3.9 + 5.6 - 0.02 = 1.68.

Now, second row: (middle left) + (-1.3) + (middle right) = -3.9 → (middle left) + (-1.3) + 1.68 = -3.9 → middle left = -3.9 + 1.3 - 1.68 = -4.28.

First row: (top left) + (top middle) + (-5.6) = -3.9 → (top left) + 4.32 - 5.6 = -3.9 → top left = -3.9 + 5.6 - 4.32 = -2.62.

Now, first column: (top left) + (middle left) + 3 = -2.62 + (-4.28) + 3 = -3.9, which matches S. Good.

Diagonal: top left (-2.62) + center (-1.3) + bottom right (0.02) = -2.62 -1.3 +0.02 = -3.9, which matches. Good.

Other diagonal: top right (-5.6) + center (-1.3) + bottom left (3) = -5.6 -1.3 +3 = -3.9, which matches. Good.

Now, let's list all cells:

Top row: -2.62, 4.32, -5.6

Middle row: -4.28, -1.3, 1.68

Bottom row: 3, -6.92, 0.02

Let's check each row:

Top row: -2.62 + 4.32 -5.6 = (-2.62 -5.6) +4.32 = -8.22 +4.32 = -3.9 ✔️

Middle row: -4.28 -1.3 +1.68 = (-4.28 -1.3) +1.68 = -5.58 +1.68 = -3.9 ✔️

Bottom row: 3 -6.92 +0.02 = (3 +0.02) -6.92 = 3.02 -6.92 = -3.9 ✔️

Columns:

First column: -2.62 -4.28 +3 = (-6.9) +3 = -3.9 ✔️

Second column: 4.32 -1.3 -6.92 = (4.32 -6.92) -1.3 = -2.6 -1.3 = -3.9 ✔️

Third column: -5.6 +1.68 +0.02 = -5.6 +1.7 = -3.9 ✔️

Diagonals:

-2.62 -1.3 +0.02 = -3.9 ✔️

-5.6 -1.3 +3 = -3.9 ✔️

So the magic square cells are:

Top left: -2.62

Top middle: 4.32

Middle left: -4.28

Middle right: 1.68

Bottom middle: -6.92

Wait, but let's check the calculations again. Maybe I made a mistake in the magic constant. Wait, the center cell is -1.3, so S = 3*(-1.3) = -3.9. That's correct for a 3x3 magic square, where the magic constant is 3 times the center number.

So let's recheck the third row: 3 + (-6.92) + 0.02 = 3 -6.92 +0.02 = -3.9. Correct.

Second column: 4.32 + (-1.3) + (-6.92) = 4.32 -1.3 -6.92 = (4.32 -6.92) -1.3 = -2.6 -1.3 = -3.9. Correct.

Third column: -5.6 +1.68 +0.02 = -5.6 +1.7 = -3.9. Correct.

Middle row: -4.28 + (-1.3) +1.68 = -4.28 -1.3 +1.68 = -5.58 +1.68 = -3.9. Correct.

Top row: -2.62 +4.32 -5.6 = ( -2.62 -5.6 ) +4.32 =…

Answer:

-2.624.32-5.6
3-6.920.02