Complete the blank cells in the table below with fractions in such a way that the sum of the three numbers in any row, any column, and the two diagonals is always the same.

|  |  | $3 / 5$ |
| :--- | :--- | :--- |
|  | $1 / 2$ |  |
| 0.4 | 0.5 |  |
To facilitate our calculations, it is convenient to reduce all the fractions that appear in the table to a common denominator. Since $0.4=4/10$ and $0.5=5/10$, we can rewrite the table as follows, where we indicate with the letters $a, b, c, d$, and $e$ the numbers that need to be calculated.

| $a$ | $c$ | $6 / 10$ |
| :---: | :---: | :---: |
| $b$ | $5 / 10$ | $d$ |
| $4 / 10$ | $5 / 10$ | $e$ |

Looking at the ascending diagonal, we see that the sum of the elements of this diagonal is $4 / 10 + 5 / 10 + 6 / 10 = 15 / 10$. Since the sum of the elements of the third row must be equal to this sum of the elements of the diagonal, we get $4 / 10 + 5 / 10 + e = 15 / 10$, from which $e = 6 / 10$. We also get, in the second column, $5 / 10 + 5 / 10 + c = 15 / 10$, from which $c = 5 / 10$. Placing these values of $c$ and $e$ in the table, we get

| $a$ | $5 / 10$ | $6 / 10$ |
| :---: | :---: | :---: |
| $b$ | $5 / 10$ | $d$ |
| $4 / 10$ | $5 / 10$ | $6 / 10$ |

Now, the first row provides $a + 5 / 10 + 6 / 10 = 15 / 10$, from which $a = 4 / 10$. From the third column, we get $6 / 10 + d + 6 / 10 = 15 / 10$, from which $d = 3 / 10$; similarly, we get $b = 7 / 10$ and the table is complete.

| $4 / 10$ | $5 / 10$ | $6 / 10$ |
| :--- | :--- | :--- |
| $7 / 10$ | $5 / 10$ | $3 / 10$ |
| $4 / 10$ | $5 / 10$ | $6 / 10$ |