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Algebra Difficulty 3.7 AMC 10/12 Find the answer Canada

Arushi wrote an integer in each cell of a $3\times 4$ grid so that the sum of the
numbers in each row and in each column was the same. Some of the
integers that Arushi wrote are shown in the grid while the others remain
hidden.

$7$

$3$

$4$

$c$

$-1$

$d$

The value of $c-d$ is

Pick one

Solution

Suppose the common sum of the numbers in each row and in each
column is $S$ .

The total of all $12$ integers in the
grid is $3S$ , since it is the sum of
the $3$ row sums. On the other hand,
the total of all $12$ integers in the
grid is also $4S$ since it is the sum
of all $4$ column sums.

Therefore, $3S=4S$ , which forces
$S=0$ .

The second column is shown to contain a $7$ and a $-1$ , so the integer in the second row and
second column must be $-6$ in order
to make the column have a sum of $S=0$ .

Suppose the integer in the second row and fourth column is $x$ . Then $4+(-6)+c+x=0$ , or $x=2-c$ .

The sum in the fourth column is $3+(2-c)+d=5+(d-c)$ , but this sum must be
$S=0$ , so $5+(d-c)=0$ . Rearranging gives $5=c-d$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.