CEMC Gauss (Grade 7), 2016

The sum of the units column is $Q+Q+Q=3Q$.

Since $Q$ is a single digit, and $3Q$ ends in a 6, then the only possibility is $Q=2$.

Then $3Q=3\times2=6$, and thus there is no carry over to the tens column.

The sum of the tens column becomes $2+P+2=P+4$, since $Q=2$.

Since $P$ is a single digit, and $P+4$ ends in a 7, then the only possibility is $P=3$.

Then $P+4=3+4=7$, and thus there is no carry over to the hundreds column.

We may verify that the sum of the hundreds column is $3+3+2=8$, since $P=3$ and $Q=2$.

The value of $P+Q$ is $3+2=5$, and the final sum is shown.