Mind Your Ps And Qs

The sum of $Q+Q+Q=3Q$ is in the units column. Since $Q$ is a single digit, and $3Q$ ends in a 6, then the only possibility is $Q = 2.$ Then $3Q = 3 × 2 = 6,$ 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 $3 + 2 = 5.$

$322 + 332 + 222 = 876.$

We got = Q + Q + Q = 6

So,3Q = 6. Thats Mean Q = 6 : 3 = 2

P + Q + Q = 7

= P + 2 + 2 = 7..

So,P = 7 - 2 - 2 = 3

P + P + Q = 8
3 + 3 + 2 = 8

So, P + Q = 3 + 2 = 5.

Converting the equations into algebra, we get that $(110P + Q) + (100P + 11Q) + 111Q = 876$ . This gives us

$210 P + 123 Q = 876$

Looking at the units digit, since $Q$ is from 0 to 9, we conclude that $Q = 2$ . Thus,

$P = \frac{ 876 - 123 \times 2 } { 210} = \frac{ 630 } { 210} = 3 .$

Hence $P + Q = 5$ .