Consider the following multiplication problem:

(PQ) × 3 = RQQ, where P, Q and R are different digits and R ≠ 0.

What is the value of (P + R) ÷ Q?

The correct option is 2.

Step-by-step Explanation:

We are given the multiplication equation:

$\left(PQ\right)\times 3=RQQ$

where $P$, $Q$, and $R$ are distinct single digits and $R\ne 0$.

Let's represent the numbers in expanded decimal form:
$PQ=10P+Q$
$RQQ=100R+10Q+Q=100R+11Q$

Substituting these expressions into the given multiplication problem:

$3\times (10P+Q)=100R+11Q$

$30P+3Q=100R+11Q$

Subtracting $3Q$ from both sides:

$30P=100R+8Q$

Dividing the entire equation by 2:

$15P=50R+4Q$

Since $15P$ and $50R$ are both divisible by 5, the term $4Q$ must also be divisible by 5. Since $Q$ is a single digit (0 through 9), $Q$ must be either 0 or 5.

Case 1: If $Q=0$
$15P=50R\Rightarrow 3P=10R$
Since $P\le 9$, $3P\le 27$, which means $10R$ cannot exceed 27. The only possibilities for $R$ would be 1 or 2, but neither yields an integer value for $P$. Thus, $Q\ne 0$.

Case 2: If $Q=5$
Substitute $Q=5$ into the simplified equation:
$15P=50R+4\left(5\right)$
$15P=50R+20$
Dividing by 5 gives:
$3P=10R+4$

Testing integer values for $R$ (since $R\ne 0$):
If $R=1$: $3P=14$ (not an integer).
If $R=2$: $3P=24\Rightarrow P=8$.

So, we find:
$P=8$
$Q=5$
$R=2$

Checking the multiplication:
$85\times 3=255$
All digits $P,Q,R$ (8, 5, 2) are different, and $R\ne 0$.

Now, calculate the value of $(P+R)÷Q$:

$(P+R)÷Q=(8+2)÷5=10÷5=2$