What could be the maximum value of $Q$ in the following equation? $$5P9 + 3R7 + 2Q8 = 1114$$

### Concept & Logic This is a cryptarithmetic puzzle based on column addition. To find the maximum value of one digit variable ($Q$) in a sum, you must determine the combined sum of all variables and then set the other unknown variables ($P$ and $R$) to their minimum possible integer value, which is $0$. ### Step-by-Step Solution * **Deduction:** Analyze the addition column by column, starting from the units place (right to left). * **Units Column:** Add the known digits: $9 + 7 + 8 = 24$. The result writes down $4$ and carries over $2$ to the tens column. * **Tens Column:** The sum of the tens column includes the carry-over ($2$) plus the unknown digits ($P, R, Q$). We know the final total $1114$ has a $1$ in the tens place. * **Hundreds Column Check:** The hundreds digits are $5 + 3 + 2 = 10$. However, the final sum is $11$ (from $1114$). This confirms the tens column generated a carry-over of exactly $1$. * **Formulating the Equation:** Since the tens column ends in $1$ and carries over $1$, its total sum is exactly $11$. $$ 2 \text{ (carry)} + P + R + Q = 11 $$ $$ P + R + Q = 9 $$ * **Maximizing Q:** To make $Q$ as large as possible, $P$ and $R$ must be as small as possible. Since $P$, $Q$, and $R$ are digits, their minimum value is $0$. * Setting $P = 0$ and $R = 0$: $$ 0 + 0 + Q = 9 \Rightarrow Q = 9 $$ ### Exam Strategy & Shortcut Immediately sum the units: $9+7+8 = 24$. Note the carry $2$. The total tens sum must be $11$ because the hundreds sum ($5+3+2=10$) needs $+1$ to reach the $11$ in $1114$. Therefore, $2 + P + Q + R = 11$, so $P+Q+R = 9$. To maximize one variable in a sum of non-negative integers, make the others zero. Max $Q = 9$. ### Common Pitfall The most common mistake is forgetting to add the carry-over from the units place ($2$) to the tens place equation. This leads to the incorrect assumption that $P + R + Q = 11$, which makes students wrongly assume a digit could theoretically be greater than 9 if the question asked for a combined total. ### Final Answer Therefore, the correct answer is **9**.