A number is doubled and $9$ is added. If the resultant is trebled, it becomes $75$. What is that number?

### Concept & Logic This problem requires forming an equation with nested operations. We must use parentheses carefully to ensure the order of operations described in the text translates correctly into algebra. ### Step-by-step Solution **Given:** * Let the original number be $x$. * "Doubled and 9 is added": $2x + 9$ * "Resultant is trebled": Multiply the entire previous expression by $3$. **Calculation:** Set up the equation: $$ 3(2x + 9) = 75 $$ Divide both sides by $3$ to simplify the equation immediately: $$ 2x + 9 = \frac{75}{3} $$ $$ 2x + 9 = 25 $$ Subtract $9$ from both sides: $$ 2x = 25 - 9 $$ $$ 2x = 16 $$ Divide by $2$: $$ x = 8 $$ ### Exam Strategy & Shortcut **Reverse Operations (Backtracking):** Instead of using algebra, work backward from the final answer by doing the opposite operations. 1. Final result is $75$. 2. Before that, it was trebled (multiplied by $3$). So, divide by $3 \rightarrow 75 \div 3 = 25$. 3. Before that, $9$ was added. So, subtract $9 \rightarrow 25 - 9 = 16$. 4. Before that, it was doubled (multiplied by $2$). So, divide by $2 \rightarrow 16 \div 2 = 8$. The number is $8$. This takes less than 10 seconds mentally. ### Common Pitfall The most common algebraic error is failing to use parentheses for the "resultant". Students often write $2x + 9 \times 3 = 75$ or $3 \times 2x + 9 = 75$, which only trebles a portion of the expression. "The resultant" always implies evaluating everything before that point as a single block. ### Final Answer **Therefore, the correct answer is 8.**