$7$ is added to a certain number ; the sum is multiplied by $5$ ; the product is divided by $9$ and $3$ is subtracted from the quotient. Thus, if the remainder left is $12$, what was the original number?

### Concept & Logic This problem involves a straightforward sequence of arithmetic operations. The most efficient way to find the unknown starting number is to apply "reverse engineering," working backwards from the final result using inverse operations. ### Step-by-Step Solution * **Given:** * Let the original number be $x$. * Sequence of operations: Add $7$, multiply by $5$, divide by $9$, subtract $3$. * Final result (referred to as "remainder left" in the text) is $12$. * **Calculation / Deduction:** * **Method 1: Algebraic Setup** 1. Write the sequence as a single equation: $$\frac{(x + 7) \times 5}{9} - 3 = 12$$ 2. Solve for $x$. First, add $3$ to both sides: $$\frac{(x + 7) \times 5}{9} = 15$$ 3. Multiply both sides by $9$: $$(x + 7) \times 5 = 135$$ 4. Divide both sides by $5$: $$x + 7 = 27$$ 5. Subtract $7$: $$x = 20$$ * **Method 2: Reverse Operations (Mental Math)** 1. Start with final result: $12$. 2. Inverse of subtracting $3$ is adding $3$: $12 + 3 = 15$. 3. Inverse of dividing by $9$ is multiplying by $9$: $15 \times 9 = 135$. 4. Inverse of multiplying by $5$ is dividing by $5$: $135 / 5 = 27$. 5. Inverse of adding $7$ is subtracting $7$: $27 - 7 = 20$. ### Exam Strategy & Shortcut Using the inverse operation method (Method 2) is the fastest strategy. You can execute steps 1 through 5 entirely in your head without writing a single algebraic equation. Read the prompt backward and invert the verbs: $12 \rightarrow +3 \rightarrow \times 9 \rightarrow \div 5 \rightarrow -7 \rightarrow 20$. ### Common Pitfall The word "remainder" in the final sentence is poorly phrased in the original text; it means the *final mathematical result*, not the remainder of a division operation. If a student interprets "remainder" as a modulo operation, they will be unable to construct the proper equation. ### Final Answer Therefore, the correct answer is 20.