A number is doubled and then 9 is added. If the resulting value is multiplied by 3, the final result is 75. What is the original number?

Introduction / Context:
This problem is a simple linear equation word problem involving a sequence of operations performed on an unknown number. The number is doubled, increased by a fixed amount, and then the result is multiplied by 3 to give 75. These kinds of questions are designed to test whether learners can translate multi-step verbal descriptions into algebraic equations and then solve them correctly.

Given Data / Assumptions:

• Let the original number be x.
• The number is doubled to give 2x.
• Then 9 is added, giving 2x + 9.
• This result is multiplied by 3, giving 3(2x + 9).
• The final result equals 75.
• We need to find the value of x.

Concept / Approach:
We systematically reflect each step in the operations as algebraic expressions. Doubling corresponds to multiplying by 2, then adding 9 is a simple addition, and multiplying the result by 3 corresponds to distributing 3 across the expression. Setting the final expression equal to 75 yields a solvable linear equation. After solving for x, we verify by reapplying the steps to check whether the final result is indeed 75.

Step-by-Step Solution:
Step 1: Let the original number be x. Step 2: The number is doubled, giving 2x. Step 3: Add 9 to this result to obtain 2x + 9. Step 4: The problem says that if this result is multiplied by 3, the final answer is 75, so write 3(2x + 9) = 75. Step 5: Expand the left side: 3 * 2x + 3 * 9 = 6x + 27. Step 6: So we have the equation 6x + 27 = 75. Step 7: Subtract 27 from both sides to isolate the term with x: 6x = 75 − 27. Step 8: Compute the difference 75 − 27 = 48, so 6x = 48. Step 9: Divide both sides by 6: x = 48 / 6 = 8. Step 10: Hence, the original number is 8.

Verification / Alternative check:
Starting with x = 8, double it to get 16. Add 9 to obtain 25. Multiply 25 by 3 to get 75. This matches the final result given in the problem, confirming that our solution is correct and that the original number is 8.

Why Other Options Are Wrong:
Option (a) 2: Doubling gives 4, plus 9 is 13, and 13 * 3 = 39, not 75.
Option (b) 4: Doubling gives 8, plus 9 is 17, and 17 * 3 = 51, not 75.
Option (c) 6: Doubling gives 12, plus 9 is 21, and 21 * 3 = 63, not 75.

Common Pitfalls:
Learners may misread the phrase and incorrectly interpret the order of operations, for example adding 9 after multiplying by 3. It is important to follow the steps in the exact order described: double, then add 9, then multiply by 3. Another common error arises when expanding 3(2x + 9) incorrectly. Careful reading and correct expansion prevent such mistakes.

Final Answer:
The original number is 8.