If 1/2 is added to a certain real number and the resulting sum is multiplied by 3 to give a result of 21, what is the value of the original number before these operations were performed?

Introduction / Context:
This is a straightforward linear equation problem framed in words. It checks the ability to translate a verbal description of operations on a number into an algebraic equation and then solve that equation to retrieve the original number. Such problems are standard in basic arithmetic and algebra sections of aptitude tests.

Given Data / Assumptions:

• Half (1/2) is added to the original number.
• The resulting sum is then multiplied by 3.
• The final result after multiplication is 21.
• The original number is a real number we must find.

Concept / Approach:
Assign a variable, say x, to represent the original number. Translate the description step by step into an algebraic equation. First express the addition of 1/2, then the multiplication by 3, and set this equal to 21. Solve the resulting linear equation in x using basic algebra operations to isolate x.

Step-by-Step Solution:
Let x be the original number.Adding 1/2 to x gives x + 1/2.Multiplying this sum by 3 gives 3(x + 1/2).According to the problem statement, 3(x + 1/2) = 21.Divide both sides by 3: x + 1/2 = 21 / 3 = 7.Subtract 1/2 from both sides: x = 7 − 1/2 = 6.5.

Verification / Alternative check:
Substitute x = 6.5 back into the original description of the operations. First add 1/2: 6.5 + 0.5 = 7. Then multiply by 3: 7 * 3 = 21. This matches the final result specified in the question, confirming that the original number is indeed 6.5.

Why Other Options Are Wrong:

• 5.5 and 4.5, when used in the same operations, produce results 18 and 15 respectively, which are not equal to 21.
• −6.5 gives a sum of −6 when 1/2 is added, and 3 * (−6) = −18, not 21.
• 7.5 yields (7.5 + 0.5) * 3 = 8 * 3 = 24, which is also incorrect.

Common Pitfalls:

• Interpreting the operations in the wrong order, for example multiplying by 3 first and then adding 1/2.
• Making mistakes with fractional arithmetic when subtracting 1/2 from 7.
• Not checking the solution by substituting it back into the original word description.

Final Answer:
6.5