If 75% of a certain number is added to 75, the result is equal to the number itself. What is the value of this number?

Introduction / Context:
This is a simple algebraic percentage problem where a number is related to its own percentage plus a constant. Specifically, 75% of the number is added to 75 and the result equals the original number. We need to find that number. Problems of this type are common in basic algebra and percentage topics and help students translate verbal statements into equations involving percentages and solve them systematically.

Given Data / Assumptions:
- Let the required number be x.- 75% of x is added to 75.- The result equals x itself.- We are asked to find the numerical value of x.

Concept / Approach:
The key concept is expressing the verbal description as an algebraic equation. "75% of x" can be written as (75 / 100) * x or 0.75x. The condition "75% of x added to 75 equals x" becomes 0.75x + 75 = x. Solving this equation for x involves basic algebraic manipulation: moving terms involving x to one side and constants to the other, then dividing to isolate x.

Step-by-Step Solution:
Step 1: Let the number be x.Step 2: 75% of x = (75 / 100) * x = 0.75x.Step 3: According to the problem, 75% of x plus 75 equals x.Step 4: Write the equation: 0.75x + 75 = x.Step 5: Move 0.75x to the right-hand side to collect x terms together.75 = x - 0.75x.Step 6: x - 0.75x = 0.25x.Step 7: Therefore, 75 = 0.25x.Step 8: Solve for x by dividing both sides by 0.25.x = 75 / 0.25.Step 9: 0.25 = 1/4, so dividing by 0.25 is the same as multiplying by 4.x = 75 * 4 = 300.Therefore, the number is 300.

Verification / Alternative check:
Check the condition using x = 300.75% of 300 = (75 / 100) * 300 = 0.75 * 300 = 225.Add 75 to this: 225 + 75 = 300.This equals the original number x, confirming that x = 300 satisfies the condition given in the problem.

Why Other Options Are Wrong:
- 100: 75% of 100 = 75; 75 + 75 = 150, which is not equal to 100.- 200: 75% of 200 = 150; 150 + 75 = 225, not 200.- 400: 75% of 400 = 300; 300 + 75 = 375, not 400.

Common Pitfalls:
- Misinterpreting "75% of the number is added to 75" as "75% of (number plus 75)" which would give a different equation.- Using 75x instead of 0.75x or (75 / 100) * x.- Making algebraic errors when moving 0.75x to the other side of the equation.- Treating 0.25 as 1/5 instead of the correct 1/4 when solving for x.

Final Answer:
The required number is 300.