Solving the Equation: If 75% of a Number Plus 75 Equals the Number
In this article, we will explore a mathematical problem where we need to find a number that, when 75% of it is added to 75, equals the number itself. This article breaks down the problem step-by-step, providing a clear and detailed solution.
Problem Definition
Let's define the number as X. According to the problem statement, we have:
75% of X plus 75 is equal to X.
Mathematically, this can be expressed as:
0.75X 75 X
Solution
Step 1: Isolate the Variable Term
To solve for X, we first need to isolate the variable term on one side of the equation. Subtract 0.75X from both sides:
0.75X 75 - 0.75X X - 0.75X
This simplifies to:
75 0.25X
Step 2: Solve for the Variable
To solve for X, divide both sides by 0.25:
75 / 0.25 X
This simplifies to:
x 300
Conclusion
The number X that satisfies the given condition is 300. This can be verified by substituting X back into the original equation:
0.75 * 300 75 300
225 75 300
300 300
Alternative Methods
Method 1: Direct Multiplication and Analysis
Using an alternative approach, let's consider the same number X again. We know that 75% of X can be represented as 3/4 of X or 0.75X. Adding 75 to this fraction should yield X.
Mathematically, this becomes:
(3/4)X 75 X
Isolating X leads to:
(3/4)X - X -75
-1/4X -75
X 300
Method 2: Unit Analysis
Let's denote the number as 100 units (100u). If 75% of the number is added to 75, it should equal the number itself:
Mathematically, this becomes:
(75/100) * 100u 75 100u
Simplifying, we get:
75u 75 100u
75 25u
u 3
100u 300
Therefore, the number is 300.
Conclusion
The final answer is X 300.
Verification
Let's verify the solution:
(0.75 * 300) 75 300
225 75 300
300 300
Thus, the solution is correct and consistent with the problem statement.
Related Topics
This problem touches on fundamental concepts in algebra, including:
Algebraic equation solving: Solving equations involving variables and percentages. Percentage calculation: Converting between percentages and decimal form. Number solving: Finding the value of a variable given a specific relationship. Mathematical proof: Verifying the solution through substitution and simplification.By understanding these concepts, one can solve a wide range of similar problems in algebra and percentages.