Solving the Equation: If 75% of a Number Plus 75 Equals the Number

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.