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The key to solving percentage problems like the one given in the exercise is a clear understanding of the percent equation. The percent equation is a fundamental concept that helps to connect parts of a whole using percentages. It typically takes the form:\[\text{Part} = \text{Percent} \times \text{Whole} \]In our problem, the part is the number 45, and the percentage is given as 25%. The unknown whole is represented by \(x\). By substituting into the percent equation, we have:\[ 45 = 0.25 \times x \]This equation means that 45 is 25% of some number \(x\). The transformation of the percentage into a decimal (by dividing 25 by 100) is a crucial step, as it allows for straightforward calculation using multiplication.

Solving an algebra problem often involves expressing a relationship in the form of an equation and then isolating the variable. In this exercise, the given relationship is expressed as the equation:\[ 45 = 0.25 \times x \]The main goal in algebra problem-solving is to find the value of the unknown variable \(x\). To isolate \(x\), you must perform operations that will simplify the equation. Here, we divided both sides by 0.25, the coefficient of \(x\):\[ x = \frac{45}{0.25} \]This technique of dividing both sides by the same number ensures that the equality holds while simplifying the equation. This step-by-step approach of logical reasoning and arithmetic operations highlights the elegance of algebra problem-solving.

Calculating percentages is a practical skill that can simplify solving real-world problems. In this exercise, calculating the percentage involves multiplying a number by the percentage in decimal form. Once we have the value of \(x\), we can verify our answer by performing the percentage calculation again. This checks if 45 is indeed 25% of 180.Let's calculate 25% of 180:

• Convert 25% to its decimal form: 0.25
• Multiply it by 180: \(0.25 \times 180 = 45\)

This calculation confirms that our solution is correct. Being adept at percentage calculations helps ensure you can check your work and understand the mathematical relationships within various percentage problems.