Question

Identify the unknown amount as the percent, part, or whole. 12 is \(25 \%\) of what number?

Short answer

Answer: The unknown number is 48.

Step-by-step solution

Write down the percent equation

The percent equation is \(Percent = \frac{Part}{Whole}\). In this problem, we have the percent (25%) and the part (12), and we need to find the whole.

Convert the percentage to a decimal

To convert the given percentage (25%) to a decimal, we need to divide it by 100. So, \(25 \% = \frac{25}{100} = 0.25\).

Set up the equation

We will now set up the equation with the values we have. Our percent is 0.25, and our part is 12. We will replace those in our percent equation: \(0.25 = \frac{12}{Whole}\)

Solve for the whole

To solve for the whole, let's first multiply both sides of the equation by the whole: \(0.25 \times Whole = \frac{12}{Whole} \times Whole\) This simplifies to: \(0.25 \times Whole = 12\) Now, we will divide both sides of the equation by 0.25 to find the value of the whole: \(Whole = \frac{12}{0.25}\) \(Whole = 48\)

State the answer

12 is \(25 \%\) of the number 48. So, the unknown number (the whole) is 48.

Key concepts

Percent Equation

The percent equation is a fundamental concept in percent problems, helping us understand the relationship between three key elements: the percent, the part, and the whole. It's expressed with the formula: \[ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \] Given one or two values, this equation allows us to solve for the missing value, providing a systematic way to tackle percentage-based problems. In the given exercise, we used this equation to find the whole when the part and the percent are known. Start by identifying which values you have in your problem.

• **Percent**: Represents the percentage given in the problem (e.g., 25%)
• **Part**: The portion or part of the whole that you know (e.g., 12)
• **Whole**: The total or complete amount that you need to find

Understanding this relationship will make it easier to solve any percent problem with confidence.

Converting Percentages to Decimals

When dealing with percentages, it's often necessary to convert them into decimals for easier calculation. This process simplifies setting up and solving equations involving percentages. To convert a percentage into a decimal:

• Take the percentage number (e.g., 25%),
• Divide it by 100.

For example, converting 25% into a decimal involves dividing 25 by 100, resulting in 0.25. This conversion helps us to homogenize different parts of mathematical problems, allowing us to directly substitute these values into our equations. This way, calculations are straightforward, and the results are accurate. Once the percentage is converted to a decimal, it can easily integrate into algebraic computations.

Solving Equations

Once the equation is set up using known values and converted decimals, solving the equation becomes the next step. You must isolate the unknown variable to find its value. In the example problem, the equation \( 0.25 = \frac{12}{\text{Whole}} \) is set up to find the 'whole'. This involves simple algebraic manipulation:

• Multiply both sides by the 'whole' to remove the fraction: \[ 0.25 \times \text{Whole} = 12 \]
• Then, solve for 'Whole' by dividing both sides by 0.25: \[ \text{Whole} = \frac{12}{0.25} \]
• Complete the calculation to find \( \text{Whole} = 48 \).

This step-by-step process illustrates the method of handling equations involving percent problems. Follow these steps methodically to solve similar problems, ensuring clarity and precision at each stage. As you become more practiced, solving these equations will become intuitive.