Problem 74 - Elementary Algebra

Question

Twenty-five percent of a number is \(12.32 .\) What is the number?

Step-by-Step Solution

Answer

Answer: The original number is 49.28.

Step 1: Set up the equation

We have the equation to solve: 0.25 * x = 12.32

Step 2: Solve for x

To solve for x, we must isolate it on one side of the equation. To do this, we can divide both sides of the equation by 0.25: x = 12.32 / 0.25

Step 3: Calculate the result

Now we perform the division to find x: x = 49.28 So, the original number is 49.28.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that helps us to describe relationships using symbols and letters. These symbols and letters represent numbers or quantities in equations and expressions. In an algebra problem like the one given, we use the variable \(x\) to represent the unknown number.

When working with algebra, it's crucial to understand that variables can take on any value that makes the equation true. So in our problem, the equation is \(0.25 \times x = 12.32\). Here, \(0.25\) is a constant that represents 25%, a common way to express fractions as decimals, and \(x\) is the variable we are trying to find.

Solving Equations

Solving equations is all about finding the value of the unknown variable that makes the equation true. We achieve this by isolating the variable on one side of the equation.

In our exercise, we start with the equation \(0.25 \times x = 12.32\). To solve for \(x\), we need to get \(x\) by itself. We do this by dividing both sides of the equation by \(0.25\). This is because division is the inverse operation of multiplication, and it effectively "cancels out" the \(0.25\) attached to \(x\).

After dividing, we get \(x = 12.32/0.25\). This step transitions us from the algebraic manipulation to the actual calculation needed to solve the problem.

Using Mathematical Operations

Mathematical operations such as multiplication, division, addition, and subtraction are the basic tools used to manipulate numbers and solve equations. They help simplify expressions, make comparisons, and find unknowns.

In our example, solving the equation involves dividing \(12.32\) by \(0.25\). Division is used here to "undo" the multiplication of \(x\) by \(0.25\), allowing us to isolate \(x\).

Breaking this operation down: Division helps determine how many times one number (0.25, in this case) fits into another (12.32). Performing this division gives us \(x = 49.28\), showing us that 25% of \(49.28\) is indeed \(12.32\). These fundamental operations are the stepping stones to mastering more advanced mathematical concepts.