Problem 62 - Elementary Algebra Within Reach

Question

Structure Write an equation that can be used to find the number \(x\) that is \(25 \%\) of a number \(y\).

Step-by-Step Solution

Answer

The equation you are looking for is \(x = 0.25y\)

Step 1: Understand the problem

The problem states that \(x\) is 25% of \(y\). In mathematical terms, this means that \(x\) is equal to 25% times \(y\). The expression '25% of \(y\)' translates to '25/100 times \(y\)', or '\(0.25y\)'

Step 2: Write the equation

The equation that translates this situation is \(x = 0.25y\)

Key Concepts

Percent Equation

When we talk about a percent equation, we refer to an algebraic expression that relates a percentage to its equivalent fractional or decimal form when applied to a number. In essence, it's the translation of a percentage relationship into an algebraic statement.

For example, if we are given that a number, let's call it 'x', is 25% of another number 'y', we express this in algebraic terms. Since 'percent' means 'per hundred', 25% is the same as the fraction \( \frac{25}{100} \) or the decimal 0.25. Therefore, 25% of 'y' is written as \( 0.25 \times y \) in equation form, which visually represents the portion of 'y' that 'x' constitutes. This simplifies to the equation \( x = 0.25y \).

Percent equations are integral in various fields, including finance for calculating interest, in statistics for probability, and in everyday scenarios like computing discounts or sales tax.

Algebraic Representation

Algebraic representation is the process of modeling real-world situations or problems using algebraic symbols and expressions. This method is crucial because it allows us to solve problems that involve unknown quantities by using variables.

In the context of our example, 'x' and 'y' are variables. 'x' is the unknown we want to determine, and 'y' symbolizes another number that 'x' is a certain percentage of. Algebraic representation involves assigning symbols to these quantities and expressing the relationship between them mathematically. This representation helps visualize the problem and provides a clear pathway to the solution.

By representing the percentage relationship as \( x = 0.25y \), we have laid down a foundation that can be used to calculate 'x' given any value of 'y'. This kind of algebraic translation is a basic tool in problem-solving across many applications.

Equation Solving

Equation solving is the procedure of finding the value(s) of variable(s) that satisfy an equation. For our percent equation, \( x = 0.25y \), solving it would involve finding the value of 'x' for a given 'y', or vice versa.

To solve an equation, we perform operations to isolate the variable we're solving for. If additional information is given, such as the value of 'y', you would substitute that number in place of 'y' and multiply it by 0.25 to find 'x'. Suppose no specific value is provided; the equation as it stands represents the relationship between 'x' and 'y', with 'x' being always one-quarter of 'y'.

The step-by-step process of equation solving requires an understanding of basic algebraic principles, including the order of operations and inverse operations. Mastering these principles facilitates the solving of not just simple percentage problems but more complex algebraic equations as well.