Problem 24 - Algebra 1: Concepts and Skills

Question

Solve the percent problem. 45% of what distance is 135 miles?

Step-by-Step Solution

Answer

The full distance is 300 miles.

Step 1: Understand the Problem

In the problem, it's given that 45% of an unknown distance is equal to 135 miles. The goal is to find this unknown distance.

Step 2: Set Up the Equation

The problem can be translated into the algebraic equation as: 0.45 * x = 135. Here, x is the unknown distance that needs to be found.

Step 3: Solve for x

To solve for x, divide both sides of the equation by 0.45. Thereby, the equation becomes x = 135 / 0.45.

Step 4: Calculate x

Finally, calculating the right side of the equation, digits 135 divided by 0.45, gives us the full distance.

Key Concepts

Algebraic Equations

Algebraic equations are expressions that represent relationships between different quantities. They are formed by combining numbers, variables, and arithmetic operations. In the context of percent problems, an algebraic equation helps us model real-world situations into a mathematical context.

In our percent problem, the phrase "45% of what distance is 135 miles?" needs translation into a mathematical statement. This is done by expressing the percentage as a decimal (45% becomes 0.45) and setting it up in an equation like this:

• 0.45 * x = 135

Where x represents the unknown distance we want to solve for. Algebraic equations such as this one are crucial because they provide a structured method to approach and solve problems.

Solving for Variables

Solving for variables is a key skill in algebra that involves isolating the unknown value on one side of the equation. This is achieved through various mathematical operations which maintain the equation's balance.

In the example problem, the variable "x" represents the unknown distance. The equation is linear, which means it can be solved in a straightforward manner:

• Start with the equation: 0.45 * x = 135
• To isolate x, we need to undo the multiplication by 0.45, achieved by dividing both sides by 0.45
• This transforms the equation into: x = 135 / 0.45

Thus, solving for variables involves performing arithmetic operations that allow x to stand alone, thereby revealing its value.

Percentage Calculation

Percentage calculation is essential for converting problems into solvable algebraic equations. It describes a number as a fraction of 100. In many practical applications, such as this exercise, percentages are used to detail parts of a whole.

To tackle the problem statement, "45% of what distance is 135 miles?", it's necessary to convert the percentage into a decimal. The conversion is straightforward:

• Divide the percent by 100: 45% becomes 0.45

This decimal makes it possible to set up an equation which can then be re-arranged to solve for the unknown distance. By mastering percentage calculations, students can easily transition from word problems to mathematical solutions.