Problem 98 - Intermediate Algebra

Question

Answer the question with an algebraic expression. The length of a line segment is \(d\) decimeters. How long is the line segment expressed in meters?

Step-by-Step Solution

Answer

\(\frac{d}{10}\) meters.

Step 1: Understand the Relationship between Units

Recognize that the problem involves converting a measurement from decimeters to meters. Recall that 1 meter is equal to 10 decimeters.

Step 2: Set up the Conversion Equation

Given that 1 meter = 10 decimeters, we can express the length of the line segment in meters by dividing the length in decimeters by the conversion factor 10.

Step 3: Write the Algebraic Expression

The length of the segment in meters, if it is originally given as \(d\) decimeters, is calculated by the expression \(\frac{d}{10}\). This expression represents the conversion of decimeters to meters.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. In this exercise, our variable is \( d \), which represents the length of a line segment in decimeters. We need to convert this length into meters. An algebraic expression helps us perform operations on variables to solve a problem, often involving unknown quantities. By using mathematical operations, we can manipulate these expressions to find the desired form or solution. In our current situation, we aim to transform the given length \( d \) from decimeters into meters. This process only requires a simple arithmetic operation: division by the conversion factor. It's essential to understand how to set up and solve algebraic expressions because it forms the basis for more complex mathematics. As we get more comfortable manipulating these variables and expressions, we can solve wide-ranging real-world problems.

Units of Measurement

Units of measurement are standards for expressing quantities. They are crucial in ensuring clarity and accuracy when we need to convey physical quantities, like length in this exercise.Here, we're dealing with two units of length: decimeters and meters. These units belong to the metric system, which is widely used globally for most scientific and everyday situations. The relation between decimeters and meters is simple:

• 1 meter is equivalent to 10 decimeters.
• Conversely, 1 decimeter equals 0.1 meters (or \( \frac{1}{10} \) of a meter).

This relationship is pivotal because it directly influences how we construct our algebraic expression for conversion. By understanding these units and how to convert between them, we can communicate and solve scientific problems with greater precision, avoiding confusion that can arise from unclear measurements.

Step by Step Solutions

The step-by-step approach breaks down a complex problem into manageable pieces. We start by identifying what we know and then proceed through logical operations until we reach the required solution.In this exercise:

• **Step 1**: Recognize the initial and final units (from decimeters to meters).
• **Step 2**: Set up the conversion equation. We reason that since 1 meter = 10 decimeters, to convert from decimeters to meters, we must use the relationship that 1 decimeter is \( \frac{1}{10} \) of a meter.
• **Step 3**: Derive the algebraic expression. This involves expressing the length \( d \) in meters, which we achieve by dividing \( d \) by 10, resulting in \( \frac{d}{10} \).

Following these steps helps clarify the conversion process, highlighting the path from understanding the unit relationship to expressing it algebraically. It's an approach that can be applied across various mathematical problems, emphasizing the importance of logical progression and clarity when tackling any mathematical task.