Problem 24 - Algebra Form and Function

Question

Find a formula for \(n\) in terms of \(m\) where: \(n\) is a length in feet and \(m\) is the length in inches.

Step-by-Step Solution

Answer

Answer: The formula is n = \frac{m}{12}.

Step 1: Identify the conversion factor between feet and inches

Since 1 foot is equal to 12 inches, we can express this conversion factor as: 1 foot = 12 inches

Step 2: Express the length in feet (n) in terms of the length in inches (m)

To express the length in feet in terms of the length in inches, we will divide the length in inches (m) by the conversion factor (12 inches) as follows: n = \frac{m}{12 inches}

Step 3: Simplify the expression

Lastly, simplify the expression: n = \frac{m}{12} Therefore, the formula for the length in feet (n) in terms of the length in inches (m) is n = \frac{m}{12}.

Key Concepts

Feet to Inches Conversion

To understand how to convert feet to inches, you first need to know the relationship between these two units of measurement. Feet and inches are both units used to measure length in the imperial system. The conversion factor is straightforward:

• 1 foot is equivalent to 12 inches.

This means that if you have a measurement in feet, you multiply it by 12 to find out how many inches it represents. Conversely, to convert from inches to feet, you divide the number of inches by 12. This simple mathematical relationship helps in converting measurements easily without any complex calculations. For example, if you want to find out how many inches are in 3 feet, you simply calculate: \[ 3 \times 12 = 36 \text{ inches} \].
This fundamental transformation is crucial, especially when dealing with situations where precise length measurements are necessary.

Length Measurement

Length is a basic physical quantity that expresses the size or extent of objects. In our everyday lives, we often use rulers, tape measures, or yardsticks to measure length. These tools might display measurements in different units, such as:

• Inches
• Feet
• Centimeters (metric system)

Understanding how to convert between units, like inches and feet, is important for accurate communication and calculation in both domestic tasks and many professional fields.
Length measurement is not just about understanding numbers and units; it involves knowing which unit is most appropriate for your specific measurement task. For example, feet might be used for larger objects (room dimensions), while inches are better suited for smaller objects (book size).
Being comfortable with these conversions aids in making precise measurements, which is essential in fields like construction, design, and engineering.

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operations (+, –, ×, ÷) to describe relationships and calculate values. In this exercise, we are looking at an algebraic expression that shows how to calculate feet from inches: \[ n = \frac{m}{12} \], where:

• \( n \): length in feet
• \( m \): length in inches

This expression represents the conversion factor, where the length in inches is divided by 12 to determine the equivalent length in feet. Such expressions are valuable because they provide a systematic way to solve problems. By substituting different values for \( m \), you can easily find out different lengths in feet.
Algebraic expressions like this one form the backbone of mathematical problem-solving. They allow you to express real-world situations in a form that can be manipulated mathematically to find solutions. So, being comfortable working with these expressions is highly beneficial for anyone looking to develop strong analytical skills.