Problem 28 - Algebra Form and Function

Question

Find a formula for \(n\) in terms of \(m\) where: \(n\) is an elapsed time in hours and \(m\) the time in minutes.

Step-by-Step Solution

Answer

Answer: The formula is \(n = \frac{m}{60}\).

Step 1: Define the relationship between hours and minutes

There are 60 minutes in an hour. So, to convert the time in minutes to hours, we need to divide the time in minutes by 60.

Step 2: Derive the formula for \(n\) in terms of \(m\)

Since \(n\) is the elapsed time in hours, and we need to find the formula in terms of \(m\), we will divide the time in minutes (\(m\)) by 60 (minutes in an hour). So, \(n = \frac{m}{60}\).

Step 3: Simplify the formula

The formula is already simplified as \(n = \frac{m}{60}\). This means that to find the elapsed time, \(n\) (in hours), the time in minutes, \(m\), must be divided by 60.

Key Concepts

Units of Measurement

Understanding units of measurement is crucial when converting time between different formats such as minutes to hours. In our daily routines, we rely on standard units like hours and minutes to organize our schedules.

• Hours: A larger unit used commonly as a reference for periods in a typical day, with 24 hours completing a full day cycle.
• Minutes: A smaller unit, vital for more precise time intervals, where 60 minutes equal one hour.

When converting from one unit to another, it's essential to understand the relationship between them so the conversion is accurate. For time, this relationship is particularly clear: 1 hour = 60 minutes. This means that each unit can be broken down into smaller parts (minutes), allowing us to express any elapsed time in a chosen unit effectively.

Elapsed Time Formula

The elapsed time formula enables you to translate the time expressed in one unit into another. It is especially useful when you have a variable measured in minutes, but you need to know how many hours this time represents. To derive this formula, we use the concept that there are 60 minutes in an hour. Thus, if you have a time value in minutes (let's call it "m"), you need to convert this to hours by dividing by 60. The elapsed time in hours, represented by "n", is calculated using:\[ n = \frac{m}{60} \]This formula is helpful whenever you need to determine how lengthy a specific time is in hours when it is initially given in minutes. It's an essential conversion for various fields like logistics, aviation, and even in everyday planning.

Mathematical Relationships

In mathematics, defining the relationship between different quantities is fundamental to solving conversion problems like time conversion. In this context, identifying the relationship allows us to construct a mathematical formula that satisfies the problem requirements. Consider the conversion from minutes to hours. Here, the relationship is linear because it relies on division by a constant (60).

• When we denote minutes as "m" and hours as "n", we derive the relationship as: \[ n = \frac{m}{60} \]
• This becomes our conversion formula, signifying that every value of "m" - minutes, directly converts to "n" - hours, by dividing by 60.

Such mathematical relationships facilitate understanding and implementing conversions accurately, ensuring that we maintain proportionality and consistency in values during calculations. It helps develop a stronger grasp of how different quantities interconnect and influence each other in practical scenarios.