Problem 25 - Algebra Form and Function

Question

Find a formula for \(n\) in terms of \(m\) where: \(n\) is a distance in \(\mathrm{km}\) and \(m\) the distance in meters.

Step-by-Step Solution

Answer

Answer: The formula to convert the distance in meters to kilometers is n = m/1000, where n is the distance in kilometers and m is the distance in meters.

Step 1: Set up the relationship between km and meters

We know for sure that 1 km equals 1,000 meters. So the relationship between the two can be expressed as a simple proportion: \(n\) kilometers contain \(m = 1000n\) meters.

Step 2: Isolate \(n\) on one side of the equation

Since we want to express \(n\) in terms of \(m\), we need to isolate \(n\) in the equation \(m = 1000n\). We do that by dividing both sides of the equation by 1000: \[ n = \frac{m}{1000} \]

Step 3: Simplify if possible

In this case, our equation \(n = \frac{m}{1000}\) is already in its simplest form. Therefore, we have found the formula to convert meters to kilometers.

Step 4: Conclusion

The formula for \(n\) in terms of \(m\) is \(n = \frac{m}{1000}\), where \(n\) is the distance in kilometers and \(m\) is the distance in meters.

Key Concepts

Kilometers to Meters

When converting distances from kilometers to meters, it's important to understand the basic unit conversion involved. The metric system, which is widely used around the world, defines 1 kilometer as equivalent to 1,000 meters. This means that the two units are directly proportional to each other. To convert kilometers to meters, simply multiply the number of kilometers by 1,000.
For example:

• If you have 5 kilometers, you multiply 5 by 1,000 to get 5,000 meters.
• 10 kilometers would equal 10,000 meters.

This straightforward multiplication is because the metric system uses powers of ten, making conversions neat and predictable. Remembering this conversion factor helps in many real-world applications, like measuring distances in different units.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as addition or multiplication). In the context of our exercise, the expression used is derived from the relationship between kilometers and meters. We start with:

• The known equation: \( m = 1000n \)

Here, "\(n\)" represents a number of kilometers, while "\(m\)" denotes the equivalent number of meters.
To express "\(n\)" in terms of "\(m\)," we manipulated this expression by isolating "\(n\):"

• We divide both sides by 1,000: \( n = \frac{m}{1000} \)

Doing so provides a formula that can be directly used to convert meters into kilometers. Algebraic expressions like this allow for easy calculations, showing how interconnected variables can be.

Proportional Relationship

A proportional relationship exists when two quantities increase or decrease in a consistent ratio. In our distance conversion exercise, the relationship between kilometers and meters is proportional. This means that doubling the kilometers will result in double the meters.
Some characteristics of proportional relationships include:

• A constant ratio or rate, which in this case is 1 kilometer : 1,000 meters.
• A linear relationship when graphed, showing a straight line through the origin.

Understanding this concept is crucial for many mathematical tasks. It helps explain why the conversion factor between units remains consistent and allows us to dependably scale distances up or down with simple multiplication or division.