Problem 68 - Algebra 1

Question

Find the resulting unit of measure. (meters) \(\cdot\) (kilometers per meter)

Step-by-Step Solution

Answer

The resulting unit of measure is kilometers (km).

Step 1: Identify relevant units and operations

In this case, we are dealing with the multiplication of two units of measure: meters (m) and kilometers-per-meter (km/m).

Step 2: Apply mathematical operations

The unit for kilometers-per-meter is essentially a fraction with kilometers on top and meters on the bottom, divided by the per symbol. When we multiply the meters (from the left) by kilometers-per-meter (from the right), we are basically multiplying meters by the fraction kilometers divided by meters. This results in the meters on top (from the left) and the meters on the bottom (from the right) canceling out. This leaves us with the unit: kilometers.

Step 3: Present resulting unit of measure

With the meters canceled out during the multiplication, only kilometers is left. Therefore, the resulting unit of measure is kilometers (km).

Key Concepts

Units of Measure

Units of measure are essential for expressing the quantity of a physical quantity, like length, mass, or volume. Each unit is part of a measurement system, such as the metric or imperial systems. In this exercise, we deal with meters and kilometers:

• **Meters**: A base unit of length in the metric system, meters are used internationally to measure distance.
• **Kilometers**: A derivative from meters, kilometers are used for longer distances and are equal to 1,000 meters.

These units help quantify how much or how many. Understanding units enables you to interpret measurements correctly and apply them in calculations, like conversions or comparisons.

Multiplication of Units

When multiplying units, you're essentially multiplying the measurements they represent. This involves carrying over the units into the result and understanding whether they're being added or balanced out by each other. For example, when you multiply meters by kilometers-per-meter, you effectively perform the following:

• Combine the units explicitly: \(\text{meters} \times \frac{\text{kilometers}}{\text{meter}}\).
• Understand how these units interact: Units like meters in the numerator and the denominator cancel out.

The multiplication of these units results in different combinations that may cancel out or create new units, depending on the mathematical operation being performed. This is crucial for correctly interpreting the output in terms of units.

Canceling Units

Canceling units is a fundamental concept in unit conversion, particularly useful in simplifying the resulting unit after performing an operation. In the case of multiplying \(\text{meters} \times \frac{\text{kilometers}}{\text{meter}}\), here's how canceling works:

• **Identify a common unit**: Notice that meters appear in both the numerator and the denominator.
• **Cancel out the common units**: Since a unit appears on both sides, it simplifies or "cancels" each other out. This leaves you with the remaining unit, which is kilometers in this case.

Understanding how to cancel units helps ensure accuracy in scientific and mathematical computations by reducing the complexity of unit combinations and focusing on what's left over.