Problem 40 - Prealgebra

Question

Metric System. Another very small unit of measure in the metric system is the angstrom (abbreviated \(\AA\) ). There are \(10,000,000 \AA\) in 1 millimeter. How many angstroms are in 15 decimeters?

Step-by-Step Solution

Answer

15 decimeters equals 15,000,000,000 angstroms.

Step 1: Understanding the question

We need to find out how many angstroms () are present in 15 decimeters.

Step 2: Convert decimeters to millimeters

Since we know the conversion from millimeters to angstroms, we first need to convert decimeters into millimeters. Recall that 1 decimeter equals 100 millimeters. Thus, 15 decimeters is equivalent to \( 15 \times 100 = 1500 \) millimeters.

Step 3: Convert millimeters to angstroms

Now that we have the length in millimeters, we use the conversion factor that there are 10,000,000 angstroms in one millimeter. Therefore, 1500 millimeters is equivalent to \( 1500 \times 10,000,000 = 15,000,000,000 \) angstroms.

Key Concepts

Unit Conversion

Unit conversion is essential in scientific and everyday scenarios. It involves changing a measurement from one unit to another, keeping the same value, just represented differently. In the metric system, which is widely used and consistent, conversions are made a bit simpler due to its decimal nature. Understanding this can help in various subjects, including science and mathematics.

To convert between units, you use conversion factors. A conversion factor is a ratio that expresses how many of one unit are equivalent to another unit. In this exercise, converting between different metric units like decimeters, millimeters, and angstroms involves two main conversion factors:

• 1 decimeter = 100 millimeters
• 1 millimeter = 10,000,000 angstroms

Start by converting to an intermediate unit. Here, we first converted decimeters to millimeters, as it's a direct conversion with an easy-to-remember factor of 100. Once in millimeters, another multiplication by the conversion factor gives us the number of angstroms as needed for the question.

Length Measurement

Length measurement refers to determining how long, wide, or tall something is. It helps us understand the size of objects or the distance between points. In the metric system, length is measured using units like meters, centimeters, millimeters, and even angstroms for extremely small lengths.

The hierarchy of metric units for length goes:

• Kilometer (km)
• Meter (m)
• Decimeter (dm)
• Centimeter (cm)
• Millimeter (mm)
• angstrom (Å)

Each step to a smaller unit involves multiplying by 10. This decimal nature makes it easy to understand and perform calculations. For measuring very minute sizes, like atomic structures, we use angstroms, known in scientific fields as an essential unit for very small measurements. When you know how to convert measurements, you can work with all these different units fluently.

Prealgebra

Prealgebra serves as the foundation for higher levels of math. It introduces essential concepts like arithmetic operations, fractions, decimals, and simple equations, which are critical in performing unit conversions.

In our example, prealgebra skills come into play through multiplication and understanding ratios. Converting decimeters to millimeters, then millimeters to angstroms, we perform basic multiplication:

• First, to get from decimeters to millimeters: Multiply by 100
• Second, to convert these millimeters to angstroms: Multiply by 10,000,000

These operations in sequence illustrate the workings of unit conversion, relying heavily on multiplication and handling large numbers. Practicing prealgebra enhances your ability to work efficiently and accurately with such conversions. Understanding these steps makes it easier to tackle larger problems and enhances logical mathematical thinking.