Question

Perform the following metric-metric conversions. (a) \(5.00 \mathrm{~m}\) to \(\mathrm{km}\) (b) \(5.00 \mathrm{~g}\) to \(\mathrm{cg}\) (c) \(5.00 \mathrm{~L}\) to \(\mathrm{dL}\) (d) \(5.00 \mathrm{~s}\) to \(\mathrm{ns}\)

Short answer

(a) 0.00500 km, (b) 500 cg, (c) 50.0 dL, (d) 5,000,000,000 ns.

Step-by-step solution

Convert meters to kilometers

1 kilometer is equal to 1000 meters. To convert 5.00 meters to kilometers, use the conversion factor: \[ 1 \text{ km} = 1000 \text{ m} \] Therefore, divide 5.00 m by 1000: \[ 5.00 \text{ m} \times \frac{1 \text{ km}}{1000 \text{ m}} = 0.00500 \text{ km} \]

Convert grams to centigrams

1 gram is equal to 100 centigrams. To convert 5.00 grams to centigrams, use the conversion factor: \[ 1 \text{ g} = 100 \text{ cg} \] Therefore, multiply 5.00 g by 100: \[ 5.00 \text{ g} \times 100 = 500 \text{ cg} \]

Convert liters to deciliters

1 liter is equal to 10 deciliters. To convert 5.00 liters to deciliters, use the conversion factor: \[ 1 \text{ L} = 10 \text{ dL} \] Therefore, multiply 5.00 L by 10: \[ 5.00 \text{ L} \times 10 = 50.0 \text{ dL} \]

Convert seconds to nanoseconds

1 second is equal to 1,000,000,000 nanoseconds. To convert 5.00 seconds to nanoseconds, use the conversion factor: \[ 1 \text{ s} = 1,000,000,000 \text{ ns} \] Therefore, multiply 5.00 s by the conversion factor: \[ 5.00 \text{ s} \times 1,000,000,000 = 5,000,000,000 \text{ ns} \]

Key concepts

Units of Measurement

Units of measurement are essential for describing quantities in various fields such as science, engineering, and daily life. They provide a standard way to communicate size, amount, or magnitude of a physical quantity. In the metric system, which is widely used around the world, units are based on multiples of ten, making calculations straightforward. For instance, when we think of distance, the meter is a basic unit, while mass is measured in grams, volume in liters, and time in seconds.

• Length: Meter (m)
• Mass: Gram (g)
• Volume: Liter (L)
• Time: Second (s)

These units can have prefixes to indicate very large or very small quantities, like kilo (thousand), centi (hundredth), or nano (billionth). Understanding these prefixes is crucial when converting between units.

Conversion Factors

Conversion factors are used to switch from one unit of measurement to another. They are ratios that express how many of one unit are equal to another.
When performing conversions, it is handy to remember the basic equivalent:

• 1 kilometer (km) = 1000 meters (m)
• 1 gram (g) = 100 centigrams (cg)
• 1 liter (L) = 10 deciliters (dL)
• 1 second (s) = 1,000,000,000 nanoseconds (ns)

By using these conversion factors, you can multiply or divide as needed to convert a quantity from one unit to another.
The key is to choose a conversion factor that will cancel out the original unit, leaving you with the desired unit. For example, to change meters to kilometers, you would divide by 1000 because there are 1000 meters in a kilometer. Conversely, if moving from grams to centigrams, you multiply by 100 because there are 100 centigrams in a gram.

Metric System

The metric system is an international system of measurement that is decimal-based, meaning it increases or decreases in powers of ten. It is the most widely used measurement system worldwide.
One of its main advantages is its simplicity and ease of conversion, making international communication more straightforward. Each unit scale is based on factors of 10, allowing conversions to be performed by simply moving the decimal point.
For example, converting 5 meters to kilometers involves dividing by 1000, resulting in 0.005 kilometers. This ease of conversion stems from the consistent use of metric prefixes:

• Kilo: Factor of 1000
• Centi: Factor of 0.01
• Deci: Factor of 0.1
• Nano: Factor of 0.000000001

This structure allows anyone to quickly adapt and convert units as needed, simply by shifting the decimal point.

Problem-Solving Steps

Approaching metric conversions systematically can greatly enhance your efficiency and accuracy. Here's a sequence of steps to ensure a smooth conversion process:
1. **Identify the units involved.** Understand which unit you are converting from and which unit you will convert to.
2. **Choose the correct conversion factor.** Consult a reliable list of conversion factors to ensure accuracy.
3. **Set up the conversion equation.** Ensure that units cancel out, leaving only the desired unit. This usually involves multiplying or dividing by the conversion factor.
4. **Perform the calculation.** Carefully execute the multiplication or division.
5. **Review your result.** Double-check to see if the converted value makes sense in the context of the original problem.
By following these steps, you can effectively manage conversions, develop a better understanding of unit relationships, and avoid common pitfalls.