Question

Perform the following metric-metric conversions: (a) \(7.50 \mathrm{~km}\) to \(\mathrm{Gm}\) (b) \(750 \mathrm{Mg}\) to \(\mathrm{Tg}\) (c) \(0.750 \mathrm{pL}\) to \(\mu \mathrm{L}\) (d) \(0.000750 \mathrm{~ms}\) to \(\mathrm{ns}\)

Short answer

(a) 0.00750 Gm, (b) 0.000750 Tg, (c) 0.000000750 μL, (d) 750 ns.

Step-by-step solution

Conversion of kilometers to gigameters

To convert kilometers to gigameters, we need to understand the conversion factor between these units. One kilometer is equal to \(10^{-3}\) gigameters, because: \ - Kilo- means \(10^3\) - Giga- means \(10^9\)Thus, \(1 \mathrm{~km} = 10^{-3} \mathrm{~Gm}\). Therefore, \(7.50 \mathrm{~km} = 7.50 \times 10^{-3} \mathrm{~Gm} = 0.00750 \mathrm{~Gm}.\)

Conversion of megagrams to teragrams

Megagrams and teragrams relate through powers of ten: \ - Mega- means \(10^6\) - Tera- means \(10^{12}\)Hence, \(1 \mathrm{~Mg} = 10^{-6} \mathrm{~Tg}\). Therefore, \(750 \mathrm{Mg} = 750 \times 10^{-6} \mathrm{~Tg} = 0.000750 \mathrm{~Tg}.\)

Conversion of picoliters to microliters

To convert picoliters to microliters, use the following conversions: \ - Pico- means \(10^{-12}\) - Micro- means \(10^{-6}\)Thus, \(1 \mathrm{pL} = 10^{-6} \mathrm{~1} \mathrm{L}\). Therefore, \(0.750 \mathrm{pL} = 0.750 \times 10^{-6} \mathrm{~1} \mathrm{L} = 0.000000750 \mathrm{~1} \mathrm{L}.\)

Conversion of milliseconds to nanoseconds

The conversion factors between these units are: \ - Milli- means \(10^{-3}\) - Nano- means \(10^{-9}\)Thus, \(1 \mathrm{~ms} = 10^{6} \mathrm{~ns}\). Therefore, \(0.000750 \mathrm{~ms} = 0.000750 \times 10^{6} \mathrm{~ns} = 750 \mathrm{~ns}.\)

Key concepts

Metric Units

In the world of measurements, the metric system is a universal language, enabling us to understand and communicate quantities efficiently. The metric system is based on decimals, making it easy to convert between different units. This system uses a variety of units, each represented by prefixes that denote a power of ten.

Some common metric units are meters for length, grams for weight, and liters for volume. Distinguishing metric units from the imperial system—such as feet, pounds, and gallons—is important as they are not interchangeable without conversion.

To illustrate, a kilometer (\( ext{km} \)) and a gigameter (\( ext{Gm}\)) both measure length but on different scales. Similarly, milligrams (\( ext{mg} \)) and kilograms (\( ext{kg} \)) are both units of mass within the metric system.

Conversion Factors

Conversion factors are the multipliers that allow us to shift from one unit to another without losing the magnitude of our original measurement. They are crucial in fields like science, engineering, and everyday tasks that require precise measurement.

For example, when converting kilometers to gigameters, we use the conversion factor \( 10^{-3} \), because one kilometer equals \( 0.001 \) gigameters. To perform this conversion, multiply the number of kilometers by \( 10^{-3} \).

Similarly, going from megagrams to teragrams involves a conversion factor of \( 10^{-6} \), where you multiply the original number of megagrams by this factor to find the equivalent in teragrams.

• Step-by-step, identify the original unit and the target unit.
• Determine the conversion factor from prefix meanings or a conversion chart.
• Multiply or divide, as necessary, using the conversion factor to make the unit shift.

Prefixes in Metric System

Prefixes in the metric system help us express very large or very small quantities in a manageable way. These prefixes represent powers of ten and are standardized globally, ensuring consistent scientific communication.

Some common prefixes include:

• Kilo- (\(10^3\)): 1 kilometer (km) = 1,000 meters.
• Mega- (\(10^6\)): Commonly used in terms like megabytes (MB).
• Tera- (\(10^{12} \)): Often appears in the context of computer storage, like terabytes (TB).
• Giga- (\(10^9 \)): In large distances, such as gigameters (Gm).
• Milli- (\(10^{-3}\)): As in milliliters (mL) or milliseconds (ms).
• Nano- (\(10^{-9}\)): Nanoseconds (ns) are an example.

By understanding these prefixes, interpreting measurements within different scales becomes straightforward. For conversions, simply replace the original prefix with another while adjusting by the difference in powers of ten.