Question

Set up the problem using dimensional analysis and make the conversion as indicated. \(5 \mathrm{mcg}=\) ______ \(\mathrm{mg}\)

Short answer

5 mcg = 0.005 mg

Step-by-step solution

Understand the Conversion

Dimensional analysis is a method to convert between different units by using conversion factors. Here, we want to convert from micrograms ( mcg ) to milligrams ( mg ).

Identify the Conversion Factor

Identify the relationship between micrograms and milligrams: 1 mg = 1000 mcg . This means 1 microgram ( mcg ) is equal to 0.001 milligrams ( mg ).

Set up the Conversion Equation

Start with the given quantity, 5 mcg, and multiply it by the conversion factor to change the units:\[5 \, \text{mcg} \times \left( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \right)\]

Perform the Calculation

Cancel the micrograms unit and perform the multiplication:\[5 \, \text{mcg} \times \left( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \right) = 0.005 \, \text{mg}\]

Verify the Result

Check the logic of the conversion: You have correctly divided the number of micrograms by 1000 to get the equivalent number in milligrams. The unit conversion was done by using the fact that 1000 mcg = 1 mg.

Key concepts

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. This is often necessary in scientific and real-world settings where measurements need to be understood in different systems or scales.

Dimensional analysis is a straightforward method used for unit conversion, utilizing conversion factors to calculate the equivalent value in a different unit. A conversion factor is a ratio that expresses how many of one unit is equivalent to another.

For example, if you know that 1 mile is equivalent to 1.60934 kilometers, this conversion factor can be used to convert distances measured in miles to kilometers and vice versa.

To perform a unit conversion, you follow these steps:

• Identify the given value and its unit.
• Determine the conversion factor that relates the given unit to the desired unit.
• Multiply the given value by the conversion factor, ensuring the original unit cancels out, leaving only the desired unit.
• Double-check the calculation to ensure the conversion was carried out correctly.

Dimensional analysis ensures that units are systematically converted by making sure numerator and denominator units cancel appropriately.

Micrograms to Milligrams

Converting from micrograms (mcg) to milligrams (mg) is a common task, especially in fields like medicine or chemistry, where precise measurements are crucial.

To understand this conversion, it is important to realize the relationship between these metric units. In the metric system:

• 1 milligram is equal to 1000 micrograms.
• This means that the ratio or conversion factor between milligrams and micrograms is 1 mg = 1000 mcg.
• Conversely, 1 mcg is equal to 0.001 mg, since dividing both sides of the 1 mg = 1000 mcg equation by 1000 gives the reversed conversion factor.

For pedagogical precision, always remember that you are moving to a larger unit (from mcg to mg), so your numerical value must decrease accordingly.

When converting micrograms to milligrams, such as in the exercise where 5 mcg was converted to mg, you standardly multiply the microgram value by the conversion factor of \( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \).

This calculated: \[ 5 \, \text{mcg} \times \left( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \right) = 0.005 \, \text{mg} \], is straightforward and ensures precise scientific communication.

Mathematical Calculations

Mathematical calculations form the backbone of solving problems involving unit conversions and require meticulous attention to accuracy to ensure correct results.

When handling dimensional analysis, it is important to use multiplication and division correctly to navigate between differing units.

Here’s a simplified approach to mathematical calculations in this context:

• Understand the numbers: Begin with the original measurement, such as 5 mcg in our example.
• Follow the conversion: Use the established ratio (conversion factor) like \( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \).
• Execute the multiplication: Multiply the original quantity by the conversion factor, ensuring clear cancellation of the original unit.
• Double-check results: Verify whether the final number has been correctly calculated to the desired precision, here resulting in 0.005 mg.

By practicing these steps, confidence in performing unit conversions alongside arithmetic operations increases, simplifying the integration of mathematics in everyday applied contexts.