Question

Set up the problem using dimensional analysis and make the conversion as indicated. \(1.25 \mathrm{mcg}=\) ______ \(\mathrm{mg}\)

Short answer

1.25 mcg is 0.00125 mg.

Step-by-step solution

Identify the Units Given and Required

We start by identifying the units provided in the problem and the units we need to convert to. Here, we have micrograms (mcg) that need to be converted into milligrams (mg).

Determine the Conversion Factor

Next, we determine the conversion factor between these units. We know that 1 milligram (mg) is equivalent to 1000 micrograms (mcg). This conversion factor can be expressed as \[1 \text{ mg} = 1000 \text{ mcg}\].

Set Up the Dimensional Analysis Equation

Using the conversion factor, set up the dimensional analysis equation to convert 1.25 mcg to mg. It looks like this: \[1.25 \text{ mcg} \times \frac{1 \text{ mg}}{1000 \text{ mcg}}\].

Perform the Calculation

Perform the multiplication and division to complete the conversion: \[1.25 \times \frac{1}{1000} = 0.00125 \text{ mg}\].

Confirm Unit Cancellation

Ensure the unit mcg cancels out, leaving the desired unit mg. After performing the calculation, the unit mcg in the numerator and the denominator cancels out, confirming our conversion is correct.

Key concepts

Unit Conversion

Unit conversion is an essential concept in dimensional analysis and occurs frequently when working with measurements in different units. It allows you to translate a quantity into another unit without changing the actual value. This skill is crucial in science and everyday tasks like cooking or calculating medication doses. To perform a unit conversion, follow these steps:

• Identify the initial unit and the unit you want to convert to.
• Determine the conversion factor between these two units.
• Use dimensional analysis to perform the conversion efficiently.

Dimensional analysis helps ensure that each step of your conversion makes mathematical sense, as it relies on multiplication that converts units without errors. By mastering unit conversions, you can seamlessly navigate between different measurement systems.

Micrograms to Milligrams

Converting from micrograms (mcg) to milligrams (mg) is a common task in chemistry and medicine. Since micrograms are smaller than milligrams, you need to multiply by a conversion factor that accounts for this size difference. The conversion factor is straightforward:

• 1 milligram equals 1000 micrograms.

This indicates that micrograms are a much smaller unit compared to milligrams, as it takes 1000 micrograms to make up just one milligram. When performing this conversion, you will multiply the number of micrograms by \( \frac{1 \, \text{mg}}{1000 \, \text{mcg}} \) to get milligrams. Thus, converting micrograms to milligrams simplifies working with large values by moving the decimal point to the left.

Calculation Steps

Calculating conversions using dimensional analysis involves a series of structured steps to ensure accuracy. Let’s look at how to convert 1.25 micrograms to milligrams:

**Step 1:** Start by identifying the units you have (mcg) and the units you need (mg).
**Step 2:** Use the known conversion factor, where 1 mg equals 1000 mcg.
**Step 3:** Set up your dimensional analysis equation. Here, it would be \(1.25 \, \text{mcg} \times \frac{1 \, \text{mg}}{1000 \, \text{mcg}}\).
**Step 4:** Perform the calculation: multiply 1.25 by the fraction \( \frac{1}{1000} \) to obtain 0.00125 mg.
**Step 5:** Verify your results; ensure the units cancel appropriately in your equation, leaving you with milligrams.

Following these calculation steps correctly allows for reliable translations between units. This structured approach reduces errors in scientific calculations and guarantees unit accuracy throughout.