Question

Set up the problem using dimensional analysis and make the conversion as indicated. \(1.75 \mathrm{mg}=\) ______ \(\mathrm{mcg}\)

Short answer

1.75 mg = 1750 mcg.

Step-by-step solution

Understanding the Problem

We are tasked with converting 1.75 milligrams (mg) into micrograms (mcg). Dimensional analysis can help us understand the conversion relationship between these two units.

Identify Conversion Factor

Know that 1 milligram is equal to 1000 micrograms. This conversion factor will be used to change the units from milligrams to micrograms.

Set Up the Conversion

Write the initial quantity and multiply it by a fraction representing the conversion factor, which allows us to cancel out the mg units and introduce mcg units. \[ 1.75 \, \text{mg} \times \frac{1000 \, \text{mcg}}{1 \, \text{mg}} \]

Perform the Calculation

Multiply 1.75 by 1000 to convert milligrams to micrograms: \[ 1.75 \times 1000 = 1750 \, \text{mcg} \]

Check Units

Ensure that the units are correctly converted. After the calculation, mg units are canceled, leaving us with mcg as intended.

Key concepts

Unit Conversion

Unit conversion is an essential concept in many scientific and everyday applications. It allows us to switch between different units of measurement without changing the actual quantity. In this exercise, we're converting milligrams (mg) to micrograms (mcg). These units are both part of the metric system and are commonly used to measure mass.
The fundamental idea of unit conversion is to multiply by a ratio that equals 1. This ratio, known as the conversion factor, enables the units we're converting from to cancel out while introducing the new units. In this instance, our conversion factor is 1 mg = 1000 mcg.
When performing unit conversions, it's essential to keep track of your units by writing out each step meticulously. This eliminates mistakes and ensures accurate results.

Metric System

The metric system is a standardized system of measurement used worldwide, making it easier to perform conversions, especially in scientific contexts. It uses the base-10 system, which means each step in the metric scale differs by a power of ten.
This uniformity simplifies conversions, like those from milligrams to micrograms. In the metric system, kilo, milli, and micro are prefixes that correspond to factors of ten:

• Kilo- represents 1000 units.
• Milli- represents 1/1000 of a unit.
• Micro- represents 1/1,000,000 of a unit.

Understanding these prefixes and how they relate to each other is crucial for executing conversions accurately using dimensional analysis.

Step-by-Step Solution

When solving conversion problems, a step-by-step approach ensures that every part of the calculation is handled with precision. Let's break down the process used in the solution:
1. **Understanding the Problem:** Begin by identifying what you have (1.75 mg) and what you need (in mcg).2. **Identify Conversion Factor:** Use known relationships. Here, 1 mg = 1000 mcg is used.3. **Set Up the Conversion:** Multiply the original quantity by a conversion factor formatted as a fraction, enabling units to cancel out. \[ 1.75 \, \text{mg} \times \frac{1000 \, \text{mcg}}{1 \, \text{mg}} \] 4. **Perform the Calculation:** Carry out the multiplication to find the equivalent quantity in the new units. In this case: \[ 1.75 \times 1000 = 1750 \, \text{mcg} \]5. **Check Units:** Confirm that the original units have been canceled correctly, leaving you with the desired units. This is a critical final step to avoid unit errors.