Let's break down the problem step by step: ### **Given:** \[ 32 \frac{\mu g}{dL} \times [\text{?}] = \frac{g}{mL} \] We want to convert **micrograms per deciliter (\(\mu g/dL\))** into **grams per milliliter (\(g/mL\))**. --- ## **Step 1: Write out the units to convert** - \( \mu g \) (micrograms) to \( g \) (grams) - \( dL \) (deciliters) to \( mL \) (milliliters) --- ## **Step 2: Conversion factors** **Micrograms to grams:** - \( 1 \, g = 1,000,000 \, \mu g \) - So, \( 1 \, \mu g = 10^{-6} \, g \) **Deciliters to milliliters:** - \( 1 \, dL = 100 \, mL \) - So, \( 1 \, mL = 0.01 \, dL \) --- ## **Step 3: Set up the conversion** We want to cancel out \( \mu g \) and \( dL \) and get \( g \) and \( mL \): \[ 32 \frac{\mu g}{dL} \times \left( \frac{1\,g}{1,000,000\,\mu g} \right) \times \left( \frac{1\,dL}{100\,mL} \right) \] --- ## **Step 4: Combine the fractions** Multiply the conversion factors: \[ \left( \frac{1\,g}{1,000,000\,\mu g} \right) \times \left( \frac{1\,dL}{100\,mL} \right) \] --- ## **Step 5: Simplify** When multiplying: - Numerators: \(1\,g \times 1\,dL = g\,dL\) - Denominators: \(1,000,000\,\mu g \times 100\,mL = 100,000,000\,\mu g\,mL\) So, \[ \frac{g\,dL}{100,000,000\,\mu g\,mL} \] But for unit cancellation, let's see it with the original factor: \[ \frac{1\,g}{1,000,000\,\mu g} \times \frac{1\,dL}{100\,mL} = \frac{1\,g}{1,000,000\,\mu g} \times \frac{1}{100}\frac{dL}{mL} \] So, the complete conversion factor is: \[ \left( \frac{1\,g}{1,000,000\,\mu g} \right) \times \left( \frac{1\,dL}{100\,mL} \right) \] --- ## **Step 6: Final Answer** **The missing part of the equation is:** \[ \boxed{ \left( \frac{1\,g}{1,000,000\,\mu g} \right) \left( \frac{1\,dL}{100\,mL} \right) } \] --- ### **Optional: Calculate the final value** \[ 32 \frac{\mu g}{dL} \times \left( \frac{1\,g}{1,000,000\,\mu g} \right) \times \left( \frac{1\,dL}{100\,mL} \right) = 32 \times \frac{1}{1,000,000} \times \frac{1}{100} \frac{g}{mL} \] \[ = 32 \times \frac{1}{100,000,000} \frac{g}{mL} \] \[ = \frac{32}{100,000,000} \frac{g}{mL} \] \[ = 3.2 \times 10^{-7} \frac{g}{mL} \] --- ### **Summary Table** | Step | Description | Conversion Factor | |------|-------------|------------------| | 1 | \(\mu g \to g\) | \(\frac{1\,g}{1,000,000\,\mu g}\) | | 2 | \(dL \to mL\) | \(\frac{1\,dL}{100\,mL}\) | --- ## **Final Answer (for the missing part):** \[ \boxed{ \left( \frac{1\,g}{1,000,000\,\mu g} \right) \left( \frac{1\,dL}{100\,mL} \right) } \]