# 3.9: Metric Unit Conversions

You are training for a 10-kilometer run by doing laps on a 400-meter track. You ask yourself "How many times do I need to run around this track in order to cover ten kilometers?" (more than you realize). By using dimensional analysis, you can easily determine the number of laps needed to cover the $$10 \: \text{km}$$ distance.

### Metric Unit Conversions

The metric system's many prefixes allow quantities to be expressed in many different units. Dimensional analysis is useful to convert from one metric system unit to another.

Example 3.9.1

A particular experiment requires $$120 \: \text{mL}$$ of a solution. The teacher knows that he will need to make enough solution for 40 experiments to be performed throughout the day. How many liters of solution should he prepare?

Solution

Step 1: List the known quantities and plan the problem.

Known

• 1 experiment requires $$120 \: \text{mL}$$
• $$1 \: \text{L} = 1000 \: \text{mL}$$

Unknown

• $$\text{L}$$ of solution for 40 experiments

Since each experiment requires $$120 \: \text{mL}$$ of solution and the teacher needs to prepare enough for 40 experiments, multiply 120 by 40 to get $$4800 \: \text{mL}$$ of solution needed. Now you must convert $$\text{mL}$$ to $$\text{L}$$ by using a conversion factor.

Step 2: Calculate

$4800 \: \text{mL} \times \frac{1 \: \text{L}}{1000 \: \text{mL}} = 4.8 \: \text{L}$

Note that the conversion factor is arranged so that the $$\text{mL}$$ unit is in the denominator and thus cancels out, leaving $$\text{L}$$ as the remaining unit in the answer.

A liter is much larger than a milliliter, so it makes sense that the number of liters required is less than the number of milliliters.

### Two-Step Metric Unit Conversions

Some metric conversion problems are most easily solved by breaking them down into more than one step. When both the given unit and the desired unit have prefixes, one can first convert to the simple (unprefixed) unit, followed by a conversion to the desired unit. An example will illustrate this method.

Example 3.9.2

Convert $$4.3 \: \text{cm}$$ to $$\mu \text{m}$$.

Solution

Step 1: List the known quantities and plan the problem.

Known

• $$1 \: \text{m} = 100 \: \text{cm}$$
• $$1 \: \text{m} = 10^6 \: \mu \text{m}$$

Unknown

• $$4.3 \: \text{cm} = ? \: \mu \text{m}$$

You may need to consult a table for the multiplication factor represented by each metric prefix. First convert $$\text{cm}$$ to $$\text{m}$$, followed by a conversion of $$\text{m}$$ to $$\mu \text{m}$$.

Step 2: Calculate

$4.3 \: \text{cm} \times \frac{1 \: \text{m}}{100 \: \text{cm}} \times \frac{10^6 \: \mu \text{m}}{1 \: \text{m}} = 43,000 \: \mu \text{m}$

Each conversion factor is written so that unit of the denominator cancels with the unit of the numerator of the previous factor.