Lesson 2: Converting Between SI Units - Using a Proportion

   Constructing Knowledge

Conversion Strategy 1: Use a Proportion

A proportion is a statement of equality between two ratios. Proportions can be used to convert between any pair of units for which a conversion ratio is known.

Proportion
A statement of equality between two ratios.

Ratio
A relationship between two quantities. A ratio is often expressed as a fraction.

To convert from one SI unit to another using proportions follow the steps below.

  1. Determine a conversion ratio for the two units, using the SI Prefix Table.
  2. Assign a variable for the unknown and create a second ratio using the information from the question.
  3. Write a proportion with both ratios and solve for the unknown (the variable).

   Multimedia

A video demonstrating unit conversion using a proportion is provided.


EXAMPLE 1

Convert 75 kL into cL.

Solution


Step 1: Determine a conversion ratio for the two units.

Using the SI Prefix Table, note that 0.001kL = 100cL, which can also be writtten as either \(\frac{0.001\,\text{kL}}{100\,\text{cL}}\) and \(\frac{0.001\,\text{kL}}{100\,\text{cL}}=1\).

Step 2: Assign a variable for the unknown and create a second ratio using the information from the question.

Let x represent the unknown number of centiliters.

Then, \(x\,\,\text{cL}=75\,\text{kL}\) and \(\frac{x\,\text{cL}}{75\,\text{kL}}=1\).

The expression \(\frac{75\,\text{kL}}{x\,\,\text{cL}}=1\) can also be used here. However, expressions with a variable in the denominator are often harder to work with.

Step 3: Write a proportion with both ratios, and solve for the variable.

Now there are unique ratios equal to 1, so they must be equal to each other. Set up a proportion. Choose the conversion ratio format that has the same unit location as the ratio containing the variable. This will allow for a proper conversion.


\(\begin{align} \frac{x\,\,\text{cL}}{75\,\text{kL}}&=\frac{100\,\text{cL}}{0.001\,\text{kL}} \\ \\ \frac{x\,\,\text{cL}}{75\,\text{kL}}\times {\color{red}{75\,\text{kL}}}&=\frac{100\,\text{cL}}{0.001\,\text{kL}}\times {\color{red}{75\,\text{kL}}} \\ \end{align}\)

Often, the units are not shown with the variable, and "x" is written instead of "x cL". When the same value or unit is in the numerator and in the denominator, they are eliminated.

\(\begin{align} \frac{x\,\,\text{cL}}{\cancel{75}\cancel{\text{kL}}}\times \cancel{75}\cancel{\text{kL}}&=\frac{100\,\text{cL}}{0.001\cancel{\text{kL}}}\times 75\cancel{\text{kL}} \\ \\ x\,\,\text{cL}&=7\,500\,000\,\text{cL} \end{align}\)

Notice that the kilolitre units are eliminated on both sides, leaving just the centilitre units.


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