# Unit Conversions

During your studies and as part of everyday life, you will be required convert units that are within the same system, for example, from centimetres to metres.

There are two common methods to convert units; using proportion equations and the 'unit cancelling method'.

__Proportion method of unit conversion__

Let's take a look at a hypothetical question – If a student weights 80 kg, how much does he weight in pounds?

**Step 1:** Find a conversion factor that relates the units of interest – in this case, 1 kg = 2.205 pounds

**Step 2:** Express the conversion factor as a ratio (fraction)

$$ \newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}} 1\ctext{ kg} \;/\; 2.204 \ctext{ pounds} $$

**Step 3:** Express the problem as a ratio with an unknown:

$$ 80 \ctext{ kg}\;/\; ? \ctext{ pounds} $$

If we let *x* be the unknown number of pounds, then this ratio can be written as:

$$ 80 \ctext{ kg}\;/\; x \ctext{ pounds} $$

**Step 4:** Set up a proportion equation such that the units in both numerators are the same and the units in both denominators are the same:

$$ \frac{1\;kg}{2.204\ctext{ pounds}} = \frac{80\;kg}{x \ctext{ pounds}} $$

**Step 5:** Solve for the unknown:

$$ \begin{align*} x &= \frac{80\;kg (2.205 \ctext{ pounds})}{1\;kg} \cr &= 176.4 \ctext{ pounds} \end{align*} $$

Here is a summary for the proportion method of unit conversion:

- Find a conversion factor that relates the units of interest to one another
- Express the conversion factor as a ratio in fraction form
- Express the problem to be solved as a ratio with an unknown.
- Set up a proportion equation such that the units in both numerators are the same and the units in both denominators are the same.
- Solve for the unknown.

Unit cancelling method of unit conversion

Unit cancelling method of unit conversion

Let's take a look at the same hypothetical question above – If a student weights 80 kg, how much does he weigh in pounds?

**Step 1:** Write the problem as an equation, 80kg = ? pounds

**Step 2:** Find a conversion factor such that the units cancel. In this case, a pound is 0.454 kg. This conversion factor is expressed as a ratio:

$$ 1\;lb\; /\; 0.454\;kg $$

*Note: remember that all conversion factors compare a quantity to one unit of another quantity.*

**Step 3:** Place the conversion factor into the equation:

$$ 80\; kg \times 1\;lb\; /\; 0.454\;kg = ? $$

**Step 4:** Solve the equation – multiply 80 kg by the conversion factor:

$$ 80\; kg \times 1\;lb\; /\; 0.454\;kg = 176.2\;lb $$

*Note: the difference in both calculations is simply due to round up of the decimal place.*

Here is a summary for the unit cancelling method:

- Write the problem as an equation
- Find one or more conversion factors such that the units cancel, leaving only desired units.
- Put the conversion factor(s), expressed as ratio(s), into the equation.
- Solve the equation.

A good tip to remember is that when we change a large unit to a small unit, the numerical part of the answer is larger than the original answer. But when we change a small unit to a large unit, the numerical part of the answer is smaller than the original number.

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