4.08 Convert units

Converting units allows us to compare two quantities that have different units, such as a number of minutes compared to a number of hours, a distance in kilometers to a distance in meters, a duration in days to a duration in weeks, and so on.

To compare these types of quantities, we will need to convert one of the quantities to use the same units as the other. It does not matter which one we convert, we will end up with exactly the same ratio in the end.

We can use a conversion factor, a number used to change one set of units to another, by multiplying or dividing. We must use the appropriate conversion factor when converting. For example, to convert miles to kilometers, the appropriate conversion value is 1 mile equals 1.6 kilometers. To convert minutes to hours, the appropriate conversion value is 60 minutes equals 1 hour.

When using the conversion factor which is written as a ratio in fraction form, it is important that we have the desired unit as the numerator. This will ensure that if we are going from a smaller unit to a larger unit that we will divide, and if we are going from a larger unit to a smaller unit, we will multiply.

For example, if we want to convert 8 feet into inches, the conversion value is 1 \text{ ft} = 12 \text{ in}.

Notice that the unit of the given number is \text {ft} and the conversion factor has the unit \text {ft} in the denominator to cancel the unit \text {ft} and the result will be the desired unit, \text {in}.

However, if we want to convert 96 inches into feet, the conversion value is the same, but we put the feet in the numerator.

Let's look at the following examples.

Examples

Example 1

The ratio of kilograms to pounds is 1\text{:}2.2. Use this fact to complete the workings below for finding out how many pounds are equal to 10 kilograms.

Worked Solution

Create a strategy

To get from 1 kilogram up to 10 kilograms we need to mulitply by 10. To keep the ratio equivalent, we need to do this to both sides of the ratio.

Apply the idea

Multiply both sides by 10.

Reflect and check

Using the conversion factor 1 \text{ kg}= 2.2 \text{ lbs}:

\displaystyle 10 \text{ kg} \times \dfrac{2.2\text{ lbs}}{1\text{kg}}

\displaystyle =

\displaystyle 22 \text{ lbs}

Mutltiply the given by the conversion factor

Notice that the unit kg of the given can be cancelled by the kg in the denominator of the conversion factor leaving the desired unit, lbs.

Watch question walkthrough

Example 2

Burj Al Arab, one of the tallest hotels in the world is 321\text{ m} tall. Using the conversion 1 \text{ ft} = 30 \text{ cm}, find the height of the hotel in feet.

Worked Solution

Create a strategy

Multiply the given measurement by the conversion factor to convert meters to centimeters and then divide to go from centimeters to feet.

Apply the idea

\displaystyle 321\text{ m} \times \dfrac{100 \text{ cm}}{1 \text{m}}	\displaystyle =	\displaystyle ⬚ \text{ cm}	Set up the conversion with the desired unit in the numerator. We first need to convert \text {m} to \text{cm}
\displaystyle 321\text{ m} \times \dfrac{100 \text{ cm}}{1 \text{m}}	\displaystyle =	\displaystyle 32\,100 \text{ cm}	Multiply 100 in the numerator to convert meters to centimeters
\displaystyle 32\,100 \text{ cm} \times \dfrac{1 \text{ ft}}{30 \text{ cm}}	\displaystyle =	\displaystyle ⬚ \text{ ft}	Set up another conversion to convert \text {cm} to \text {ft}. This time we are converting to a larger unit, so we need to divide.
\displaystyle 32\,100 \text{ cm} \times \dfrac{1 \text{ ft}}{30 \text{ cm}}	\displaystyle =	\displaystyle 1070 \text{ ft}	Divide by 30 to convert centimeters to feet

The hotel's height is 1070 \text{ ft}.