How to Do Unit Conversions – Ladder Method 3

When defining a unit, you first define a means to measure that physical quantity. Measuring means a method of calculating the quantity from other known quantities. People have developed standard units for this purpose. Science today uses an international system of units known as SI units. (SI comes from the French “Système International d’unités”). While SI is the standard for most cases, occasionally other systems are used. The old Imperial system is still in use in the United States for many common day to day uses. The metric system was the precursor to the modern SI system but has some minor differences.

It is common to have to convert a measurement between measuring systems. This is done using conversion factors. An example of a conversion factor is

1 km = 1000 m

There are 1000 meters in 1 kilometer. If you want to know how many meters are in 3 kilometers, you use this conversion factor to calculate the result.

unit conversion ladder method step 1

This is possible because you are just multiplying the measurement by 1. This can be seen by dividing both sides of the conversion factor equation by one of the units.

Divide both sides of the conversion factor by 1 km

unit conversion ladder method step 2
unit conversion ladder method step 3

The same is true if you divide both sides by 1000 m

unit conversion ladder method step 4
unit conversion ladder method step 5

It doesn’t change the value of the measurement when you multiply by the conversion factor. It just changes the units the measurement is compared to. This method of unit conversion is called “Unit Cancelling” because the undesired unit is canceled out by the conversion factor.

Take the 3 kilometer example above. The kilometer unit is canceled out leaving only meters on both sides of the equation.

Occasionally it will be necessary to perform multiple conversions in a row to get the units you need on a measurement. Since all you are doing is multiplying the base measurement by 1, you can perform all of the conversions as one long chain of conversion factors. This method is known as the “Ladder Method”. Each conversion step is another run of a ladder to cancel out the undesired units.

For example, if you were to convert 18 km/hr to m/s, you would have to use the following conversion factors:

1000 m = 1 km
60 min = 1 hr
60 s = 1 min

The goal in this conversion is to end up with meters in the numerator and seconds in the denominator. Set up your conversions so the undesired units are canceled out.

unit conversion ladder method step 6

If you go ahead and perform the final calculation, the above equation reduces to:

unit conversion ladder method step 7

I have marked the cancellation of each different undesired measurement in different colors to illustrate the ladder of calculations to arrive at the correct units. In this case, 18 km/hr = 5 m/s.