Multiple Unit Multipliers: Metric System of Measurement

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions, such as Grand Rapids Community College, Pikes Peak Community College, and Austin Peay State University.

Unit multipliers are a great tool for when you want to convert from one unit to another. This lesson will review unit multipliers in the metric system, and we will look at how to use multiple unit multipliers when unit conversion is more involved.

Unit Multipliers and the Metric System

Suppose Jesse bought an encyclopedia, and the salesperson said that it weighs 1 kilogram. Meanwhile, Jack bought the same encyclopedia, but his salesperson said that it weighs 1000 grams. Hmmm…which salesperson is right? As it turns out, they both are, because:

  • 1000g = 1kg

They are both right

Relationships between units like these result in unit multipliers. Unit multipliers are fractions, with units, that equal 1.

Notice that for any number a,

  • a/a = 1

In other words, if we divide a quantity by an equal quantity, we get 1. Consider the relationship 1 kilogram = 1000 grams again. Since 1 kilogram and 1,000 grams are equal quantities, we have:

  • 1 kg/1000g = 1 and 1000g/1kg = 1

The fractions 1 kg/1000g and 1000g/1kg are examples of unit multipliers.

Great, so why is this useful? Unit multipliers can be used to convert units. In the metric system, the four main units of measurement are seconds for time, kilograms for mass, liters for capacity, and meters for length. We can use metric prefixes to find relationships that lead to unit multipliers.


From this table, we can come up with relationships that lead to unit multipliers. For instance, notice that the prefix milli- corresponds to thousandths (0.001). Therefore, putting the prefix milli- in front of one of our standard units means it is one-thousandth of that unit. Consider milliseconds. Because the prefix is milli-, we have that 1 millisecond is 0.001 seconds, or 1/1000 seconds, giving the relationship 1 millisecond = 1/1000 seconds, or 1000 milliseconds = 1 second. From this, we have the unit multipliers 1000ms/1s and 1s/1000ms.

Suppose we want to convert 5 kilograms to grams. Again, if we multiply 5kg by a unit multiplier, it's the same as multiplying it by 1, so it doesn't change the value of 5 kilograms. If we multiply 5kg by 1000g/1kg, the kilogram units cancel out.


We see that multiplying by the unit multiplier, that 5 kilograms = 5000 grams. That's how we use unit multipliers to convert units in the metric system:

  1. Use relationships to find appropriate unit multipliers.
  2. Multiplying by the unit multiplier will cancel out the old unit you're trying to change. If the old unit is in the numerator, multiply by the unit multiplier with the old unit in the denominator. If the old unit is in the denominator, multiply by the unit multiplier with the old unit in the numerator.


Multiple Unit Multipliers

Suppose you have plans with your friend in 30 minutes. Since you have 30 minutes to kill, you decide to convert those 30 minutes to milliseconds. Hmmm…this one's not as obvious. We know there are 60 seconds in one minute and our metric prefixes tell us that there are 1000 milliseconds in 1 second, but we're not sure how many milliseconds are in a minute.

When this is the case, we can use multiple unit multipliers. To do this, we first convert 30 minutes to seconds using the unit multiplier that corresponds to the relationship 60 seconds = 1 minute. Then, we convert that to milliseconds using the unit multiplier that corresponds to 1000 milliseconds = 1 second.

Because 60 seconds = 1 minute, we have two unit multipliers:

  • 60s/1min and 1min/60s

Since we are converting 30 minutes to seconds, we want to use the unit multiplier that will cause the minutes to cancel out, so we want to use 60s/1min.


We get that 30 minutes = 1800 seconds. Now, we want to convert 1800 seconds to milliseconds using either 1000ms/1s or 1s/1000ms. Which one should we use? Well, we want to cancel out the second units when we multiply, so we use 1000ms/1s.


We end up with 1,800,000 milliseconds, so there are 1,800,000 milliseconds in 30 minutes. That's a lot of milliseconds!

We can also do this in one multiplication problem by multiplying the units we are trying to convert by both unit multipliers in one step.


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