Performing Arithmetic with Various Units of Measure

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Different units of measure can make our arithmetic problems much more complicated. In this lesson, we'll learn how to perform arithmetic using various units of measure and work through some examples.

Units of Measure

Units of measure can really complicate an arithmetic problem because you can't add apples and oranges together (unless you're making a fruit salad!). You must make your units match before you can combine numbers in an arithmetic problem. For example, say that John, Jan, and Marty want to buy a gift for their mother, a beautiful scarf that will perfectly match her favorite coat. It's $45.75 and they're trying to figure out if they have enough money.

  1. John has a $20 bill from his birthday and six quarters from washing cars.
  2. Jan has two $5 bills for walking dogs, 12 dimes from tips, and 123 pennies in a sack from her penny collection.
  3. Marty has 160 nickels from his paper route and 13 $1 bills from tips.

They pour out all their money on the table and stare at it. Is it enough money to buy the scarf?

Arithmetic is the group of methods including adding, subtracting, multiplying, and dividing used to combine numbers together and solve problems. Units of measure are the word or symbol labels that say how you are measuring things, such as inches, ounces, kilograms, or cents.

Converting One Unit to Another

The rule in arithmetic is that you can't combine different units of measure to come up with an answer. For example, if you add five tablespoons of flour to two cups of flour, how much do you have? If you can convert one of the units to the other, you can give a simple answer. Let's figure out how to do that.

Unit conversion means to change a quantity from one unit into the equivalent quantity (same total amount) using a different unit. Unit conversions use a conversion factor, a statement of equality between two different units, such as 3 feet = 1 yard, 16 ounces = 1 pound, or (more to the point) 16 tablespoons = 1 cup. You can convert from tablespoons to cups in four steps:

  1. Find a conversion factor (16 tablespoons = 1 cup).
  2. Multiply by the unit you want to end up with (multiply by 1 cup).
  3. Divide by the unit you want to get rid of (divide by 16 tablespoons).
  4. Remove the old label and put in the new one (drop the 'tablespoons' label and put in the 'cups' label).

So, let's solve our flour example. Our conversion factor says that 16 tablespoons = 1 cup. To convert 5 tablespoons to cups we'll multiply by the number of cups (the unit we want) and divide by the number of tablespoons (the unit we don't want) in the conversion factor.

Converting tablespoons to cups

Now we can calculate the total amount of flour.

Total flour in cups


Let's go back to our original problem with John, Jan, and Marty. Since the scarf is priced in dollars ($45.75), they decide to convert all their different money units into dollars and then add it all together. John has a $20 bill and six quarters. Since the $20 bill is already in dollars, he just needs to convert the quarters. One dollar = 4 quarters.

Converting quarters to dollars

Adding up his $20 plus the $1.50, he realizes he has $21.50 to pay toward the scarf. Now, it's Jan's turn. Jan has two $5 bills, 12 dimes, and 123 pennies. The two $5 bills add up to $10, but what about the coins? There are 10 dimes in a dollar, and 100 pennies in a dollar.

Converting dimes to dollars

Converting pennies to dollars

Adding up her dollar amounts, Jan has $10.00 + $1.20 + $1.23 = $12.43 to add in for the scarf. Okay, how about Marty? Marty has 160 nickels and 13 $1 bills. The 13 bills just make $13, but what about the nickels? There are 20 nickels in a dollar.

Converting nickels to dollars

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