1. Addends’ Units Must Agree When does 2 + 2 not equal 4? When the units don’t agree. 2 hours + 2 minutes does not equal 4 of anything. So, when trying to add quantities whose units do not agree, you must apply conversion factors to make the addends’ units agree. What is 2 dollars + 2 dimes + 2 nickels? This problem is so easy and commonplace that we don’t even need to write it out. We do similar problems in our head every day. When we do these problems, though, we are implicitly converting the units to agree. Some people probably think of dimes and nickels in terms of.10 and.05 dollars, respectively. Others might convert the dimes and nickels to cents, then add the dollars to that sum. We will set up our solution to obtain an answer in dollars. 2 dollars 2 dimes 1 dollar 2 nickels 1 dollar__ 1 1 10 dimes 1 20 nickels + () + ()() * ()() *=

2. Addends’ Units Must Agree 2 dollars 1 * 2 dimes dollar 1 * 2 nickels dollar__ 1 1 * 10 dimes 1 * 20 nickels + () + () * () *= 2 dollars 2 dollars 2 dollars_ 1 10 20 + () + ()() = 2 dollars + 0.2 dollars + 0.1 dollars = 2.3 dollars = $2.30 Let’s try another problem where we need to units to agree before adding. A partially-filled fish tank contains 12 gallons of water. The owner has a five-liter bucket. After he adds two bucketfuls of water, how much does the fish tank contain? () + ()() * () *= 12 gallons 2 buckets 5 liters 1 gallon 1 1 1 bucket 3.79 liters

3. Addends’ Units Must Agree 12 gallons 1 * 2 * 5 buckets liters gallon 1 1 * 1 * 3.79 bucket liters () + () *= There are several things to note in this solution: We used a known conversion factor from liters to gallons. If this conversion is not known by the student, it must be looked up. For the second addend, we could have left out the conversion from buckets to liters. We could have done that conversion in our heads, leading us to set up the solution as adding 10 liters to 12 gallons. This would in no way change our final result. We could have done the conversion from liters to gallons in two (or more) steps. For example, we could have converted from liters to quarts, then from quarts to gallons. Again, our final result would have been the same. 12 gallons 10 gallons 1 3.79 () + () = 12 gallons + 2.64 gallons = 14.64 gallons

4. Addends’ Units Must Agree When there are units in both the numerator and the denominator, all the units must agree for addition to be correct. Consider the case where Car A is traveling at 50 miles/hour, and Car B is traveling 5 miles/hour faster. How fast is Car B going? 50 miles 5 miles 55 miles 1 hour 1 hour 1 hour () + () == 55 miles/hour Use explicit conversion factors to solve the following problems: 1) Shawn is 6 feet, 8 inches tall. How many inches tall is Shawn?

5. Addends’ Units Must Agree 2) A standard marathon is 26 miles, 385 yards. What is this distance in yards?