One basic concept in math is that if we multiply a number by 1, the result is equal to the original number. For example,


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1 MA 35 Lecture  Introduction to Unit Conversions Tuesday, March 24, 205. Objectives: Introduce the concept of doing algebra on units. One basic concept in math is that if we multiply a number by, the result is equal to the original number. For example, () 7 = 7. A related concept is that any number (other than zero) divided by itself is equal to. For example, (2) 3 3 =. We use these ideas when we simplify fractions. For example, (3) 5 2 = = = 5 7 = 5 7. Of course, we don t usually write it all out like that. The shorthand way is to see a 3 in the top of the fraction, a 3 in the bottom, and those cancel. It s a good idea to remind ourselves that when we cancel, we re really noticing that we re simply multiplying by. (4) 5 2 = = 5 7. Cancelling units. When I talk about units, I m talking about things like feet, meters, miles per hour, pounds, etc. All of the units we use multiply and divide very naturally. For example, a unit for speed, like miles per hour, is thought of in terms of division, and we ll write (5) miles per hour = miles hour. If you walked 4 miles in 2 hours (i.e. you traveled 4 miles per 2 hours), then your speed would be (6) or 2 miles per hour. 4 miles 2 hours = 4 miles 2 hours = 2 miles hours, We ll extend our concepts of multiplying by and dividing a number by itself to units. This doesn t really make sense, but doing it will always work out right. Here are our two basic rules. Basic Principle. A measurement divided by an equal measurement is equal to. For example, we know that (7) foot = 2inches, so (8) and also (9) foot 2 inches =, 2 inches foot Basic Principle 2. A unit divided by itself is equal to. In other words, we can cancel units. =.
2 MA 35 Lecture  Introduction to Unit Conversions 2 For example, (0) foot foot =. We ll use these as follows. If we want to convert 2.5 feet into inches, we ll multiply by 2inches. That is, foot () 2.5 feet = 2.5 feet 2 inches foot = inches = 30 inches. The trick is to make sure your units cancel out right, and that ll help you to keep things straight. Quiz A. Convert 3.72 feet into inches. 2. Convert 07 inches into feet. More Examples What you ll find is that in your job, you ll do the same conversions over and over. If you re a nurse, you might convert milliliters to teaspoons all the time, and you ll have a single conversion formula for that (which you ll probably have memorized). Then occasionally you ll have to do a weird conversion, and you can t find a conversion formula for it. You should be able to find a chain of formulas that get you from one unit to the other, however. Let s look at a simple situation to illustrate how we can put the formulas together. Perhaps you have a refrigerator magnet with the following conversion formulas on it. mile = 5280 feet yard = 3 feet foot = 2 inches inch = 2.54 centimeters meter = 00 centimeters kilometer = 000 meters Example. Using only these formulas, convert 440 yards into miles. There isn t a formula for yards to miles, but there is a yards to feet equation, and a feet to miles equation. That s enough. (2) 440 yards = 440 yards 3 feet yard mile 5280 feet = miles = 0.25 miles Example 2. Using only these formulas, convert 0.5 miles into kilometers. We re converting English to metric, and we only have the inches to centimeters equation that relates English to metric units. So it looks like we can go miles to feet, feet to inches, inches to centimeters, centimeters to meters, and finally meters to kilometers. In other words, 0.5 miles = 0.5 miles = feet mile = kilometers 2 inches foot centimeters inch 000 kilometers meter 00 centimeters kilometer 000 meters
3 MA 35 Lecture  Introduction to Unit Conversions 3 Use only the equations listed above.. Convert 20 meters to yards. 2. Convert 5 yards to centimeters. 3. Convert 0 kilometers to miles. Quiz B Virtually all units that combine simpler units, like miles per hour, combine the simpler units with division or multiplication. Anything with a per is division, and multiplications are usually indicated with dashes. Example 3. If you were to walk 6 miles in 2 hours, we might describe this as a rate of 6 miles per 2 hours (every 2 hours, you ll travel 6 miles), and think of it in terms of division. 6 miles (3) 2 hours = 6 2 miles = 3 miles per hour. hours Example 4. Since the hours in this last example go in the denominator, we can cancel them out the same way we did in the conversions before. Let s say we want to convert 3 miles per hour into feet per second, and we had the following equations available feet = mile 60 minutes = hour 60 seconds = minute Our conversion will go (4) 3 miles hour = 3 miles hour hour 60 minutes minute 5280 feet feet = seconds mile second. To get rid of the hour, we need to use a fraction that has hours in the numerator. The equation we used put minutes in the denominator, so we next need a fraction with minutes in the numerator. This leaves seconds in the denominator, which is what we want. To get rid of the miles in the numerator, we use a fraction that has miles in the denominator, and this leaves feet in the numerator, which is what we want. Densities The density of water is about one ounce per fluid ounce. Note that an ounce is a measure of weight, and a fluid ounce is a measure of volume. This is equivalent to a density of one gram per cubic centimeter in the metric system. In this case, the density compares the weight and volume of a substance (although density can also compare other things like population density might be measured in people per square mile). Some relevant conversion equations are 6 ounces = pound 6 fluid ounce = 2 s = quart 4 quarts = gallon Since there are 6 ounces in a pound and 6 fluid ounces in a, a of water will weigh one pound. A gallon will weigh 8 pounds. Water ties our measures of weight and volume together, at least approximately. Other fluids have different densities. For example, gasoline is less dense than water. Example 5. How much does a of gasoline weigh? The density of gasoline depends on things like temperature and atmospheric pressure, but Wikipedia gives a value of pounds per U.S. gallon. Let s convert that to pounds per. (5) pounds gallon pounds = gallon gallon 4 quarts quart 2 s pounds =
4 MA 35 Lecture  Introduction to Unit Conversions 4 We can see from this that one of gasoline weighs pounds. If we had 3.26 s of gasoline, we would multiply. Look how the units work out. (6) pounds 3.26 s = 2.47 pounds. The units will help you keep things straight. Homework I know there are unit converters on the internet, and that s fine, but I want you to be able to make conversions from any set of conversion equations. Make sure you can do these using a conversion chart and a calculator. Use only the equations on the last page of these lecture notes.. Convert 6 feet into meters. 2. Convert 5 kilometers into yards. 3. Convert 600 meters into miles. 4. Convert 80 centimeters into feet. 5. Convert a speed of 30 miles per hour into meters per minute. 6. Convert a speed of 5 inches per hour into millimeters per second. 7. Gasoline has a density of pounds per (according to our calculations above). Convert this to grams per liter. 8. How many grams will liters of gasoline weigh (technically, what will the mass be)? 9. According to Wikipedia, the density of air is about.225 kilograms per cubic meter. Convert this to ounces per. 0. How much would a gallon of air weigh? That is, convert to ounces per gallon. Answers on next page.
5 MA 35 Lecture  Introduction to Unit Conversions 5 Answers: Quiz A: ) 3.72 feet 2 inches feet 2) 07 inches foot 2 inches = 07 2 feet = 8.92 feet. = inches = inches. Quiz B: ) 2.87 yards. 2) centimeters. 3) 6.2 miles. HW: ).83 meters. 2) yards. 3) miles. 4) 5.9 feet. 5) 30 miles 5280 feet meters hour hour mile foot 60 minutes = meters per minute 6) 5 inches foot hour 60 minutes minute centimeters 2.54 meter millimeters seconds inch 00 centimeters meter = grams per liter. 7) pounds 8) grams liter kilogram 000 grams 2.2 pounds kilogram 2 s quart liters = grams. quart = grams per liter liters 9).225 kilograms cubic meter liters cubic meter 000 liters quart quart 2.2 pounds 6 2 s ounces = ounces per. kilogram pound 0) ounces 2 quart 4 quart = ounces. gallon
6 MA 35 Lecture  Introduction to Unit Conversions 6 Some Formulas 5280 feet = mile 2 inches = foot foot = meters 2.54 centimeters = inch 000 millimeters = meter 00 centimeters = meter acre = square feet quart = liters cubic meter = meter 3 = 000 liters centimeter 3 = milliliter 6 fluid ounce = 2 s = quart 4 quarts = gallon foot 3 = 7.48 gallons 6 ounces = pound 2.2 pounds = kilogram 000 grams = kilogram 2000 pounds = ton 60 minutes = hour 60 seconds = minute
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