Conversions. 12 in. 1 ft = 1.

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1 Conversions There are so many units that you can use to express results that you need to become proficient at converting from one to another. Fortunately, there is an easy way to do this and it works every time with no exceptions! You simply multiply by one. (The only question is: which one?) Since 12 in = 1 ft, you can get a one by dividing both sides of the equation by 1 ft, giving 12 in 1 ft = 1. (Notice that this doesn t say that 12 = 1.) You use this or any similar factor by noticing that if you multiply anything by one, you don t change it. It s easy to see with an example; each factor in parentheses equals one days 24 hr 60 min 60 sec 1 yr = 1 yr 1 yr 1 day 1 hr 1 min day / 24 hr / 60 min / 60 sec = 1 yr / 1 yr / 1 day / 1 hr / 1 min / = sec = sec = sec, where there are only 5 significant figures in the result, because that s the limit on the accuracy of the input data for the number of days in a year. (This is about π 10 7 seconds easier to remember.) Another example: 1 mi 1 hr = 1 mi 1 hr = 1 /mi 1 hr / ( ) 5280 ft 1 hr 1 min 1 mi 60 min 60 sec ( ) 5280 ft 1 hr / 1 min / = 5280 ft. = 1.47 ft/sec 1 /mi 60 min / 60 sec 3600 sec For an example of what can go wrong, what is one mile in inches? 1 mi = 1 mi 5280 ft 1 ft = 1 /mi 1 mi 12 in 5280 ft 1 ft 1 /mi 12 in = 5280 ft 2 12 in. This isn t wrong, but it s pretty silly, and isn t responsive to the original question of converting miles to inches. The key point to notice is that I kept track of the units honestly and I didn t assume that I ll never make a mistake. A common error (the common error) in doing these conversions is to write down the conversion factors and then to assume that they do what you want them to. If you skip the steps shown here, where you explicitly put in the units and cancel them one pair at a time, you have about a chance of messing it up. After all, if 12 in/1 ft = 1, then so does 1 ft/12 in = 1, and you can multiply anything by one without changing it. 1

2 2 Conversions To avoid this trap, try writing the units first and fill in the numbers next, as in ft yard 5280 ft 1 yard 1 mi THEN 1 mi = 1760 yards mi ft 1 mi 3 ft Don t forget that each factor in a conversion must be done; if a unit is squared, its conversion factor enters twice. Convert one square mile to square yards: 1 mi 2 ( 5280 ft 1 mi (I like the metric system more and more.) ) 2 ( ) 2 1 yard = yard 2. 3 ft Convert 10 cm 1/2 using meters instead of centimeters. ( 10 cm 1/2 m ) ( ) 1/2 1/2 1 m then 10 cm 1/2 = 10 m 1/2 = 1 m 1/2. cm 100 cm 100 For a particularly odd unit that I once encountered, convert 10 cm 2/3 gm 1/3 sec 2 to meters and kilograms. First, just get the units in place: ( 10 cm 2/3 gm 1/3 sec 2 10 cm 2/3 gm 1/3 sec 2 ( 1 m 100 cm m ) ( ) 2/3 1/3 kg THEN cm gm ) 2/3 ( ) 1/3 1 kg 10 3 gm 10 = 100 2/ m2/3 kg 1/3 sec 2 = /3 m2/3 kg 1/3 sec 2 = /3 m2/3 kg 1/3 sec 2

3 Conversions 3 Practice Set 1 [1] 2.54 cm = 1 in. What is 1 meter in inches? [2] g = 9.80 m/s 2. Convert this to ft/s 2. [3] The conversion between Canadian and US dollars is currently C$1.55=US$1.00. Gasoline in Canada costs C$0.80 per liter; what is the cost per gallon in US$? (1 liter is gallons.) [4] The light-year is the distance that light travels (in vacuum) in one year. The speed of light is m/s. What is one light-year in meters? [5] The standard class hour is 50 minutes. What is this in microcenturies?

4 4 Conversions Solutions for Practice Set cm 1 in [1] 2.54 cm = 1 in. What is 1 meter in inches? 1 m = 100 cm = in 1 m 2.54cm 2.54 [2] g = 9.80 m/s 2. Convert this to ft/s m ( ) 100 cm 1 in 1 ft s 2 = 32.2 ft 1 m 2.54cm 12 in s 2 [3] The conversion between Canadian and US dollars is currently C$1.55=US$1.00. Gasoline in Canada costs C$0.60 per liter; what is the cost per gallon in US$? (1 liter is gallons.) 0.80 C$ ( ) ( ) 1 liter 1 US$ = liter gallons C$1.55 US$1.95 per gallon [4] The light-year is the distance that light travels (in vacuum) in one year. The speed of light is m/s. What is one light-year in meters? m ( ) days s 1 yr 1 yr 24 hours 3600 s 1 day 1 hr = m [5] The standard class hour is 50 minutes. What is this in microcenturies? ( 1 hr 1 day 1 yr 50 min 60 min 24hr ( ) ( 1 century 10 6 ) microcenturies days ) 100 yr 1 century = 0.95 microcenturies

5 Conversions 5 Practice Set 2 [1] What is 60 mi/hr in ft/sec? [2] Given 2.54 cm = 1 in exactly, what is 1 mile in kilometers? [3] Water flows down a river at a rate of 500 cubic meters per second. What is this in gallons per hour? (1 liter = 10 3 m 3, 1 liter = gallons.) [4] g = 9.8 m/s 2 is the standard gravitational field at the surface of the Earth. Convert this to light-years/year 2. [5] Astronomers most often use the unit called a parsec, which is about 3.26 light-years. The farthest objects ever observed are measured to be about light-years away. What is this in megaparsecs?

6 6 Conversions Practice Set 3 [1] The density of water is 1 gm/cm 3. Convert this to kg/m 3? [2] Start with 10 m 3/2 and put the meters into centimeters. [3] If you go on a diet, and lose at a rate of 1 kg/week, what is this in milligrams per second? [4] The speed of light is m/s. What is it in feet per nanosecond? [5] A certain physical constant is 1.57 gm 1/2 cm 3/2. Convert the grams to kilograms and the centimeters to meters.